C-R Immersions and Sub-Riemannian Geometry

Abstract

On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form $\theta$, we consider natural $ϵ$-contractions, i.e., contractions $g_{ϵ}^M$ of the Levi form $G_{\theta }$, such that the norm of the Reeb vector field T of $(M, \theta )$ is of order $O\left({ϵ}^{-1}\right)$. We study isopseudohermitian (i.e., $f^{\ast }\mathsf{\Theta }=\theta$) Cauchy–Riemann immersions $f:M\to (A,\mathsf{\Theta })$ between strictly pseudoconvex CR manifolds M and A, where $\mathsf{\Theta }$ is a contact form on A. For every contraction $g_{ϵ}^A$ of the Levi form $G_{\mathsf{\Theta }}$, we write the embedding equations for the immersion $f:M\to \left(A,g_{ϵ}^A\right)$. A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as $ϵ\to 0^{+}$. For every isopseudohermitian immersion $f:M\to S^{2N+1}$ into a sphere $S^{2N+1}\subset {\mathbb{C}}^{N+1}$, we show that Webster’s pseudohermitian scalar curvature R of $(M, \theta )$ satisfies the inequality $R\le 2n\left[\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T,T)+n+1\right]+\frac{1}{2}\{\parallel H\left(f\right){\parallel }_{g_{\mathsf{\Theta }}^f}^2+\parallel {\mathrm{trace}}_{G_{\theta }}{\mathsf{\Pi }}_{H\left(M\right)}\left({\nabla }^{\top }-\nabla \right){\parallel }_{f^{\ast }{g}_{\mathsf{\Theta }}}^2\}$ with equality if and only if $B\left(f\right)=0$ and ${\nabla }^{\top }=\nabla$ on $H\left(M\right)\otimes H\left(M\right)$. This gives a pseudohermitian analog to a classical result by S-S. Chern on minimal isometric immersions into space forms.

1. Introduction

The present paper has two main purposes: a general one, which looks at certain problems originating in complex analysis from the point of view of pseudohermitian geometry, and a more specific purpose, which is contributing to the study of CR immersions between strictly pseudoconvex CR manifolds, from a differential geometric viewpoint. Pseudohermitian geometry was brought into mathematical practice by S.M. Webster [1] and N. Tanaka [2], and the term pseudohermitian structure was coined by S.M. Webster himself (see op. cit.). Pseudohermitian geometry soon became a popular research area, and its development up to 2006 is reported in the monographs by S. Dragomir and G. Tomassini [3] and by E. Barletta, S. Dragomir, and K.L. Duggal [4]. The further growth of the theory, though confined to the topic of subelliptic harmonic maps and vector fields on pseudohermitian manifolds, is reported in the monograph by S. Dragomir and D. Perrone [5]. The part added to the theory of CR immersions by the present paper, which is deriving a pseudohermitian analog to the Gauss equation (of an isometric immersion between Riemannian manifolds), aims to contribute applications to rigidity theory. The remainder of the Introduction is devoted to a brief parallel between rigidity within Riemannian geometry on one hand and complex analysis on the other, and to a glimpse into the main results. The authors benefit from the (partial) embedding (described in detail in [6] and adopted there for different purposes, i.e., the study of the geometry of Jacobi fields on Sasakian manifolds) of pseudohermitian geometry into sub-Riemannian geometry, and the main novelty from a methodological viewpoint is the use of methods in sub-Riemannian geometry (see [7,8]).

Rigidity in differential geometry has a long history, perhaps starting with rigidity of regular curves $\alpha :I\to {\mathbb{R}}^3$ of curvature $k\left(s\right)>0$ and torsion $\tau \left(s\right)$ ($s\in I$): any other regular curve $\overline{\alpha }:I\to {\mathbb{R}}^3$ with the same curvature $k\left(s\right)$ and torsion $\tau \left(s\right)$ differs from $\alpha$ by a rigid motion i.e., $\overline{\alpha }\left(s\right)=\rho \left[\alpha \left(s\right)\right]+c$ for some orthogonal linear map $\rho :{\mathbb{R}}^3\to {\mathbb{R}}^3$ and some vector $c\in {\mathbb{R}}^3$. See M.P. Do Carmo [9], p. 19.

As a step further, one knows about the rigidity of real hypersurfaces in Euclidean space ${\mathbb{R}}^{n+1}$, i.e., if $f:M\to {\mathbb{R}}^{n+1}$ and $\overline{f}:M\to {\mathbb{R}}^{n+1}$ are two isometric immersions of an n-dimensional orientable Riemannian manifold M, whose second fundamental forms coincide on M, then $\overline{f}=\tau \circ f$ for some isometry $\tau :{\mathbb{R}}^{n+1}\to {\mathbb{R}}^{n+1}$. See Theorem 6.4 in S. Kobayashi and K Nomizu [10], Volume II, p. 45.

A close analog to rigidity in the above sense, occurring in complex analysis of functions of several complex variables, is that of rigidity of CR immersions, and our starting point is S.M. Webster’s legacy; see [11]. A CR immersion is a map $f:M\to A$ of CR manifolds M and A such that: (i) f is a $C^{\infty }$ immersion, and (ii) f is a CR map; i.e., it maps the CR structure $T_{1,0}\left(M\right)$ onto $T_{1,0}\left(A\right)$. Let $(M,T_{1,0}\left(M\right))$ be a $(2n+1)$-dimensional CR manifold, of CR dimension n. M is a CR hypersurface of the sphere $S^{2n+3}$ if $M\subset S^{2n+3}$ is a (codimension two) submanifold and the inclusion $\iota :M\to S^{2n+3}$ is a CR immersion. A CR hypersurface M is rigid in $S^{2n+3}$ if for any other CR hypersurface $M^{\prime }\subset S^{2n+3}$, every CR isomorphism $\varphi :M\to M^{\prime }$ extends to a CR automorphism $\mathsf{\Phi }\in {\mathrm{Aut}}_{\mathrm{CR}}\left(S^{2n+3}\right)$. By a classical result of S.M. Webster (see [11]), if $n\ge 3$, every CR hypersurface $M\subset S^{2n+3}$ is rigid.

The proof (see [10], Volume II, pp. 45–46) of rigidity of real hypersurfaces in Euclidean space relies on the analysis of the Gauss–Codazzi equations for a given isometric immersion, and the treatment of rigidity of CR hypersurfaces in $S^{2n+3}$ exploits (again see [11]) in a rather similar manner CR, or more precisely pseudohermitian, analogs to Gauss–Codazzi equations, where the ambient and intrinsic Levi–Civita connections (at work within the geometry of isometric immersions between Riemannian manifolds) are replaced by the Tanaka–Webster connections. The Tanaka–Webster connection is a canonical connection (similar to the Levi–Civita connection in Riemannian geometry, and to the Chern connection in Hermitian geometry) occurring on any nondegenerate CR manifold, on which a contact form has been fixed (see [1,2]). The Tanaka–Webster connection is also due to S.M. Webster (see [1]), yet was independently discovered by N. Tanaka in a monograph (see [2]) that remained little known to Western scientists up to the end of the 1980s. The pseudohermitian analog to the Gauss equation in Webster’s theory (see [11]) is stated as:

$$\begin{array}{c}{S}_{\beta \overline{\alpha }\rho \overline{\sigma }}=-b_{\beta \rho }{b}_{\overline{\alpha }\overline{\sigma }}-\frac{1}{(n+1)(n+2)}\left[g_{\beta \overline{\alpha }}{g}_{\rho \overline{\sigma }}+g_{\rho \overline{\alpha }}{g}_{\beta \overline{\sigma }}\right]b_{\mu \nu }{b}^{\mu \nu }\\ +\frac{1}{n+2}\left[g_{\beta \overline{\alpha }}{b}_{\rho \mu }{b^{\mu }}_{,\overline{\sigma }}+g_{\rho \overline{\alpha }}{b}_{\beta \mu }{b^{\mu }}_{,\overline{\sigma }}+b_{\beta \mu }{b^{\mu }}_{,\overline{\alpha }}{g}_{\rho \overline{\sigma }}+b_{\rho \mu }{b^{\mu }}_{,\overline{\alpha }}{g}_{\beta \overline{\sigma }}\right].\end{array}$$

(1)

Insufficient computational details are furnished in [11], and the derivation of (1) remains rather obscure.

A more recent tentative approach to the (CR analog to) the Gauss–Codazzi–Ricci equations was taken up by P. Ebenfelt, X-J. Huang, and D. Zaitsev (see [12]). They introduced and made use of a CR analog to the second fundamental form (of an isometric immersion), which is naturally associated with a given CR immersion and springs from work in complex analysis by B. Lamel (see [13,14]). Their pseudohermitian (analog to) the Gauss equation

$$R^A(X,Y,Z,W)=R(X,Y,Z,V)+\langle \mathsf{\Pi }(X,Z),\mathsf{\Pi }(Y,V)\rangle ,$$

$$X,Y,Z,V\in T_{1,0}\left(M\right),$$

for a given CR immersion $f:M\to A$ depends on a particular choice of contact forms $\theta$ and $\mathsf{\Theta }$, respectively, on the submanifold M and on the ambient space A, such that: (i) $f^{\ast }\mathsf{\Theta }=\theta$, and (ii) $f\left(M\right)$ is tangent to the ambient Reeb vector field $T_A$ (the globally defined nowhere zero tangent vector field on A, transverse to the Levi distribution, uniquely determined by $\mathsf{\Theta }\left(T_A\right)=1$ and $T_A\rfloor \mathsf{\Theta }=0$). However, the proof of the existence of such $\theta$ and $\mathsf{\Theta }$ is purely local and, in general, global contact forms on M and A such that f is isopseudohermitian, and $T_A^{\perp }=0$ might not exist at all.

The class of isopseudohermitian immersions between strictly pseudoconvex CR manifolds enjoying the property $T_A^{\perp }=0$ was studied independently by S. Dragomir (see [15]). As it turns out, any CR immersion in the class is also isometric with respect to the Webster metrics, i.e., $f^{\ast }{g}_{\mathsf{\Theta }}=g_{\theta }$, and then a pseudohermitian (analog to the) geometry of the second fundamental form (of an isometric immersion) may be built by closely following its Riemannian counterpart, in a rather trivial manner. Despite the enthusiastic review by K. Spallek (see [16]) and the later development (by S. Dragomir and A. Minor [17,18]) relating the geometry of the second fundamental form (of a CR immersion in the class above) to the Fefferman metrics of $(M,\theta )$ and $(N,\mathsf{\Theta })$, the built theory of CR immersions is not general enough: it does not suggest a path towards a theory of CR immersions not belonging to the class, within which one may hope to recover Webster’s mysterious “Gauss equation” (1). It is our purpose, within the present paper, to adopt an entirely new approach to building a “second fundamental form” based theory of CR immersions, using methods coming from sub-Riemannian geometry (e.g., in the sense of R.S. Strichartz [8]).

That CR geometry (partially) embeds into sub-Riemannian geometry is a rather well-known fact: given a strictly pseudoconvex CR manifold M, endowed with a positively oriented contact form $\theta$, the pair $(H\left(M\right),G_{\theta })$, consisting of the Levi distribution $H\left(M\right)=\mathrm{Re}\left\{T_{1,0}\left(M\right)\oplus T_{0,1}\left(M\right)\right\}$ and the Levi form $G_{\theta }(X,Y)=\left(d\theta \right)(X,JY)$, $X,Y\in H\left(M\right)$, is a sub-Riemannian structure on M, and the Webster metric $g_{\theta }$ is a contraction of $G_{\theta }$ (see [6,7,8,19]).

We adopt the additional assumption that the given CR immersion $f:M\to A$ (between the strictly pseudoconvex CR manifolds M and A) is isopseudohermitian, i.e., $f^{\ast }\mathsf{\Theta }=\theta$ for some choice of contact forms $\theta$ and $\mathsf{\Theta }$ on M and A, respectively, yet we refrain from assuming that $f\left(M\right)$ is tangent to the Reeb vector field of the ambient space $(A,\mathsf{\Theta })$; rather, $T_A$ will be, relative to $f\left(M\right)$, always oblique. $f\left(M\right)$ may be looked at as a submanifold in the Riemannian manifold $(A,g_{\mathsf{\Theta }})$, yet, by our assumption $T_A^{\perp }\ne 0$, the first fundamental form (i.e., the pullback $f^{\ast }{g}_{\mathsf{\Theta }}$ to M of the ambient Webster metric $g_{\mathsf{\Theta }}$) of the given immersion $f:M\to (A,g_{\mathsf{\Theta }})$ does not coincide with the intrinsic Webster metric $g_{\theta }$. That is, $f:(M,g_{\theta })\to (A,g_{\mathsf{\Theta }})$ is not an isometric immersion, and the well-established and powerful apparatus based on the Gauss–Codazzi–Mainnardi–Ricci equations cannot be a priori applied to f.

To circumnavigate this obstacle, one endows A with the Riemannian metric $g_{ϵ}^A$, the contraction of the Levi form $G_{\mathsf{\Theta }}$ associated with each $0<ϵ<1$, given by

$$g_{ϵ}^A=g_{\mathsf{\Theta }}+\left(\frac{1}{{ϵ}^2}-1\right)\mathsf{\Theta }\otimes \mathsf{\Theta }.$$

(2)

Our strategy will be to regard $f\left(M\right)$ as a submanifold of the Riemannian manifold $(A,g_{ϵ}^A)$ and derive the Gauss–Weingarten and Gauss–Ricci–Codazzi equations of the immersion $f:M\to (A,g_{ϵ}^A)$. In the end, these will lead, as $ϵ\to 0^{+}$, to the seek after pseudohermitian analogs to the embedding equations. To illustrate the expected results, we state the pseudohermitian Gauss equation of a CR immersion into a sphere.

Corollary 1.

Let M be a strictly pseudoconvex CR manifold, of CR dimension n, equipped with the positively oriented contact form $\theta \in {\mathcal{P}}_{+}\left(M\right)$. Let $f:M\to S^{2(n+k)+1}$, $k\ge 1$, be a CR immersion of M into the standard sphere $S^{2(n+k)+1}$ carrying the CR structure induced by the complex structure of ${\mathbb{C}}^{n+k+1}$. Let $\mathsf{\Theta }=\frac{i}{2}\left(\overline{\partial }-\partial \right){\left|Z\right|}^2$ be the canonical contact form on $S^{2(n+k)+1}$. If f is isopseudohermitian(i.e., $f^{\ast }\mathsf{\Theta }=\theta$), then

$$\begin{array}{c}{g}_{\theta }(R^{\nabla }(X,Y)Z,W)=g_{\theta }(Y,Z)g_{\theta }(X,W)-g_{\theta }(X,Z)g_{\theta }(Y,W)\\ +g_{\mathsf{\Theta }}^f\left(B\left(f\right)(X,W),B\left(f\right)(Y,Z)\right)-g_{\mathsf{\Theta }}^f\left(B\left(f\right)(Y,W),B\left(f\right)(X,Z)\right)\\ +(2\lambda -1)\left\{\mathsf{\Omega }(X,W)\mathsf{\Omega }(Y,Z)-\mathsf{\Omega }(Y,W)\mathsf{\Omega }(X,Z)\right\}-2\lambda \mathsf{\Omega }(X,Y)\mathsf{\Omega }(Z,W)\\ -\mathsf{\Omega }(Y,Z)A(X,W)-\mathsf{\Omega }(X,W)A(Y,Z)+\mathsf{\Omega }(X,Z)A(Y,W)+\mathsf{\Omega }(Y,W)A(X,Z)\\ +\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left(U\left(f\right)(X,W),U\left(f\right)(Y,Z)\right)-\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left(U\left(f\right)(Y,W),U\left(f\right)(X,Z)\right)\\ -\mathsf{\Omega }(Y,Z)\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left(U\left(f\right)(X,W),T\right)-\mathsf{\Omega }(X,W)\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left(U\left(f\right)(Y,Z),T\right)\\ +\mathsf{\Omega }(X,Z)\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left(U\left(f\right)(Y,W),T\right)+\mathsf{\Omega }(Y,W)\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left(U\left(f\right)(X,Z),T\right).\end{array}$$

(3)

for any $X,Y,Z,W\in H\left(M\right)$.

Here, $R^{\nabla }$ is the curvature tensor field of the Tanaka–Webster connection ∇ of $(M,\theta )$, and $B\left(f\right)$ is the pseudohermitian second fundamental form of the given immersion $f:M\to S^{2(n+k)+1}$. A brief inspection of (3) reveals a strong formal analogy to the ordinary Gauss equation in Riemannian geometry; see B-Y. Chen [20]. At the same time, all obstructions springing from the geometric structure at hand (which is pseudohermitian, rather than Riemannian) are inbuilt in Equation (3). For instance, Equation (3) contains the (eventually nonzero) pseudohermitian torsion tensor field A of $(M,\theta )$. Additionally, (3) contains the $(1, 2)$ tensor field $U\left(f\right)$ expressing the difference between the induced connection ${\nabla }^{\top }$ and the Tanaka–Webster connection ∇ (the non-uniqueness of the canonical connection on M is of course tied to the failure of $f:(M,g_{\theta })\to (S^{2(n+k)+1},g_{\mathsf{\Theta }})$ to be isometric). Our expectation is that an analysis of the pseudohermitian Gauss–Codazzi equations will lead to rigidity theorems for isopseudohermitian CR immersions $f:M\to S^{2n+3}$ and, in particular, $f:M\to S^5$ (focusing on the case $M=S^3$).

The certitude that Riemannian objects on $(A,g_{ϵ}^A)$ (and their tangential and normal components, relative to $f\left(M\right)$) will give, in the limit as $ϵ\to 0^{+}$, the “correct” pseudohermitian analogs to the (Riemannian) embedding equations is already acquired from the following early observations: let $(M, \theta )$ be endowed with the contraction of $G_{\theta }$ given by ${\displaystyle g_{ϵ}=g_{\theta }+\left({ϵ}^{-2}-1\right)\theta \otimes \theta }$, and let ${\nabla }^{ϵ}$ and ${\mathsf{\Delta }}_{ϵ}$ be respectively the gradient and Laplace–Beltrami operators (on functions) of the Riemannian manifold $(M,g_{ϵ})$. Then,

$${\nabla }^{ϵ}u={\nabla }^{H}u+{ϵ}^2\theta (\nabla u)T,u\in C^1\left(M\right),$$

$${\mathsf{\Delta }}_{ϵ}u={\mathsf{\Delta }}_{b}u-{ϵ}^2{T}^2\left(u\right),u\in C^2\left(M\right),$$

showing that ${\nabla }^{ϵ}u$ tends, in the limit as $ϵ\to 0^{+}$, to the horizontal gradient ${\nabla }^{H}u$ (familiar in subelliptic theory; see, e.g., [19]), while ${\mathsf{\Delta }}_{ϵ}$ tends (in an appropriate Banach space topology, where second order elliptic operators such as ${\mathsf{\Delta }}_{ϵ}$ form an open set, one of whose boundary points is ${\mathsf{\Delta }}_b$) to the sublaplacian ${\mathsf{\Delta }}_b$ of $(M,\theta )$.

As an application of the pseudohermitian Gauss Equation (3) in Corollary 2, we shall establish the following result.

Theorem 1.

Let M be a strictly pseudoconvex CR manifold, of CR dimension n, equipped with the contact form $\theta \in {\mathcal{P}}_{+}\left(M\right)$. Let $f:M\to S^{2(n+k)+1}$ be an isopseudohermitian immersion of $(M,\theta )$ into the sphere $\mathbf{j}:S^{2(n+k)+1}\hookrightarrow {\mathbb{C}}^{n+k+1}$, endowed with the contact form $\mathsf{\Theta }={\mathbf{j}}^{\ast }\left[\frac{i}{2}\left(\overline{\partial }-\partial \right){\left|Z\right|}^2\right]$. Then, the pseudohermitian scalar curvature $R=g^{\alpha \overline{\beta }}{R}_{\alpha \overline{\beta }}$ of $(M,\theta )$ satisfies the inequality

$$R\le 2n[(f^{\ast }{g}_{\mathsf{\Theta }})(T,T)+n+1]+\frac{1}{2}\left\{\parallel H\left(f\right){\parallel }_{g_{\mathsf{\Theta }}^f}+\parallel {\mathrm{trace}}_{G_{\theta }}{\mathsf{\Pi }}_{H\left(M\right)}U\left(f\right){\parallel }_{f^{\ast }{g}_{\mathsf{\Theta }}}\right\}$$

with equality if and only if $B\left(f\right)=0$ and ${\nabla }^{\top }=\nabla$ on $H\left(M\right)\otimes H\left(M\right)$.

Here, $H\left(f\right)={\mathrm{trace}}_{G_{\theta }}{\mathsf{\Pi }}_{H\left(M\right)}B\left(f\right)$. Theorem 1 generalizes a classical result by S-S. Chern (see [21]) on isometric immersions of Riemannian manifolds into a space form (to the case of isopseudohermitian immersions of strictly pseudoconvex CR manifolds into a sphere).

The definitions of objects used in the present Introduction can be found in Section 2 of the present paper.

2. Sub-Riemannian Techniques in CR Geometry

All basic notions and results used through the paper are described in detail in Section 2, following the monograph by S. Dragomir and G. Tomassini [3]. Specifically, in Section 2.1, we recall the necessary material in Cauchy–Riemann (CR) and pseudohermitian geometry by essentially following monograph [3]. CR geometry is known to (partially) embed into sub-Riemannian geometry, in the sense of R. Strichartz [8]. We therefore recall the basics of sub-Riemannian geometry, at work in the present paper, in Section 2.2 by following J.P. D’Angelo and J.T. Tyson (see [7]) and [6,19], and of course [8].

2.1. CR Structures and Pseudohermitian Geometry

Let M be an orientable real $(2n+1)$-dimensional $C^{\infty }$ differentiable manifold, and let $T\left(M\right)$ be the (total space of the) tangent bundle over M.

Definition 1

([3], pp. 3–4). A CR structure is a complex rank n complex subbundle $T_{1,0}\left(M\right)\subset T\left(M\right)\otimes \mathbb{C}$ of the complexified tangent bundle such that

$$T_{1,0}{\left(M\right)}_x\cap T_{0,1}{\left(M\right)}_x=\left(0\right),x\in M,$$

$$Z,W\in C^{\infty }(U,T_{1,0}\left(M\right))⟹[Z,W]\in C^{\infty }(U,T_{1,0}\left(M\right))$$

for any open set $U\subset M$. A pair $(M,T_{1,0}\left(M\right))$ consisting of a $(2n+1)$-dimensional $C^{\infty }$ manifold M and a CR structure $T_{1,0}\left(M\right)$ on M is a CR manifold. The integer n is the CR dimension.

Here, $T_{0,1}\left(M\right)=\overline{T_{1,0}\left(M\right)}$ (an overbar denotes complex conjugation). Every real hypersurface $M\subset {\mathbb{C}}^{n+1}$ may be organized (see e.g., formula 1.12 in [3], p. 5) as a CR manifold of CR dimension n, with the CR structure

$$T_{1,0}{\left(M\right)}_x=\left[T_x\left(M\right){\otimes }_{\mathbb{R}}\mathbb{C}\right]\cap T^{\prime }{\left({\mathbb{C}}^{n+1}\right)}_x,x\in M,$$

induced by the complex structure of the ambient space. Here, $T^{\prime }\left({\mathbb{C}}^{n+1}\right)$ denotes the holomorphic tangent bundle over ${\mathbb{C}}^{n+1}$, i.e., the span of $\{\partial /\partial z^j:1\le j\le n+1\}$ where $(z^1,\cdots ,z^{n+1})$ are the Cartesian complex coordinates on ${\mathbb{C}}^{n+1}$.

Definition 2

([3], p. 4). The real rank $2n$ (hyperplane) distribution

$$H\left(M\right)=\mathrm{Re}\left\{T_{1,0}\left(M\right)\oplus T_{0,1}\left(M\right)\right\}$$

is the Levi (or maximally complex) distribution.

$H\left(M\right)$ carries the complex structure

$$J=J_M:H\left(M\right)\to H\left(M\right),J(Z+\overline{Z})=i(Z-\overline{Z}),Z\in T_{1,0}\left(M\right).$$

Definition 3

([3], p. 4). A $C^{\infty }$ map $f:M\to A$ of the CR manifold $(M,T_{1,0}\left(M\right))$ into the CR manifold $(A,T_{1,0}\left(A\right))$ is a CR map if

$$\left(d_{x}f\right)T_{1,0}\left(M\right)\subset T_{1,0}{\left(A\right)}_{f\left(x\right)},x\in M.$$

Equivalently, a CR map $f:M\to A$ is characterized by the properties

$$\left(d_{x}f\right)H{\left(M\right)}_x\subset H{\left(A\right)}_{f\left(x\right)},\left(d_{x}f\right)\circ J_{M,x}=J_{A,f\left(x\right)}\circ \left(d_{x}f\right),$$

for any $x\in M$; see formulas 1.10 and 1.11 in [3], p. 4.

Definition 4

([3], p. 5). A CR isomorphism is a $C^{\infty }$ diffeomorphism and a CR map. A CR automorphism of the CR manifold M is a CR isomorphism of M into itself.

For every CR manifold M, let ${\mathrm{Aut}}_{\mathrm{CR}}\left(M\right)$ be the group of all CR automorphisms of M.

Definition 5

([3], p. 5). The conormal bundle is the real line bundle $\mathbb{R}\to H{\left(M\right)}^{\perp }\to M$ given by

$$H{\left(M\right)}_x^{\perp }=\left\{\omega \in T_x^{\ast }\left(M\right):\mathrm{Ker}\left(\omega \right)\supset H{\left(M\right)}_x\right\},x\in M.$$

As M is orientable, and $H\left(M\right)$ is oriented by its complex structure, the quotient bundle $T\left(M\right)/H\left(M\right)$ is orientable. Moreover, there is a (non-canonical) vector bundle isomorphism $H{\left(M\right)}^{\perp }\approx T\left(M\right)/H\left(M\right)$, hence $H{\left(M\right)}^{\perp }$ is orientable as well. Any orientable real line bundle over a connected manifold is trivial (see, e.g., Remark 11.3 in [22], p. 115). Hence, $H{\left(M\right)}^{\perp }\approx M\times \mathbb{R}$ (a vector bundle isomorphism). Therefore, $H{\left(M\right)}^{\perp }$ admits globally defined nowhere zero sections.

Definition 6

([3], p. 5). A global section $\theta \in C^{\infty }(H{\left(M\right)}^{\perp })$ such that ${\theta }_x\ne 0$ for every $x\in M$ is called a pseudohermitian structure on M.

A pseudohermitian structure is a real valued differential 1-form $\theta$ on M such that $\mathrm{Ker}\left(\theta \right)=H\left(M\right)$ (and in particular ${\theta }_x\ne 0$ for any $x\in M$).

Definition 7.

A pair $(M, \theta )$ consisting of a CR manifold M and a pseudohermitian structure $\theta$ on M is a pseudohermitian manifold.

Let $\mathcal{P}=\mathcal{P}\left(M\right)$ be the set of all pseudohermitian structures on M.

Definition 8

([3], pp. 5–6). Given $\theta \in \mathcal{P}$, the Levi form is

$$G_{\theta }(X,Y)=\left(d\theta \right)(X,JY),X,Y\in H\left(M\right).$$

Definition 9

([3], p. 6). The CR structure $T_{1,0}\left(M\right)$ is nondegenerate if the (symmetric bilinear) form $G_{\theta }$ is nondegenerate for some $\theta \in \mathcal{P}$.

Any other pseudohermitian structure $\widehat{\theta }\in \mathcal{P}$ is related to $\theta$ by $\widehat{\theta }=\lambda \theta$ for some $C^{\infty }$ function $\lambda :M\to \mathbb{R}\backslash \left\{0\right\}$. Then,

$$d\widehat{\theta }=d\lambda \wedge \theta +\lambda d\theta ,$$

hence the corresponding Levi forms $G_{\theta }$ and $G_{\widehat{\theta }}$ are related by $G_{\widehat{\theta }}=\lambda G_{\theta }$. Consequently, if $G_{\theta }$ is nondegenerate for some $\theta \in \mathcal{P}$, it is nondegenerate for all. That is, nondegeneracy is a CR invariant notion; it does not depend on the choice of pseudohermitian structure. Strictly speaking:

Definition 10.

A geometric object, or a notion, on a CR manifold M is CR invariant if it is invariant with respect to the action of ${\mathrm{Aut}}_{\mathrm{CR}}\left(M\right)$.

The signature of the Levi form $G_{\theta }$ of a nondegenerate CR manifold M is a CR invariant.

Definition 11

([3], p. 43). A differential 1-form $\theta \in {\mathsf{\Omega }}^1\left(M\right)$ is a contact form if $\theta \wedge {\left(d\theta \right)}^n$ is a volume form on M.

If $T_{1,0}\left(M\right)$ is nondegenerate, then each $\theta \in \mathcal{P}$ is a contact form; see, e.g., Proposition 1.9 in [3], pp. 43–44.

For any nondegenerate CR manifold, on which a contact form $\theta \in \mathcal{P}$ has been fixed, there is a unique globally defined nowhere zero tangent vector field $T=T_M\in \mathfrak{X}\left(M\right)$, transverse to the Levi distribution, determined by the requirements

$$\theta \left(T\right)=1,T\rfloor d\theta =0.$$

See Proposition 1.2 in [3], pp. 8–9.

Definition 12.

T is called the Reeb vector field of $(M,\theta )$.

Let $\theta \in \mathcal{P}$ be a contact form on M, and let us define the $(0, 2)$ tensor field $g_{\theta }$ on M by setting

$$g_{\theta }(X,Y)=G_{\theta }(X,Y),g_{\theta }(X,T)=0,g_{\theta }(T,T)=1,$$

for any $X,Y\in H\left(M\right)$. $g_{\theta }$ is a semi-Riemannian metric on M; see [3], p. 9.

Definition 13

([3], p. 9). $g_{\theta }$ is called the Webster metric of $(M, \theta )$.

Definition 14

([3], p. 6). A CR structure $T_{1,0}\left(M\right)$ is strictly pseudoconvex (and the pair $(M,T_{1,0}\left(M\right))$ is a strictly pseudoconvex CR manifold) if the Levi form $G_{\theta }$ is positive definite for some $\theta \in \mathcal{P}$.

Let ${\mathcal{P}}_{+}={\mathcal{P}}_{+}\left(M\right)$ be the set of all $\theta \in \mathcal{P}$ such that $G_{\theta }$ is positive definite. If M is strictly pseudoconvex, then ${\mathcal{P}}_{+}\ne \varnothing$. Any strictly pseudoconvex CR manifold is nondegenerate. If $\theta \in {\mathcal{P}}_{+}$, then the Webster metric $g_{\theta }$ is a Riemannian metric on M.

Definition 15.

A contact form $\theta \in {\mathcal{P}}_{+}$ is said to be positively oriented.

Quadrics ${\displaystyle Q_n=\{(z,\zeta )\in {\mathbb{C}}^n\times \mathbb{C}:\frac{1}{2i}\left(\zeta -\overline{\zeta }\right)={\left|z\right|}^2\}}$ and odd dimensional spheres $S^{2N+1}\subset {\mathbb{C}}^{N+1}$ are organized as CR manifolds, with the CR structures naturally induced by the ambient complex structure. ${\mathrm{Aut}}_{\mathrm{CR}}(S^{2n+1})$ consists of all fractional linear, or projective, transformations preserving $S^{2n+1}$; see, e.g., [23].

Definition 16

([3], p. 11). The Heisenberg group is the non-commutative Lie group ${\mathbb{H}}_n={\mathbb{C}}^n\times \mathbb{R}\approx {\mathbb{R}}^{2n+1}$, with the group law

$$(z,t)·(w,s)=\left(z+w,t+s+2\mathrm{Im}\langle z,w\rangle \right),$$

$$z,w\in {\mathbb{C}}^n,t,s\in \mathbb{R},\langle z,w\rangle ={\delta }_{jk}{z}^j\overline{w^k}.$$

The Heisenberg group ${\mathbb{H}}_n$ is organized as a CR manifold with the CR structure spanned by

$$L_{\alpha }\equiv \frac{\partial }{\partial z^{\alpha }}+i{\overline{z}}^{\alpha }\frac{\partial }{\partial t},1\le \alpha \le n.$$

See formula (1.24) in [3], p. 12.

Definition 17

([3], p. 12, and [24]). When $n=1$, the first order differential operator ${\overline{L}}_1$ is the Lewy operator.

The mapping

$$f:{\mathbb{H}}_n\to Q_n,f(z,t)=t+i{\left|z\right|}^2,(z,t)\in {\mathbb{H}}_n$$

is a CR isomorphism. Let us set

$${\theta }_0=dt+i\sum _{\alpha =1}^n\left\{z^{\alpha }d{\overline{z}}_{\alpha }-{\overline{z}}^{\alpha }d{z}^{\alpha }\right\},\mathsf{\Theta }=\frac{i}{2}\left(\overline{\partial }-\partial \right){\left|Z\right|}^2,$$

with ${\left|Z\right|}^2={\sum }_{A=0}^N{Z}_A{\overline{Z}}_A$, $Z=(Z_0,\cdots ,Z_N)$. Both ${\mathbb{H}}_n$ and $S^{2N+1}$ are strictly pseudoconvex, and ${\theta }_0\in {\mathcal{P}}_{+}\left({\mathbb{H}}_n\right)$, $\mathsf{\Theta }\in {\mathcal{P}}_{+}\left(S^{2N+1}\right)$.

For any nondegenerate CR manifold M on which a contact form $\theta \in \mathcal{P}$ has been fixed, there is a unique linear connection ∇ on M satisfying the following requirements: (i) $H\left(M\right)$ is parallel with respect to ∇, i.e., $X\in \mathfrak{X}\left(M\right)$ and $Y\in H\left(M\right)$$⟹{\nabla }_{X}Y\in H\left(M\right)$; (ii) $\nabla g_{\theta }=0$ and $\nabla J=0$; and (iii) the torsion $T_{\nabla }$ of ∇ is pure, i.e.,

$$\tau \circ J+J\circ \tau =0,T_{\nabla }(Z,W)=0,T_{\nabla }(Z,\overline{W})=2i{G}_{\theta }(Z,\overline{W}),$$

for any $Z,W\in T_{1,0}\left(M\right)$. Here, $\tau \left(X\right)=T_{\nabla }(T,X)$ for any $X\in \mathfrak{X}\left(M\right)$. See Theorem 1.3 in [3], p. 25.

Definition 18

([3], p. 26). ∇ is the Tanaka–Webster connection of $(M, \theta )$. The vector-valued 1-form $\tau$ on M is the pseudohermitian torsion of ∇.

As a consequence of axioms (i)–(ii) $\tau \left[T_{1,0}\left(M\right)\right]\subset T_{0,1}\left(M\right)$. In particular, the pseudohermitian torsion is trace-less; i.e., $\mathrm{trace}\left(\tau \right)=0$. Moreover, if $A(X,Y)=g_{\theta }(X,\tau Y)$, then A is symmetric, i.e., $A(X,Y)=A(Y,X)$; see Lemma 1.4 in [3], pp. 38–40.

Definition 19.

For every $C^1$ function $u:M\to \mathbb{R}$, the horizontal gradient of u is

$${\nabla }^{H}u={\mathsf{\Pi }}_H\nabla u.$$

Here, ${\mathsf{\Pi }}_H:T\left(M\right)\to H\left(M\right)$ is the projection associated with the direct sum decomposition $T\left(M\right)=H\left(M\right)\oplus \mathbb{R}T$. Additionally, $\nabla u$ is the gradient of u with respect to the Webster metric, i.e.,

$$g_{\theta }\left(\nabla u,X\right)=X\left(u\right),X\in \mathfrak{X}\left(M\right).$$

Definition 20.

For every $C^1$ vector field X on M, the divergence of X is its divergence $\mathrm{div}\left(X\right)$ with respect to the contact form ${\mathsf{\Psi }}_{\theta }=\theta \wedge {\left(d\theta \right)}^n$, i.e.,

$${\mathcal{L}}_X{\mathsf{\Psi }}_{\theta }=\mathrm{div}\left(X\right){\mathsf{\Psi }}_{\theta }.$$

Here, ${\mathcal{L}}_X$ is the Lie derivative at the direction X.

Definition 21

([3], p. 111). Let $\theta \in {\mathcal{P}}_{+}$. The sublaplacian of $(M,\theta )$ is the second order differential operator ${\mathsf{\Delta }}_b$ given by

$${\mathsf{\Delta }}_{b}u=-\mathrm{div}\left({\nabla }^{H}u\right)$$

for every $C^2$ function $u:M\to \mathbb{R}$.

${\mathsf{\Delta }}_b$ is a formally self-adjoint, degenerate elliptic operator (formally similar to the Laplace–Beltrami operator of a Riemannian manifold) naturally occurring on a strictly pseudoconvex CR manifold M, on which a positively oriented contact form $\theta$ has been fixed. While ${\mathsf{\Delta }}_b$ is not elliptic (ellipticity degenerates in the cotangent directions spanned by $\theta$; see [25]), ${\mathsf{\Delta }}_b$ is subelliptic of order $1/2$, and hence it is hypoelliptic; see [3], pp. 114–116, and L. Hörmander [26].

We end the section by briefly recalling a few elements of curvature theory on a nondegenerate CR manifold M, endowed with a contact form $\theta$. Let $R^{\nabla }$ be the curvature tensor field of the Tanaka–Webster connection ∇ of $(M,\theta )$. Let $\{T_{\alpha }:1\le \alpha \le n\}$ be a local frame of $T_{1,0}\left(M\right)$, defined on the open set $U\subset M$, and let $T_{\overline{\alpha }}\equiv \overline{T_{\alpha }}$. Then,

$$\left\{T_A:A\in \{0,1,\cdots ,n,\overline{1},\cdots ,\overline{n}\}\right\}\equiv \{T,T_{\alpha },T_{\overline{\alpha }}:1\le \alpha \le n\},T_0=T$$

is a local frame of $T\left(M\right)\otimes \mathbb{C}$ on U. For all local calculations, one sets

$$g_{\alpha \overline{\beta }}=g(T_{\alpha },T_{\overline{\beta }}),[g^{\alpha \overline{\beta }}]={[g_{\alpha \overline{\beta }}]}^{-1}.$$

Let us consider the $C^{\infty }$ functions ${{R_C}^D}_{AB}:U\to \mathbb{C}$ determined by

$$R^{\nabla }(T_A,T_B)T_C={{R_C}^D}_{AB}{T}_D.$$

Definition 22.

The Ricci tensor is

$${\mathrm{Ric}}_{\nabla }(X,Y)=\mathrm{Trace}\left\{Z\mapsto R^{\nabla }(Z,Y)X\right\}.$$

The pseudohermitian Ricci tensor is $R_{\alpha \overline{\beta }}={\mathrm{Ric}}_{\nabla }(T_{\alpha },T_{\overline{\beta }})$.

Then $R_{\alpha \overline{\beta }}={{R_{\alpha }}^{\gamma }}_{\gamma \overline{\beta }}$; see [3], p. 50.

Definition 23.

The pseudohermitian scalar curvature is $R=g^{\alpha \overline{\beta }}{R}_{\alpha \overline{\beta }}$.

A pseudohermitian analog to the holomorphic sectional curvature (of a Kählerian manifold) was introduced by S.M. Webster [1] and studied in some detail by E. Barletta [27].

2.2. Sub-Riemannian Geometry

Let $(M,T_{1,0}\left(M\right))$ be a CR manifold. Let $S:x\in M\mapsto S_x\subset T_x\left(M\right)$ be a $C^{\infty }$ distribution on M.

Definition 24

([8] p. 224 and [19] p. 124). S is bracket generating if the $C^{\infty }$ sections in S, together with their commutators, span $T_x\left(M\right)$ at each point $x\in M$.

Given $v\in S_x$, let $X\in C^{\infty }\left(S\right)$ such that $X_x=v$. Let $S_x+\left[v,S_x\right]\subset T_x\left(M\right)$ be the subspace spanned by

$$S_x\cup \left\{{[X,Y]}_x:Y\in C^{\infty }\left(S\right)\right\}.$$

Next, let us inductively define the spaces ${\mathcal{D}}_k\left(v\right)\subset T_x\left(M\right)$ by setting

$${\mathcal{D}}_2\left(v\right)=S_x+\left[v,S_x\right],{\mathcal{D}}_k\left(v\right)=S_x+\left[{\mathcal{D}}_{k-1},S_x\right],k\ge 3.$$

Definition 25

([8,19]). A tangent vector $v\in S_x$ is a k-step bracket generator if ${\mathcal{D}}_k\left(v\right)=T_x\left(M\right)$. The distribution S is said to satisfy the strong bracket generating hypothesis if, for arbitrary $x\in M$, each $v\in S_x$ is a 2-step bracket generator.

Let S be a bracket generating distribution on M.

Definition 26

([8,19]). A sub-Riemannian metric on S is a Riemannian bundle metric on S, i.e., a $C^{\infty }$ positive definite section $Q\in C^{\infty }\left(S^{\ast }\otimes S^{\ast }\right)$. A pair $(S,Q)$ consisting of a bracket generating distribution S on M and a sub-Riemannian metric Q on S is called a sub-Riemannian structure on M.

Definition 27

([8], p. 229 and [7,19]). A piecewise $C^1$ curve $\gamma :I\to M$ (where $I\subset \mathbb{R}$ is an interval) is horizontal if $\dot{\gamma }\left(t\right)\in H{\left(M\right)}_{\gamma \left(t\right)}$ for all values of the parameter t (for which $\dot{\gamma }\left(t\right)$ makes sense).

Assume M to be strictly pseudoconvex. Let $\theta \in {\mathcal{P}}_{+}$ be a positively oriented contact form on M. Then, $(H\left(M\right),G_{\theta })$ is a sub-Riemannian structure on M; see [19], p. 125.

Definition 28.

The sub-Riemannian length of a horizontal curve $\gamma :I\to M$ is

$$\ell \left(\gamma \right)={\int }_I{G}_{\theta }{\left(\dot{\gamma }\left(t\right),\dot{\gamma }\left(t\right)\right)}^{1/2}dt.$$

A piecewise $C^1$ curve $\gamma :I\to M$ joins the points $x,y\in M$ if $I=[a, b]$, $\gamma \left(a\right)=x$, and $\gamma \left(b\right)=y$. Let $\mathsf{\Omega }(x, y)$ (,respectively, ${\mathsf{\Omega }}_H(x, y)$) be the set of all piecewise $C^1$ (respectively horizontal) curves joining x and y. Let $d(x, y)$ (respectively, $d_H(x, y)$) be the distance between $x,y\in M$ induced by the Riemannian metric $g_{\theta }$ (respectively, the greatest lower bound of $\{\ell \left(\gamma \right):\gamma \in {\mathsf{\Omega }}_H(x, y)\}$). $d_H:M\times M\to [0,+\infty )$ is a distance function on M; see [8], p. 230.

Definition 29.

$d_H$ is the Carnot–Carthéodory distance function on M, induced by the sub-Riemannian structure $(H\left(M\right),G_{\theta })$.

Definition 30

([8], p. 230). A Riemannian metric g on M is said to be a contraction of the sub-Riemannian metric $G_{\theta }$ if the distance function $\rho :M\times M\to [0,+\infty )$ associated with g satisfies $\rho (x,y)\le d_H(x,y)$ for any $x,y\in M$.

As ${\mathsf{\Omega }}_H(x, y)\subset \mathsf{\Omega }(x, y)$ (a strict inclusion), one has $d(x, y)\le d_H(x, y)$ for any $x, y\in M$. Hence, the Webster metric $g_{\theta }$ is a contraction of the Levi form $G_{\theta }$. The construction of a contraction of $G_{\theta }$ by the requirement that the norm of the Reeb vector T be 1, appearing as quite natural a priori, proves to be rather restrictive later on; i.e., the Riemannian geometry of $(M,g_{\theta })$ turns out to be insufficiently related to the CR and pseudohermitian geometry on $(M,\theta )$. As shown by J. Jost and C-J. Xu (see [28]), the requirement that the norm of T be $O\left({ϵ}^{-1}\right)$ is far reaching (and related to the notion of homogeneous space in PDEs theory; see [28]). In the next section, we adopt a version of the construction in [28], referred to in the sequel as an $ϵ$-contraction of $G_{\theta }$.

2.3. $ϵ$-Contractions

Let $(M,T_{1,0}\left(M\right))$ be a strictly pseudoconvex CR manifold, and let $\theta \in {\mathcal{P}}_{+}\left(M\right)$ be a positively oriented contact form on M. Let $T\in \mathfrak{X}\left(M\right)$ be the Reeb vector field of $(M, \theta )$. Let $0<ϵ<1$, and let $g_{ϵ}=g_{ϵ}^M$ be the $(0, 2)$ tensor field on M defined by

$$g_{ϵ}(T, T)={ϵ}^{-2},$$

(4)

$$g_{ϵ}(X,Y)=G_{\theta }(X,Y),g_{ϵ}(X, T)=0,$$

(5)

for any $X,Y\in H\left(M\right)$. $g_{ϵ}$ is a Riemannian metric on M and a contraction of $G_{\theta }$. The direct sum decomposition

$$T\left(M\right)=H\left(M\right)\oplus \mathbb{R}T$$

(6)

together with (4) and (5) yields

$$g_{ϵ}=g_{\theta }+\left(\frac{1}{{ϵ}^2}-1\right)\theta \otimes \theta .$$

(7)

Definition 31.

$g_{ϵ}$ is called the $ϵ$-contraction of $G_{\theta }$.

The contraction $g_{ϵ}$ is built such that the $g_{ϵ}$ norm of T is $O\left({ϵ}^{-1}\right)$, a property of crucial importance in the further asymptotic analysis as $ϵ\to 0^{+}$. For every $0<ϵ<1$, we consider the contact form ${\theta }_{ϵ}={ϵ}^{-1}\theta$. The Reeb vector field $T_{ϵ}\in \mathfrak{X}\left(M\right)$ of $(M,{\theta }_{ϵ})$ is given by $T_{ϵ}=ϵT$.

Lemma 1.

The Webster metric $g_{{\theta }_{ϵ}}$ and the ϵ-contraction $g_{ϵ}$ of $G_{\theta }$ are related by

$$g_{{\theta }_{ϵ}}(X,Y)={ϵ}^{-1}{g}_{ϵ}(X,Y),$$

(8)

$$g_{{\theta }_{ϵ}}(X, T)=g_{ϵ}(X, T)=0,$$

(9)

$$g_{{\theta }_{ϵ}}(T, T)=g_{ϵ}(T, T)={ϵ}^{-2},$$

(10)

for any $X, Y\in H\left(M\right)$. Summing up:

$$g_{{\theta }_{ϵ}}={ϵ}^{-1}{g}_{ϵ}+{ϵ}^{-2}(1-{ϵ}^{-1})\theta \otimes \theta .$$

(11)

In particular, none of the metrics $\{g_{ϵ}:ϵ>0\}$ is a Webster metric; i.e., there is no $u_{ϵ}\in C^{\infty }(M,\mathbb{R})$ such that $g_{ϵ}=g_{exp\left(u_{ϵ}\right)\theta }$.

The proof of Lemma 1 is straightforward.

Lemma 2.

The Levi–Civita connection${\nabla }^{ϵ}$of the Riemannian manifold$(M,g_{ϵ})$and the Tanaka–Webster connection ∇ of the pseudohermitian manifold $(M,\theta )$ are related by

$${\nabla }_X^{ϵ}Y={\nabla }_{X}Y+\left\{\mathsf{\Omega }(X,Y)-{ϵ}^{2}A(X,Y)\right\}T,$$

(12)

$${\nabla }_X^{ϵ}T=\tau X+\frac{1}{{ϵ}^2}JX,$$

(13)

$${\nabla }_T^{ϵ}X={\nabla }_{T}X+\frac{1}{{ϵ}^2}JX,$$

(14)

$${\nabla }_T^{ϵ}T=0,$$

(15)

for any $X,Y\in H\left(M\right)$.

Here, $\mathsf{\Omega }(X,Y)=g_{\theta }(X,JY)$ for any $X,Y\in \mathfrak{X}\left(M\right)$. $\mathsf{\Omega }$ is a pseudohermitian analog to the fundamental 2-form in Hermitian geometry. However, $\mathsf{\Omega }=-d\theta$ so that, unlike the (perhaps more familiar) case of Kählerian geometry, $\mathsf{\Omega }$ and its exterior powers do not determine nontrivial de Rham cohomology classes on M.

The remainder of the section is devoted to the proof of Lemma 2. This requires a rather involved calculation, as follows. Given a Riemannian metric g on M, it will be useful to adopt the following:

Definition 32.

The Christoffel mapping is

$$C_g:\mathfrak{X}\left(M\right)\times \mathfrak{X}\left(M\right)\times \mathfrak{X}\left(M\right)\to C^{\infty }(M,\mathbb{R}),$$

$$C_g(X,Y,Z)=X\left(g(Y, Z)\right)+Y\left(g(Z, X)\right)-Z\left(g(X, Y)\right)$$

$$+g\left([X,Y],Z\right)-g\left([Y,Z],X\right)+g\left([Z,X],Y\right),$$

for any $X,Y,Z\in \mathfrak{X}\left(M\right)$.

Let $g_{\theta }$ and ∇ be, respectively, the Webster metric and Tanaka–Webster connection of $(M, \theta )$. As $\nabla g_{\theta }=0$, one may apply the so-called Christoffel process; i.e., starting from

$$X\left(g_{\theta }(Y, Z)\right)=g_{\theta }\left({\nabla }_{X}Y,Z\right)+g_{\theta }\left(Y,{\nabla }_{X}Z\right),$$

we produce other two identities of the sort by circular permutation of $X,Y,Z$,

$$Y\left(g_{\theta }(Z, X)\right)=g_{\theta }\left({\nabla }_{Y}Z,X\right)+g_{\theta }\left(Z,{\nabla }_{Y}X\right),$$

$$Z\left(g_{\theta }(X,Y)\right)=g_{\theta }\left({\nabla }_{Z}X,Y\right)+g_{\theta }\left(X,{\nabla }_{Z}Y\right),$$

add the first two and subtract the third, and use ${\nabla }_{X}Y-{\nabla }_{Y}X-[X,Y]=T_{\nabla }(X,Y)$ to recognize torsion terms. We obtain

$$\begin{array}{c}2g_{\theta }\left({\nabla }_{X}Y,Z\right)=X\left(g_{\theta }(Y, Z)\right)+Y\left(g_{\theta }(Z, X)\right)-Z\left(g_{\theta }(X,Y)\right)\\ +g_{\theta }\left([X,Y],Z\right)-g_{\theta }\left([Y,Z],X\right)+g_{\theta }\left([Z,X],Y\right)\\ +g_{\theta }\left(T_{\nabla }(X,Y),Z\right)-g_{\theta }\left(T_{\nabla }(Y,Z),X\right)+g_{\theta }\left(T_{\nabla }(Z,X),Y\right)\end{array}$$

(16)

for any $X,Y,Z\in \mathfrak{X}\left(M\right)$. By the purity axiom, the torsion of the Tanaka–Webster connection satisfies

$${\mathrm{Tor}}_{\nabla }=2\left\{\theta \wedge \tau -\mathsf{\Omega }\otimes T\right\}.$$

(17)

Then, by (17),

$$g_{\theta }\left({\mathrm{Tor}}_{\nabla }(X,Y),Z\right)-g_{\theta }\left({\mathrm{Tor}}_{\nabla }(Y,Z),X\right)+g_{\theta }\left({\mathrm{Tor}}_{\nabla }(Z,X),Y\right)$$

(18)

$$=2\left\{A(X,Y)\theta \left(Z\right)-A(X, Z)\theta \left(Y\right)\right\}$$

$$+\mathsf{\Omega }(X, Z)\theta \left(Y\right)+\mathsf{\Omega }(Y, Z)\theta \left(X\right)-\mathsf{\Omega }(X, Y)\theta \left(Z\right)\}$$

for any $X,Y,Z\in \mathfrak{X}\left(M\right)$. Substitution from (18) into (16) furnishes

$$\begin{array}{c}2g_{\theta }\left({\nabla }_{X}Y,Z\right)=C_{g_{\theta }}(X,Y,Z)\\ +2\left\{A(X,Y)\theta \left(Z\right)-A(X,Z)\theta \left(Y\right)\right\}\\ +\mathsf{\Omega }(X,Z)\theta \left(Y\right)+\mathsf{\Omega }(Y,Z)\theta \left(X\right)-\mathsf{\Omega }(X,Y)\theta \left(Z\right)\}.\end{array}$$

(19)

Next, we may exploit (7) (relating the $ϵ$-contraction $g_{ϵ}$ to the Webster metric $g_{\theta }$) to derive

$$\begin{array}{c}{C}_{g_{ϵ}}(X,Y,Z)=C_{g_{\theta }}(X,Y,Z)\\ +\left(\frac{1}{{ϵ}^2}-1\right)\{X\left(\theta \left(Y\right)\theta \left(Z\right)\right)+Y\left(\theta \left(Z\right)\theta \left(X\right)\right)-Z\left(\theta \left(X\right)\theta \left(Y\right)\right)\\ +\theta \left([X,Y]\right)\theta \left(Z\right)-\theta \left([Y,Z]\right)\theta \left(X\right)+\theta \left([Z,X]\right)\theta \left(Y\right)\}\end{array}$$

or, by using $2\left(d\theta \right)(X,Y)=X\left(\theta \right(Y\left)\right)-Y\left(\theta \right(X\left)\right)-\theta \left(\right[X,Y\left]\right)$,

$$\begin{array}{l}{C}_{g_{ϵ}}(X,Y,Z)=C_{g_{\theta }}(X,Y,Z)+2\left(\frac{1}{{ϵ}^2}-1\right)\{X\left(\theta \left(Y\right)\theta \left(Z\right)\right)\\ +\theta \left(X\right)\left(d\theta \right)(Y,Z)+\theta \left(Y\right)\left(d\theta \right)(X,Z)-\theta \left(Z\right)\left(d\theta \right)(X,Y)\}.\end{array}$$

(20)

Let ${\nabla }^{ϵ}$ be the Levi–Civita connection of the Riemannian manifold $(M,g_{ϵ})$. As ${\nabla }^{ϵ}$ is symmetric and ${\nabla }^{ϵ}{g}_{ϵ}=0$, the Christoffel process yields

$$2g_{ϵ}\left({\nabla }_X^{ϵ}Y,Z\right)=C_{g_{ϵ}}(X,Y,Z).$$

(21)

Then, by substitution from (20) into (21),

$$2g_{ϵ}\left({\nabla }_X^{ϵ}Y,Z\right)=C_{g_{\theta }}(X,Y,Z)+2\left(\frac{1}{{ϵ}^2}-1\right)\{X\left(\theta \left(Y\right)\right)\theta \left(Z\right)$$

$$+\theta \left(X\right)\left(d\theta \right)(Y,Z)+\theta \left(Y\right)\left(d\theta \right)(X,Z)-\theta \left(Z\right)\left(d\theta \right)(X,Y)\}$$

or, by replacing $g_{ϵ}$ in terms of $g_{\theta }$ from (7), substituting $C_{g_{\theta }}(X,Y,Z)$ from (19), and using $\mathsf{\Omega }=-d\theta$,

$$\begin{array}{c}{g}_{\theta }\left({\nabla }_X^{ϵ}Y,Z\right)+\left(\frac{1}{{ϵ}^2}-1\right)\theta \left({\nabla }_X^{ϵ}Y\right)\theta \left(Z\right)\\ =g_{\theta }\left({\nabla }_{X}Y,Z\right)+\left(\frac{1}{{ϵ}^2}-1\right)X\left(\theta \left(Y\right)\right)\theta \left(Z\right)\\ +A(X,Z)\theta \left(Y\right)-A(X,Y)\theta \left(Z\right)\\ +\frac{1}{{ϵ}^2}\left\{\mathsf{\Omega }(X,Y)\theta \left(Z\right)-\mathsf{\Omega }(Y,Z)\theta \left(X\right)-\mathsf{\Omega }(X,Z)\theta \left(Y\right)\right\}.\end{array}$$

(22)

The rather involved relation (22) holding for any $X,Y,Z\in \mathfrak{X}\left(M\right)$ can be greatly simplified by using the decomposition (6). For arbitrary $Z\in H\left(M\right)$, the relation (22) yields, $\theta \left(Z\right)=0$,

$${\mathsf{\Pi }}_H{\nabla }_X^{ϵ}Y={\mathsf{\Pi }}_H{\nabla }_{X}Y+\theta \left(Y\right)\tau X+\frac{1}{{ϵ}^2}\left\{\theta \left(X\right)JY+\theta \left(Y\right)JX\right\}$$

(23)

where ${\mathsf{\Pi }}_H={\mathsf{\Pi }}_{H\left(M\right)}:T\left(M\right)\to H\left(M\right)$ is the projection with respect to the decomposition (6). Again, by (22), for $Z=T$, we obtain

$$\theta \left({\nabla }_X^{ϵ}Y\right)=X\left(\theta \left(Y\right)\right)+\mathsf{\Omega }(X,Y)-{ϵ}^{2}A(X,Y),$$

(24)

which determines the component along T of ${\nabla }_X^{ϵ}Y$, with respect to the decomposition (6). At this point, we may use (23) and (24) to compute ${\nabla }_X^{ϵ}Y$ for any $X,Y\in \mathfrak{X}\left(M\right)$. For every $V\in \mathfrak{X}\left(M\right)$,

$$V={\mathsf{\Pi }}_{H}V+\theta \left(V\right)T,$$

(25)

hence, by (23) and (24),

$${\nabla }_X^{ϵ}Y={\mathsf{\Pi }}_H{\nabla }_X^{ϵ}Y+\theta \left({\nabla }_X^{ϵ}Y\right)T=$$

$$={\nabla }_{X}Y+\mathsf{\Omega }(X,Y)T+\theta \left(Y\right)\tau X+\frac{2}{{ϵ}^2}(\theta \odot J)(X,Y)-{ϵ}^{2}A(X,Y)T,$$

(26)

where ⊙ is the symmetric tensor productl i.e., $\alpha \odot \beta =\frac{1}{2}(\alpha \otimes \beta +\beta \otimes \alpha )$. For $X,Y\in H\left(M\right)$, Equation (26) becomes

$${\nabla }_X^{ϵ}Y={\nabla }_{X}Y+\left\{\mathsf{\Omega }(X,Y)-{ϵ}^{2}A(X,Y)\right\}T,$$

and (12) is proved. The remaining relations (13)–(15) in Lemma 2 follow from (26) for: (i) $X\in H\left(M\right)$ and $Y=T$; (ii) $X=T$ and $Y\in H\left(M\right)$; and (iii) $X=Y=T$. Q.E.D.

It will be useful to compute the covariant derivative of J with respect to ${\nabla }^{ϵ}$, where $J:H\left(M\right)\to H\left(M\right)$ is extended as customary to a $(1, 1)$-tensor field on M by requiring that $JT=0$. Note that the extension of J depends on the chosen contact form $\theta$ on M.

Lemma 3.

For any $X,Y\in H\left(M\right)$

$$\left({\nabla }_X^{ϵ}J\right)Y=-\left\{g_{\theta }(X,Y)+{ϵ}^{2}A(X,JY)\right\}T,$$

(27)

$$\left({\nabla }_X^{ϵ}J\right)T=\tau JX+\frac{1}{{ϵ}^2}X,$$

(28)

$$\left({\nabla }_T^{ϵ}J\right)X=0,$$

(29)

$$\left({\nabla }_T^{ϵ}J\right)T=0.$$

(30)

Proof. 

Lemma 3 follows from (12)–(15) together with $\nabla J=0$. For instance,

$$\left({\nabla }_X^{ϵ}J\right)Y={\nabla }_X^{ϵ}JY-J{\nabla }_X^{ϵ}Y$$

by (12), and $JT=0$ and $J^2=-I+\theta \otimes T$

$$={\nabla }_{X}JY+\left\{\mathsf{\Omega }(X,JY)-{ϵ}^{2}A(X,JY)\right\}T-J{\nabla }_{X}Y$$

by $\nabla J=0$, and $\mathsf{\Omega }(X,Y)=g_{\theta }(X,JY)$

$$=-\left\{g_{\theta }(X,Y)+{ϵ}^{2}A(X,JY)\right\}T,$$

yielding (27).    □

2.4. Gradients and the Laplace–Beltrami Operator on $(M,g_{ϵ})$

For every function $u\in C^1\left(M\right)$, let ${\nabla }^{ϵ}u$ be the gradient of u with respect to the Riemannian metric $g_{ϵ}$, i.e.,

$$g_{ϵ}({\nabla }^{ϵ}u,X)=X\left(u\right),X\in \mathfrak{X}\left(M\right).$$

Let ${\mathsf{\Delta }}_{ϵ}$ be the Laplace–Beltrami operator of $(M,g_{ϵ})$, i.e.,

$${\mathsf{\Delta }}_{ϵ}u=-{\mathrm{div}}_{ϵ}\left({\nabla }^{ϵ}u\right),u\in C^2\left(M\right).$$

Here, ${\mathrm{div}}_{ϵ}$ is the divergence operator with respect to the volume form ${\mathsf{\Psi }}_{ϵ}=d\mathrm{vol}\left(g_{ϵ}\right)$, i.e.,

$${\mathcal{L}}_X{\mathsf{\Psi }}_{ϵ}={\mathrm{div}}_{ϵ}\left(X\right){\mathsf{\Psi }}_{ϵ}$$

for every $C^1$ vector field X tangent to M. Let $(U,x^i)$ be a local coordinate system on M, and let us set

$${\mathfrak{g}}_{ϵ}=det\left[g_{ij}\left(ϵ\right)\right],g_{ij}\left(ϵ\right)=g_{ϵ}\left(\frac{\partial }{\partial x^i},\frac{\partial }{\partial x^j}\right),1\le i,j\le 2n+1.$$

The Riemannian volume form of $(M,g_{ϵ})$ is locally given by

$${\mathsf{\Psi }}_{ϵ}=\sqrt{{\mathfrak{g}}_{ϵ}}d{x}^1\wedge \cdots \wedge d{x}^{2n+1}.$$

The volume form ${\mathsf{\Psi }}_{\theta }$ is parallel with respect to the Tanaka–Webster connection (i.e., $\nabla {\mathsf{\Psi }}_{\theta }=0$), hence the divergence of a $C^1$ vector field X may be computed as

$$\mathrm{div}\left(X\right)=\mathrm{Trace}\left\{Y\mapsto {\nabla }_{Y}X\right\}.$$

(31)

See [3], p. 112.

Lemma 4.

(i) For every $u\in C^1\left(M\right)$,

$${\nabla }^{ϵ}u={\nabla }^{H}u+{ϵ}^2\theta (\nabla u)T.$$

(32)

(ii) For every $u\in C^2\left(M\right)$,

$${\mathsf{\Delta }}_{ϵ}u={\mathsf{\Delta }}_{b}u-{ϵ}^2{T}^2\left(u\right).$$

(33)

Proof. 

(i) By (7), for every $X\in \mathfrak{X}\left(M\right)$,

$$g_{\theta }(\nabla u,X)=X\left(u\right)=g_{ϵ}({\nabla }^{ϵ}u,X)$$

$$=g_{\theta }({\nabla }^{ϵ}u,X)+\left(\frac{1}{{ϵ}^2}-1\right)\theta \left({\nabla }^{ϵ}u\right)\theta \left(X\right).$$

In particular, for arbitrary $X\in H\left(M\right)$,

$${\mathsf{\Pi }}_H{\nabla }^{ϵ}u={\nabla }^{H}u.$$

(34)

Also, for $X=T$,

$$\theta \left({\nabla }^{ϵ}u\right)={ϵ}^2\theta (\nabla u).$$

(35)

Finally, by (34) and (35) and the decomposition (25),

$${\nabla }^{ϵ}u={\mathsf{\Pi }}_H{\nabla }^{ϵ}u+\theta \left({\nabla }^{ϵ}u\right)T={\nabla }^{H}u+{ϵ}^2\theta (\nabla u)T,$$

and (32) in Lemma 4 is proved.

(ii) Let $\{X_a:1\le a\le 2n\}$ be a local $g_{\theta }$-orthonormal

$$G_{\theta }(X_a,X_b)={\delta }_{ab},1\le a,b\le 2n,$$

frame of $H\left(M\right)$, defined on the open subset $U\subset M$. Then, by (7),

$$\left\{E_j^{ϵ}:1\le j\le 2n+1\right\}\equiv \left\{X_a,ϵT:1\le a\le 2n\right\}$$

is a local $g_{ϵ}$-orthonormal frame of $T\left(M\right)$. Consequently, the Laplace–Beltrami operator of $(M, g_{ϵ})$ on functions can be computed as

$${\mathsf{\Delta }}_{ϵ}u=-\sum _{j=1}^{2n+1}\left\{E_j^{ϵ}\left(E_j^{ϵ}\left(u\right)\right)-\left({\nabla }_{E_j^{ϵ}}^{ϵ}{E}_j^{ϵ}\right)\left(u\right)\right\}$$

$$=-\sum _{a=1}^{2n}\left\{X_a\left(X_a\left(u\right)\right)-\left({\nabla }_{X_a}^{ϵ}{X}_a\right)\left(u\right)\right\}-{ϵ}^2\left\{T\left(T\left(u\right)\right)-\left({\nabla }_T^{ϵ}T\right)\left(u\right)\right\}$$

by (12) and (15)

$$=-\sum _{a=1}^{2n}\left\{X_a\left(X_a\left(u\right)\right)-\left({\nabla }_{X_a}{X}_a\right)\left(u\right)\right\}$$

$$+\sum _{a=1}^{2n}\left\{\mathsf{\Omega }(X_a,X_a)-{ϵ}^{2}A(X_a,X_a)\right\}T\left(u\right)-{ϵ}^2{T}^2\left(u\right)$$

for every $u\in C^2\left(M\right)$. Finally, (33) in Lemma 4 follows from

$${\mathsf{\Delta }}_{b}u=-\sum _{a=1}^{2n}\left\{X_a\left(X_a\left(u\right)\right)-\left({\nabla }_{X_a}{X}_a\right)\left(u\right)\right\}$$

(36)

as $\mathsf{\Omega }$ is skew-symmetric, $T\rfloor A=0$, and

$$\sum _{a=1}^{2n}A(X_a,X_a)=\mathrm{trace}\left(\tau \right)=0.$$

The formula (36) is a consequence of (31).    □

Let $\mathsf{\Omega }\subset {\mathbb{H}}_n$ be a bounded domain, with boundary $S=\partial \mathsf{\Omega }$ of class $C^r$, $1\le r\le \infty$. Let us assume that $\mathsf{\Omega }$ lies on one side of its boundary; i.e., for every $(z_0,t_0)\in S$, there is a neighborhood $U\subset {\mathbb{H}}_n$ and a diffeomorphism $\psi :U\to {\mathbb{B}}^{2n+1}$ such that $\psi (z_0,t_0)=0$ and $\psi (U\cap \mathsf{\Omega })=\{(z,t)\in {\mathbb{B}}^{2n+1}:t>0\}$. Here, ${\mathbb{B}}^N=\{x\in {\mathbb{R}}^N:\left|x\right|<1\}$ is the unit ball. Let ${\mathrm{DO}}_{\ell }\left(\overline{\mathsf{\Omega }}\right)$ be the space of differential operators of order ℓ with real valued continuous coefficients

$$Lu\equiv \sum _{\left|\alpha \right|\le \ell }{a}_{\alpha }(z, t)D^{\alpha }u,a_{\alpha }\in C\left(\overline{\mathsf{\Omega }},\mathbb{R}\right),\left|\alpha \right|\le \ell .$$

${\mathrm{DO}}_{\ell }\left(\overline{\mathsf{\Omega }}\right)$ is a Banach space with the norm

$$\parallel L\parallel =\sum _{\left|\alpha \right|\le k}\underset{(z,t)\in \overline{\mathsf{\Omega }}}{sup}|a_{\alpha }(z, t)|.$$

Let ${\displaystyle L_{\ell }(z,t,\xi )=\sum _{\left|\alpha \right|=\ell }{a}_{\alpha }(z, t){\xi }^{\alpha }}$ be the symbol of $L\in {\mathrm{DO}}_{\ell }\left(\overline{\mathsf{\Omega }}\right)$. L is degenerate elliptic if

(i)

There exist $(z,t)\in \overline{\mathsf{\Omega }}$ and $\xi \in {\mathbb{R}}^{2n+1}$ such that $L_{\ell }(z,t,\xi )\ne 0$ and $\mathrm{sign}{L}_{\ell }(z,t,\xi )$ is constant on $\overline{\mathsf{\Omega }}\times S^{2n}$ [where $\mathrm{sign}:\mathbb{R}\to \{\pm 1\}$],

(ii)

The set $\{(z,t,\xi )\in \overline{\mathsf{\Omega }}\times S^{2n}:L_{\ell }(z,t,\xi )=0\}$ is nonempty.

Proposition 1.

For every bounded domain $\mathsf{\Omega }\subset {\mathbb{H}}_n$ in the Heisenberg group, the sublaplacian ${\mathsf{\Delta }}_b\in {\mathrm{DO}}_2\left(\overline{\mathsf{\Omega }}\right)$ is a degenerate elliptic operator of order $\ell =2$.

Proof. 

If $E={\mathbb{H}}_n\times \mathbb{R}$ is the trivial vector bundle, one may compute the symbol ${\sigma }_2\left({\mathsf{\Delta }}_b\right)\in \mathrm{Hom}\left({\pi }^{\ast }E,{\pi }^{\ast }E\right)$ [where $\pi :T^{\ast }({\mathbb{H}}_n)\backslash \left\{0\right\}\to {\mathbb{H}}_n$ is the projection] and show that the ellipticity of ${\mathsf{\Delta }}_b$ degenerates at the cotangent directions spanned by the canonical contact form ${\theta }_0$ (see E. Barletta and S. Dragomir [25]). Here we wish to give a “sub-Riemannian proof” to the statement. Let us recall that $L\in {\mathrm{DO}}_{\ell }\left(\overline{\mathsf{\Omega }}\right)$ is elliptic in $\overline{\mathsf{\Omega }}$ if $L_{\ell }(z,t,\xi )\ne 0$ for any $(z, t)\in \overline{\mathsf{\Omega }}$ and any $\xi \in {\mathbb{R}}^{2n+1}\backslash \left\{0\right\}$. Let ${\mathrm{EO}}_{\ell }\left(\overline{\mathsf{\Omega }}\right)$ be the set of elliptic operators of order k. Then, ${\mathsf{\Delta }}_{ϵ}\in {\mathrm{EO}}_2\left(\overline{\mathsf{\Omega }}\right)$ for every $ϵ>0$ and, by (33),

$$\parallel {\mathsf{\Delta }}_{ϵ}-{\mathsf{\Delta }}_b\parallel ={ϵ}^2\parallel T^2\parallel \to 0,ϵ\to 0^{+},$$

hence ${\mathsf{\Delta }}_b\in \partial {\mathrm{EO}}_2\left(\overline{\mathsf{\Omega }}\right)$. However, (see, e.g., N. Shimakura [29], p. 184) ${\mathrm{EO}}_2\left(\overline{\mathsf{\Omega }}\right)$ is an open subset of the Banach space ${\mathrm{DO}}_2\left(\overline{\mathsf{\Omega }}\right)$ whose boundary consists precisely of the degenerate (second order) elliptic operators on $\overline{\mathsf{\Omega }}$.    □

2.5. Curvature Properties

Let $R(\nabla )$ and $R\left({\nabla }^{ϵ}\right)$ be the curvature tensor fields of ∇ (the Tanaka–Webster connection of $(M, \theta )$) and of ${\nabla }^{ϵ}$ (the Levi–Civita connection of $(M,g_{ϵ})$).

Lemma 5.

Let M be a strictly pseudoconvex CR manifold and $\theta \in {\mathcal{P}}_{+}$ a positively oriented contact form on M. Then, $R\left({\nabla }^{ϵ}\right)$ and $R(\nabla )$ are related by

$$\begin{array}{c}R\left({\nabla }^{ϵ}\right)(X,Y)Z=R(\nabla )(X,Y)Z\\ -{ϵ}^2\left[\left({\nabla }_{X}A\right)(Y,Z)-\left({\nabla }_{Y}A\right)(X,Z)\right]T\\ +\mathsf{\Omega }(Y,Z)\tau X-\mathsf{\Omega }(X,Z)\tau Y-A(Y,Z)JX+A(X,Z)JY\\ +{ϵ}^2\left[A(X,Z)\tau Y-A(Y,Z)\tau X\right]\\ +\frac{1}{{ϵ}^2}\left[\mathsf{\Omega }(Y,Z)JX-\mathsf{\Omega }(X,Z)JY-2\mathsf{\Omega }(X,Y)JZ\right],\end{array}$$

(37)

$$R\left({\nabla }^{ϵ}\right)(X,Y)T=\left({\nabla }_X\tau \right)Y-\left({\nabla }_Y\tau \right)X+2\mathsf{\Omega }(X,\tau Y)T,$$

(38)

$$\begin{array}{c}R\left({\nabla }^{ϵ}\right)(X,T)Y=R(\nabla )(X,T)Y\\ +\{{ϵ}^2\left[\left({\nabla }_{T}A\right)(X,Y)+g_{\theta }\left(\tau X,\tau Y\right)\right]\\ -\frac{1}{{ϵ}^2}{g}_{\theta }(X,Y)-2A(X,JY)-\mathsf{\Omega }(\tau X,Y)\}T,\end{array}$$

(39)

$$R\left({\nabla }^{ϵ}\right)(X,T)T=-{\tau }^{2}X-\frac{1}{{ϵ}^2}JX+\frac{1}{{ϵ}^4}X-\left({\nabla }_T\tau \right)X,$$

(40)

for any $X,Y,Z\in H\left(M\right)$.

Proof. 

Let $X,Y,Z\in H\left(M\right)$. Then,

$$[X,Y]={\mathsf{\Pi }}_H[X,Y]+\theta \left([X,Y]\right)T$$

and

$$\theta \left([X,Y]\right)=-2\left(d\theta \right)(X,Y)=2\mathsf{\Omega }(X,Y),$$

hence

$$[X,Y]={\mathsf{\Pi }}_H[X,Y]+2\mathsf{\Omega }(X,Y)T.$$

(41)

By (12)–(15) relating ${\nabla }^{ϵ}$ to ∇, one conducts the following calculations:

$$R\left({\nabla }^{ϵ}\right)(X,Y)Z={\nabla }_X^{ϵ}{\nabla }_Y^{ϵ}Z-{\nabla }_Y^{ϵ}{\nabla }_X^{ϵ}Z-{\nabla }_{[X,Y]}^{ϵ}Z$$

by substitution from (41)

$$={\nabla }_X^{ϵ}\left\{{\nabla }_{Y}Z+\left[\mathsf{\Omega }(Y,Z)-{ϵ}^{2}A(Y,Z)\right]T\right\}$$

$$-{\nabla }_Y^{ϵ}\left\{{\nabla }_{X}Z+\left[\mathsf{\Omega }(X,Z)-{ϵ}^{2}A(X,Z)\right]T\right\}$$

$$-{\nabla }_{{\mathsf{\Pi }}_H[X,Y]}Z-\left\{\mathsf{\Omega }\left({\mathsf{\Pi }}_H[X,Y],Z\right)-{ϵ}^{2}A\left({\mathsf{\Pi }}_H[X,Y],Z\right)\right\}T$$

$$-2\mathsf{\Omega }(X,Y)\left[{\nabla }_{T}Z+\frac{1}{{ϵ}^2}JZ\right]$$

again by (41) and $\nabla \mathsf{\Omega }=0$,

$$=R(\nabla )(X,Y)Z-\frac{2}{{ϵ}^2}\mathsf{\Omega }(X,Y)JZ$$

$$+\left[\mathsf{\Omega }(Y,Z)-{ϵ}^{2}A(Y,Z)\right]\left[\tau X+\frac{1}{{ϵ}^2}JX\right]$$

$$-\left[\mathsf{\Omega }(X,Z)-{ϵ}^{2}A(X,Z)\right]\left[\tau Y+\frac{1}{{ϵ}^2}JY\right]$$

$$+{ϵ}^2\left\{\left({\nabla }_{Y}A\right)(X,Z)-\left({\nabla }_{X}A\right)(Y,Z)\right\}T,$$

thus proving (37). To prove (38), one conducts the following calculation

$$R\left({\nabla }^{ϵ}\right)(X,Y)T={\nabla }_X^{ϵ}{\nabla }_Y^{ϵ}T-{\nabla }_Y^{ϵ}{\nabla }_X^{ϵ}T-{\nabla }_{[X,Y]}^{ϵ}T$$

by (13) and (41)

$$={\nabla }_X^{ϵ}\left[\tau Y+\frac{1}{{ϵ}^2}JY\right]-{\nabla }_Y^{ϵ}\left[\tau X+\frac{1}{{ϵ}^2}JX\right]$$

$$-{\nabla }_{{\mathsf{\Pi }}_H[X,Y]}^{ϵ}T-2\mathsf{\Omega }(X,Y){\nabla }_T^{ϵ}T$$

by $\tau \circ J+J\circ \tau =0$

$$=\left({\nabla }_X\tau \right)Y-\left({\nabla }_Y\tau \right)X+\frac{1}{{ϵ}^2}\left[\left({\nabla }_{X}J\right)Y-\left({\nabla }_{Y}J\right)X\right],$$

yielding (38) by $\nabla J=0$. To prove (39), one conducts the following calculation:

$$R\left({\nabla }^{ϵ}\right)(X, T)Y={\nabla }_X^{ϵ}{\nabla }_T^{ϵ}Y-{\nabla }_T^{ϵ}{\nabla }_X^{ϵ}Y-{\nabla }_{[X,T]}^{ϵ}Y$$

by (14) and (12), and by $[X, T]\in H\left(M\right)$

$$={\nabla }_X^{ϵ}\left\{{\nabla }_{T}X+\frac{1}{{ϵ}^2}JY\right\}$$

$$-{\nabla }_T^{ϵ}\left\{{\nabla }_{X}Y+\left[\mathsf{\Omega }(X,Y)-{ϵ}^{2}A(X,Y)\right]T\right\}$$

$$-{\nabla }_{[X,T]}Y-\left\{\mathsf{\Omega }\left([X, T],Y\right)-{ϵ}^{2}A\left([X, T],Y\right)\right\}T.$$

Note that, by the very definition of the pseudohermitian torsion $\tau$,

$$[X, T]=\tau X-{\nabla }_{T}X,X\in H\left(M\right).$$

Then, by $\nabla J=0$ and $\nabla \mathsf{\Omega }=0$,

$$R\left({\nabla }^{ϵ}\right)(X, T)Y=R^{\nabla }(X, T)Y$$

$$+\frac{1}{{ϵ}^2}\left[\mathsf{\Omega }(X, JY)-{ϵ}^{2}A(X, JY)\right]T$$

$$-\mathsf{\Omega }(\tau X,Y)T+{ϵ}^{2}A(\tau X,Y)T+{ϵ}^2({\nabla }_{T}A)(X,Y)T,$$

thus yielding (39). Finally, (40) follows from

$$R\left({\nabla }^{ϵ}\right)(X, T)T={\nabla }_X^{ϵ}{\nabla }_T^{ϵ}T-{\nabla }_T^{ϵ}{\nabla }_X^{ϵ}T-{\nabla }_{[X,T]}^{ϵ}T$$

by ${\nabla }_T^{ϵ}T=0$ and (13)

$$=-{\nabla }_T^{ϵ}\left\{\tau X+\frac{1}{{ϵ}^2}JX\right\}-\tau [X, T]-\frac{1}{{ϵ}^2}J[X, T]$$

by $\nabla J=0$ and $\tau \circ J+J\circ \tau =0$

$$=-{\tau }^{2}X+\frac{1}{{ϵ}^4}X-\left({\nabla }_T\tau \right)X.$$

□

3. First Fundamental Forms

Let $(M,T_{1,0}\left(M\right))$ be a strictly pseudoconvex CR manifold of CR dimension n. Through this section, given a positive integer $k\ge 1$ and another strictly pseudoconvex CR manifold $(A,T_{1,0}\left(A\right))$ of CR dimension $N=n+k$, we study the geometry of the second fundamental form of Cauchy–Riemann (CR) immersions $f:M\to A$.

Definition 33.

A CR immersion $f:M\to A$ is a $C^{\infty }$ immersion of $f:M\to A$, which is a CR map.

Our approach to the study of CR immersions is to establish pseudohermitian analogues to the Gauss–Weingarten formulas and to the Gauss–Ricci–Codazzi equations. Let $\theta \in {\mathcal{P}}_{+}\left(M\right)$ and $\mathsf{\Theta }\in {\mathcal{P}}_{+}\left(A\right)$ be positively oriented contact forms on M and A, respectively.

Lemma 6.

Let $f:M\to A$ be a CR immersion. There is a unique function $\mathsf{\Lambda }\in C^{\infty }\left(M\right)$ such that

$$f^{\ast }\mathsf{\Theta }=\mathsf{\Lambda }\theta .$$

(42)

Consequently,

$$f^{\ast }{G}_{\mathsf{\Theta }}=\mathsf{\Lambda }{G}_{\theta }$$

(43)

and $\mathsf{\Lambda }\left(x\right)\ge 0$ for any $x\in M$.

Proof. 

For every $x\in M$ and $X\in H{\left(M\right)}_x$,

$${\left(f^{\ast }\mathsf{\Theta }\right)}_{x}X={\mathsf{\Theta }}_{f\left(x\right)}\left(d_{x}f\right)X=0$$

because of

$$\left(d_{x}f\right)H{\left(M\right)}_x\subset H{\left(A\right)}_{f\left(x\right)}.$$

(44)

Hence,

$$H{\left(M\right)}_x\subset \mathrm{Ker}\left[{\left(f^{\ast }\mathsf{\Theta }\right)}_x\right].$$

(45)

Let $\{E_a:1\le a\le 2n\}$ be a local frame of $H\left(M\right)$, defined on an open neighborhood $U\subset M$ of x. Then,

$$\{E_j:0\le j\le 2n\},E_0=T,$$

is a local frame of $T\left(M\right)$ on U. Let us set

$$\mathsf{\Lambda }=\left(f^{\ast }\mathsf{\Theta }\right)T\in C^{\infty }\left(M\right).$$

(46)

For every $X=a^j{E}_{j,x}\in T_x\left(M\right)$ [by $\theta \left(T\right)=1$]

$${\left(f^{\ast }\mathsf{\Theta }\right)}_{x}X=a^0{\left(f^{\ast }\mathsf{\Theta }\right)}_x{T}_x=a^0\mathsf{\Lambda }\left(x\right)=\mathsf{\Lambda }\left(x\right){\theta }_x\left(X\right),$$

yielding (42). As f is a CR map, aside from (44), one has

$$\left(d_{x}f\right)\circ J_x=J_{f\left(x\right)}^A\circ \left(d_{x}f\right),x\in M,$$

(47)

where J and $J^A$ are the complex structures along the Levi distributions $H\left(M\right)$ and $H\left(A\right)$. Also, by exterior differentiation of (42),

$$d\mathsf{\Lambda }\wedge \theta +\mathsf{\Lambda }d\theta =f^{\ast }\left(d\mathsf{\Theta }\right).$$

Hence, for any $X,Y\in H\left(M\right)$,

$$\mathsf{\Lambda }{G}_{\theta }(X,Y)=\mathsf{\Lambda }\left(d\theta \right)(X, JY)$$

$$=\left(f^{\ast }\left(d\mathsf{\Theta }\right)\right)(X,JY)={\left(d\mathsf{\Theta }\right)}^f\left(f_{\ast }X,f_{\ast }JY\right)$$

by (47)

$$={\left(d\mathsf{\Theta }\right)}^f\left(f_{\ast }X,{\left(J^A\right)}^f{f}_{\ast }Y\right)={\left(G_{\mathsf{\Theta }}\right)}^f(f_{\ast }X,f_{\ast }Y),$$

proving (43). An upper index f denotes composition with f, e.g., ${\left(J^A\right)}^f=J^A\circ f$, where $J^A$ is thought of as a section $J^A:A\to T^{\ast }\left(A\right)\otimes T\left(A\right)$. Finally, for every $X\in T_x\left(M\right)$, $X\ne 0$ (as $d_{x}f$ is a monomorphism, and $G_{\mathsf{\Theta },f\left(x\right)}$ and $G_{\theta ,x}$ are positive definite),

$$0<G_{\mathsf{\Theta },f\left(x\right)}\left(\left(d_{x}f\right)X,\left(d_{x}f\right)X\right)={\left(f^{\ast }{G}_{\mathsf{\Theta }}\right)}_x(X,X)=\mathsf{\Lambda }\left(x\right)G_{\theta ,x}(X,X),$$

yielding $\mathsf{\Lambda }\left(x\right)\ge 0$.    □

Definition 34.

The function $\mathsf{\Lambda }=\mathsf{\Lambda }\left(f\right)=\mathsf{\Lambda }(f;\theta ,\mathsf{\Theta })$ given by (46) is the dilation of f relative to the choice of contact forms $(\theta ,\mathsf{\Theta })$.

Definition 35.

A CR immersion $f:M\to A$ of $(M, \theta )$ into $(A,\mathsf{\Theta })$ is said to be isopseudohermitian if $\mathsf{\Lambda }(f;\theta ,\mathsf{\Theta })\equiv 1$.

Proposition 2.

Let $f:M\to A$ be a CR immersion between the strictly pseudoconvex CR manifolds M and A. Let $\mathsf{\Theta }\in {\mathcal{P}}_{+}\left(A\right)$. If

$${\mathsf{\Theta }}_{f\left(x\right)}\circ \left(d_{x}f\right)\ne 0,x\in M,$$

then there is a contact form $\widehat{\theta }\in {\mathcal{P}}_{+}\left(M\right)$ such that f is an isopseudohermitian immersion of $(M,\widehat{\theta })$ into $(A,\mathsf{\Theta })$.

Proof. 

Let $\theta \in {\mathcal{P}}_{+}\left(M\right)$ and let $\mathsf{\Lambda }$ be the dilation of the CR immersion f relative to the pair $(\theta ,\mathsf{\Theta })$. Then, $\mathsf{\Lambda }\left(M\right)\subset (0,+\infty )$, and we may set $\widehat{\theta }=\mathsf{\Lambda }\theta$.    □

For every CR immersion $f:M\to A$, we may look at M as an immersed submanifold of the Riemannian manifold $(A,g_{\mathsf{\Theta }})$. However, in general, the first fundamental form, i.e., the pullback $f^{\ast }{g}_{\mathsf{\Theta }}$ of the ambient Webster metric $g_{\mathsf{\Theta }}$, of the given immersion $f:M\to (A,g_{\mathsf{\Theta }})$ does not coincide with the intrinsic Webster metric $g_{\theta }$, not even if f is isopseudohermitian. To circumnavigate this obstacle, we endow A with the Riemannian metric $g_{ϵ}^A$, the contraction of the Levi form $G_{\mathsf{\Theta }}$ associated with every $ϵ>0$ as in Section 2, given by

$$g_{ϵ}^A=g_{\mathsf{\Theta }}+\left(\frac{1}{{ϵ}^2}-1\right)\mathsf{\Theta }\otimes \mathsf{\Theta }$$

(48)

and derive the Gauss–Codazzi–Ricci equations of the immersion $f:M\to (A,g_{ϵ}^A)$. As a consequence of (48),

$$g_{ϵ}^A=G_{\mathsf{\Theta }}\mathrm{on}H\left(A\right)\otimes H\left(A\right),$$

(49)

$$g_{ϵ}^A(X,T_A)=0,X\in H\left(A\right),$$

(50)

$$g_{ϵ}^A(T_A,T_A)={ϵ}^{-2},$$

(51)

where $T_A\in \mathfrak{X}\left(A\right)$ is the Reeb vector field of $(A,\mathsf{\Theta })$. Let

$$g_{ϵ}\left(f\right)=f^{\ast }{g}_{ϵ}^A$$

(52)

be the induced metric, i.e., the first fundamental form of the given immersion $f:M\to (A,g_{ϵ}^A)$. Then, for any $X,Y\in H\left(M\right)$,

$$g_{ϵ}\left(f\right)(X,Y)={\left(g_{ϵ}^A\right)}^f(f_{\ast }X,f_{\ast }Y)$$

by (49), as $f_{\ast }X,f_{\ast }Y\in C^{\infty }\left(f^{-1}H\left(M\right)\right)$

$$=G_{\mathsf{\Theta }}^f(f_{\ast }X,f_{\ast }Y)=\left(f^{\ast }{G}_{\mathsf{\Theta }}\right)(X,Y)=\mathsf{\Lambda }{G}_{\theta }(X,Y)$$

by (43). Throughout, an upper index f denotes composition with f. Summing up:

$$g_{ϵ}\left(f\right)=\mathsf{\Lambda }{G}_{\theta }\mathrm{on}H\left(M\right)\otimes H\left(M\right).$$

(53)

For every $x\in M$, let us decompose $\left(d_{x}f\right)T_x\in T_{f\left(x\right)}\left(A\right)$ with respect to

$$T\left(A\right)=H\left(A\right)\oplus \mathbb{R}{T}_A,$$

(54)

which is

$$\left(d_{x}f\right)T_x=v+\mu T_{A,f\left(x\right)}$$

(55)

for some $v\in H{\left(A\right)}_{f\left(x\right)}$ and $\mu \in \mathbb{R}$. If $X\in H\left(M\right)$, then

$$g_{ϵ}\left(f\right){(X, T)}_x=\left(f^{\ast }{g}_{ϵ}^A\right){(X, T)}_x$$

$$=g_{ϵ,f\left(x\right)}^A\left(\left(d_{x}f\right)X_x,\left(d_{x}f\right)T_x\right)=G_{\mathsf{\Theta },f\left(x\right)}\left(\left(d_{x}f\right)X_x,v\right)$$

by (49) and (50), which can be applied because $\left(d_{x}f\right)X_x\in H{\left(A\right)}_{f\left(x\right)}$. We have shown that

$$g_{ϵ}\left(f\right){(X, T)}_x=G_{\mathsf{\Theta },f\left(x\right)}\left(\left(d_{x}f\right)X_x,v\right)$$

(56)

for any $X\in H\left(M\right)$, where

$$v={\mathsf{\Pi }}_{H\left(A\right),f\left(x\right)}\left(d_{x}f\right)T_x.$$

(57)

Let $E_H{\left(f\right)}_x$ be the orthogonal complement of $\left(d_{x}f\right)H{\left(M\right)}_x$ in the inner product space $\left(H{\left(A\right)}_{f\left(x\right)},G_{\mathsf{\Theta },f\left(x\right)}\right)$ so that

$$H{\left(A\right)}_{f\left(x\right)}=\left[\left(d_{x}f\right)H{\left(M\right)}_x\right]\oplus E_H{\left(f\right)}_x,x\in M.$$

(58)

Lemma 7.

$E_H\left(f\right)\to M$ is a $J_A$-invariant real rank $2k$ subbundle of $f^{-1}H\left(A\right)\to M$.

Proof. 

Let $V\in E_H\left(f\right)$. Then,

$$G_{\mathsf{\Theta }}^f(J_A^{f}V,f_{\ast }X)=-G_{\mathsf{\Theta }}^f(V,J_A^f{f}_{\ast }X)$$

by (47)

$$=-G_{\mathsf{\Theta }}^f(V,f_{\ast }JX)=0,$$

yielding $J_A^{f}V\in E_H\left(f\right)$.    □

Definition 36.

The real vector bundle $E_H\left(f\right)\to M$ is called the Levi normal bundle of the given CR immersion $f:M\to (A,\mathsf{\Theta })$. A section $\xi \in C^{\infty }\left(E_H\left(f\right)\right)$ is a Levi normal field.

The tangent vector $v\in H{\left(A\right)}_{f\left(x\right)}$, first appearing in the decomposition (55), may be further decomposed, with respect to (58), as

$$v=\left(d_{x}f\right)Y_x+v^{\perp }$$

(59)

for some $Y\in H\left(M\right)$ and some $v^{\perp }\in E_H{\left(f\right)}_x$. The Levi normal vector $v^{\perp }$ and the value of Y at x (but not Y) are uniquely determined by the decomposition (59). With (53) and (56), we started the calculation of the first fundamental form of $f:M\to (A,g_{ϵ}^A)$. Let us substitute from (59) into (56) and take into account (43). We obtain

$$g_{ϵ}\left(f\right)(X,T)=\mathsf{\Lambda }{G}_{\theta }(X,Y),X\in H\left(M\right).$$

(60)

Let

$${\tan }_{H,x}:H{\left(A\right)}_{f\left(x\right)}\to H{\left(M\right)}_x,{\mathrm{nor}}_{H,x}:H{\left(A\right)}_{f\left(x\right)}\to E_H{\left(f\right)}_x,$$

be the projections associated with the direct sum decomposition (58), so that

$$w=\left(d_{x}f\right){\tan }_{H,x}w+{\mathrm{nor}}_{H,x}w,w\in H{\left(A\right)}_{f\left(x\right)}.$$

(61)

Then, by (59),

$${\tan }_{H,x}v=Y_x,{\mathrm{nor}}_{H,x}v=v^{\perp }.$$

Moreover, by (25) and (57),

$$Y_x={\tan }_{H,x}\left\{{\mathsf{\Pi }}_{H\left(A\right),f\left(x\right)}\left(d_{x}f\right)T_x\right\}$$

$$={\tan }_{H,x}\left\{\left(d_{x}f\right)T_x-{\mathsf{\Theta }}_{f\left(x\right)}\left[\left(d_{x}f\right)T_x\right]T_{A,f\left(x\right)}\right\}$$

or

$$Y_x={\tan }_{H,x}\left\{\left(d_{x}f\right)T_x-\mathsf{\Lambda }\left(x\right)T_{A,f\left(x\right)}\right\}$$

(62)

as, by (42),

$${\mathsf{\Theta }}_{f\left(x\right)}\left[\left(d_{x}f\right)T_x\right]={\left(f^{\ast }\mathsf{\Theta }\right)}_x{T}_x=\mathsf{\Lambda }\left(x\right){\theta }_x\left(T_x\right)=\mathsf{\Lambda }\left(x\right).$$

For further use, let us set

$${\mathfrak{X}}_{\mathsf{\Theta }}\left(x\right):=\left(d_{x}f\right)T_x-\mathsf{\Lambda }\left(x\right)T_{A,f\left(x\right)},x\in M.$$

(63)

As a byproduct of the calculations leading to (62), we have ${\mathfrak{X}}_{\mathsf{\Theta },x}\in H{\left(A\right)}_{f\left(x\right)}$ for any $x\in M$; i.e., (63) defines a section ${\mathfrak{X}}_{\mathsf{\Theta }}$ in the pullback bundle $f^{-1}H\left(A\right)\to M$

$${\mathfrak{X}}_{\mathsf{\Theta }}=f_{\ast }T-\mathsf{\Lambda }{T}_A^f\in C^{\infty }\left(f^{-1}H\left(A\right)\right).$$

Next, by (60) and $Y_x={\tan }_{H,x}{\mathfrak{X}}_{\mathsf{\Theta },x}$, i.e., by (62) and (63),

$$g_{ϵ}\left(f\right){(X,T)}_x=\mathsf{\Lambda }\left(x\right)G_{\theta }{(X,Y)}_x=\left(f^{\ast }{G}_{\mathsf{\Theta }}\right){(X,Y)}_x$$

$$=G_{\mathsf{\Theta },f\left(x\right)}\left(\left(d_{x}f\right)X_x,\left(d_{x}f\right)Y_x\right)$$

$$=G_{\mathsf{\Theta },f\left(x\right)}\left(\left(d_{x}f\right)X_x,\left(d_{x}f\right){\tan }_{H,x}{\mathfrak{X}}_{\mathsf{\Theta },x}\right)$$

by (61) for $w={\mathfrak{X}}_{\mathsf{\Theta },x}$

$$=G_{\mathsf{\Theta },f\left(x\right)}\left(\left(d_{x}f\right)X_x,{\mathfrak{X}}_{\mathsf{\Theta },x}-{\mathrm{nor}}_{H,x}{\mathfrak{X}}_{\mathsf{\Theta },x}\right)$$

as $\left(d_{x}f\right)X_x$ and ${\mathrm{nor}}_{H,x}{\mathfrak{X}}_{\mathsf{\Theta },x}$ are $G_{\mathsf{\Theta },f\left(x\right)}$-orthogonal

$$=G_{\mathsf{\Theta },f\left(x\right)}\left(\left(d_{x}f\right)X_x,{\mathfrak{X}}_{\mathsf{\Theta },x}\right)=-{\left(d\mathsf{\Theta }\right)}_{f\left(x\right)}\left(J_{A,f\left(x\right)}\left(d_{x}f\right)X_x,{\mathfrak{X}}_{\mathsf{\Theta },x}\right)$$

as f is a CR map

$$=-{\left(d\mathsf{\Theta }\right)}_{f\left(x\right)}\left(\left(d_{x}f\right)J_x{X}_x,{\mathfrak{X}}_{\mathsf{\Theta },x}\right)$$

by $T_A\rfloor d\mathsf{\Theta }=0$

$$=-{\left(d\mathsf{\Theta }\right)}_{f\left(x\right)}\left(\left(d_{x}f\right)J_x{X}_x,\left(d_{x}f\right)T_x\right).$$

Next, note that, by taking the exterior differential of (42),

$$f^{\ast }\left(d\mathsf{\Theta }\right)=d\left(f^{\ast }\mathsf{\Theta }\right)=d\mathsf{\Lambda }\wedge \theta +\mathsf{\Lambda }d\theta$$

so that

$$g_{ϵ}\left(f\right){(X,T)}_x=-\left(d\mathsf{\Lambda }\wedge \theta \right){(JX,T)}_x+\mathsf{\Lambda }\left(x\right)\left(T\rfloor d\theta \right){\left(JX\right)}_x$$

by $\mathrm{Ker}\left(\theta \right)=H\left(M\right)$, $\theta \left(T\right)=1$, and $T\rfloor d\theta =0$

$$=-\left(JX\right){\left(\mathsf{\Lambda }\right)}_x.$$

Summing up:

$$g_{ϵ}\left(f\right)(X, T)=-\left(JX\right)\left(\mathsf{\Lambda }\right),X\in H\left(M\right).$$

(64)

Together with (53), this determines the first fundamental form $g_{ϵ}\left(f\right)=f^{\ast }{g}_{ϵ}^A$ on $H\left(M\right)\otimes H\left(M\right)$ and $H\left(M\right)\otimes \mathbb{R}T$. By taking into account the decomposition (6), to fully determine $g_{ϵ}\left(f\right)$, we ought to compute

$$g_{ϵ}\left(f\right){(T, T)}_x=\left(f^{\ast }{g}_{ϵ}^A\right){(T, T)}_x=g_{ϵ,f\left(x\right)}^A\left(\left(d_{x}f\right)T_x,\left(d_{x}f\right)T_x\right)$$

by substitution from (55)

$$=g_{ϵ,f\left(x\right)}^A\left(v+\mu T_{A,f\left(x\right)},v+\mu T_{A,f\left(x\right)}\right)$$

by (50) i.e., by $v\in H{\left(A\right)}_{f\left(x\right)}\perp T_{A,f\left(x\right)}$ with respect to $g_{ϵ,f\left(x\right)}^A$

$$=g_{ϵ,f\left(x\right)}^A(v,v)+{\mu }^2{g}_{ϵ}^A{(T_A,T_A)}_{f\left(x\right)}$$

by (49) i.e., $g_{ϵ}^A=G_{\mathsf{\Theta }}$ on $H\left(A\right)\otimes H\left(M\right)$ and by (51) i.e., $g_{ϵ}^A(T_A,T_A)={ϵ}^{-2}$

$$={\left(\frac{\mu }{ϵ}\right)}^2+G_{\mathsf{\Theta },f\left(x\right)}(v,v).$$

On the other hand, by going back to (55),

$$\mu ={\mathsf{\Theta }}_{f\left(x\right)}\left(d_{x}f\right)T_x={\left(f^{\ast }\mathsf{\Theta }\right)}_x{T}_x=\mathsf{\Lambda }\left(x\right)\theta {\left(T\right)}_x=\mathsf{\Lambda }\left(x\right)$$

so that (55) de facto reads

$$\left(d_{x}f\right)T_x=v+\mathsf{\Lambda }\left(x\right)T_{A,f\left(x\right)}$$

or, by (63),

$$v={\mathfrak{X}}_{\mathsf{\Theta },x}.$$

(65)

Our calculations so far lead to

$$g_{ϵ}\left(f\right){(T,T)}_x={\left[\frac{\mathsf{\Lambda }\left(x\right)}{ϵ}\right]}^2+\parallel {\mathfrak{X}}_{\mathsf{\Theta },x}{\parallel }_{\mathsf{\Theta }}^2.$$

(66)

Here we have set ${\parallel v\parallel }_{\mathsf{\Theta }}=G_{\mathsf{\Theta },f\left(x\right)}{(v,v)}^{1/2}$. Another useful expression of the norm ${\parallel v\parallel }_{\mathsf{\Theta }}$ may be obtained as follows. Note first that, as a consequence of our key observation ${\mathfrak{X}}_{\mathsf{\Theta }}\in C^{\infty }\left(f^{-1}H\left(M\right)\right)$,

$$g_{\mathsf{\Theta }}^f\left(f_{\ast }T,T_A^f\right)=g_{\mathsf{\Theta }}^f\left(f_{\ast }T-\mathsf{\Lambda }{T}_A^f,T_A^f\right)+\mathsf{\Lambda }{g}_{\mathsf{\Theta }}{(T_A,T_A)}^f$$

$$=g_{\mathsf{\Theta }}^f\left({\mathfrak{X}}_{\mathsf{\Theta }},T_A^f\right)+\mathsf{\Lambda }$$

or

$$g_{\mathsf{\Theta }}^f\left(f_{\ast }T,T_A^f\right)=\mathsf{\Lambda }.$$

(67)

Then, by (65),

$$G_{\mathsf{\Theta },f\left(x\right)}(v,v)=g_{\mathsf{\Theta },f\left(x\right)}\left({\mathfrak{X}}_{\mathsf{\Theta },x},{\mathfrak{X}}_{\mathsf{\Theta },x}\right)$$

$$=g_{\mathsf{\Theta },f\left(x\right)}\left(\left(d_{x}f\right)T_x,\left(d_{x}f\right)T_x\right)+\mathsf{\Lambda }{\left(x\right)}^2{g}_{\mathsf{\Theta },f\left(x\right)}\left(T_{A,f\left(x\right)},T_{A,f\left(x\right)}\right)$$

$$-2\mathsf{\Lambda }\left(x\right)g_{\mathsf{\Theta },f\left(x\right)}\left(\left(d_{x}f\right)T_x,T_{A,f\left(x\right)}\right)$$

or, by (67),

$$G_{\mathsf{\Theta },f\left(x\right)}(v,v)=\left(f^{\ast }{g}_{\mathsf{\Theta }}\right){(T,T)}_x-\mathsf{\Lambda }{\left(x\right)}^2.$$

(68)

Finally, by substitution from (68) into (66),

$$g_{ϵ}\left(f\right)(T,T)=\left(\frac{1}{{ϵ}^2}-1\right){\mathsf{\Lambda }}^2+\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T, T).$$

(69)

The first fundamental form of $f:M\to (A,g_{ϵ}^A)$ is fully determined. Summing up, we have established:

Proposition 3.

Let us set ${\mathfrak{X}}_{\mathsf{\Theta }}=f_{\ast }T-\mathsf{\Lambda }{T}_A^f$. Then

(i) ${\mathfrak{X}}_{\mathsf{\Theta }}\in C^{\infty }\left(f^{-1}H\left(A\right)\right)$,

(ii) For any $X\in H\left(M\right)$

$$G_{\mathsf{\Theta }}^f\left(f_{\ast }X,{\mathfrak{X}}_{\mathsf{\Theta }}\right)=-\left(JX\right)\left(\mathsf{\Lambda }\right)$$

(70)

or equivalently

$$\mathsf{\Lambda }{\tan }_H{\mathfrak{X}}_{\mathsf{\Theta }}=J{\nabla }^H\mathsf{\Lambda }.$$

(71)

In particular, if $\mathsf{\Lambda }\left(x\right)\ne 0$ for any $x\in M$, then

$${\tan }_H{\mathfrak{X}}_{\mathsf{\Theta }}=J{\nabla }^{H}log\left|\mathsf{\Lambda }\right|,$$

while if $\mathsf{\Lambda }=$ constant (e.g., f is isopseudohermitian, i.e., $\mathsf{\Lambda }\equiv 1$), then ${\mathfrak{X}}_{\mathsf{\Theta }}$ is a Levi normal vector field on M i.e., ${\mathfrak{X}}_{\mathsf{\Theta }}\in C^{\infty }\left(E_H\left(f\right)\right)$.

(iii) The norm of vector field ${\mathfrak{X}}_{\mathsf{\Theta }}$ is

$${\parallel {\mathfrak{X}}_{\mathsf{\Theta }}\parallel }_{\mathsf{\Theta }}={\left[\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T,T)-{\mathsf{\Lambda }}^2\right]}^{1/2}.$$

(iv) The first fundamental form $g_{ϵ}\left(f\right)$ of the immersion $f:M\to \left(A,g_{ϵ}^A\right)$ is given by

$$g_{ϵ}\left(f\right)=\mathsf{\Lambda }{G}_{\theta }\mathrm{on}H\left(M\right)\otimes H\left(M\right),$$

(72)

$$g_{ϵ}\left(f\right)(X,T)=-\left(JX\right)\left(\mathsf{\Lambda }\right),X\in H\left(M\right),$$

(73)

$$g_{ϵ}\left(f\right)(T,T)=\left(\frac{1}{{ϵ}^2}-1\right){\mathsf{\Lambda }}^2+\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T,T).$$

(74)

Consequently,

$$\begin{array}{c}{g}_{ϵ}\left(f\right)=\mathsf{\Lambda }{g}_{\theta }+\left\{{\left(\frac{\mathsf{\Lambda }}{ϵ}\right)}^2-\mathsf{\Lambda }(\mathsf{\Lambda }+1)+\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T,T)\right\}\theta \otimes \theta \\ +2\theta \odot g_{\theta }\left(J{\nabla }^H\mathsf{\Lambda },·\right)\end{array}$$

(75)

and, in particular,

$$\begin{array}{c}{g}_{ϵ}\left(f\right)=\mathsf{\Lambda }{g}_{ϵ}^M+2\theta \odot g_{\theta }\left(J{\nabla }^H\mathsf{\Lambda },·\right)\\ +\left\{\frac{1}{{ϵ}^2}\left({\mathsf{\Lambda }}^2-1\right)-\mathsf{\Lambda }(\mathsf{\Lambda }+1)+1+\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T,T)\right\}\theta \otimes \theta .\end{array}$$

(76)

Corollary 2.

Let $f:M\to A$ be an isopseudohermitian CR immersion of $(M,\theta )$ into $(A,\mathsf{\Theta })$. Then, the first fundamental form $g_{ϵ}\left(f\right)=f^{\ast }{g}_{ϵ}^A$ and the Webster metric $g_{\theta }$, respectively, the ϵ-contraction $g_{ϵ}^M$ of $G_{\theta }$, are related by

$$g_{ϵ}\left(f\right)=g_{\theta }+\left\{\frac{1}{{ϵ}^2}-2+\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T,T)\right\}\theta \otimes \theta ,$$

(77)

$$g_{ϵ}\left(f\right)=g_{ϵ}^M+\left\{\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T,T)-1\right\}\theta \otimes \theta .$$

(78)

Let $E\left(f\right)\to M$ and $E_{ϵ}\left(f\right)\to M$ be, respectively, the normal bundles of the immersions $f:M\to (A,g_{\mathsf{\Theta }})$ and $f:M\to (A,g_{ϵ}^A)$, so that

$$T_{f\left(x\right)}\left(A\right)=\left[\left(d_{x}f\right)T_x\left(M\right)\right]\oplus E{\left(f\right)}_x,$$

(79)

$$T_{f\left(x\right)}\left(A\right)=\left[\left(d_{x}f\right)T_x\left(M\right)\right]\oplus E_{ϵ}{\left(f\right)}_x,$$

(80)

for every $x\in M$.

Lemma 8.

The normal bundle $E_{ϵ}\left(f\right)\to M$ and the Levi normal bundle $E_H\left(f\right)\to M$ are related by

$$E_{ϵ}\left(f\right)\subset E_H\left(f\right)\oplus \mathbb{R}{T}_A^f.$$

(81)

A dimension count shows that the inclusion is strict.

Proof. 

Let

$$\xi \in E_{ϵ}\left(f\right)\subset f^{-1}T\left(A\right)=\left[f^{-1}H\left(M\right)\right]\oplus \mathbb{R}{T}_A^f$$

so that

$$\xi =W+\mu T_A^f$$

for some $W\in f^{-1}H\left(A\right)$ and some $\mu \in C^{\infty }\left(M\right)$. Then, for any $X\in H\left(M\right)$

$$G_{\mathsf{\Theta }}(W,f_{\ast }X)=g_{ϵ}^A(W,f_{\ast }X)=g_{ϵ}^A(\xi -\mu T_A^f,f_{\ast }X)$$

as $T_A^f\perp C^{\infty }\left(f^{-1}H\left(M\right)\right)\supset f_{\ast }{C}^{\infty }\left(H\left(M\right)\right)$

$$=g_{ϵ}^A(\xi ,f_{\ast }X)=0$$

because of $\xi \perp f_{\ast }\mathfrak{X}\left(M\right)\supset f_{\ast }{C}^{\infty }H\left(M\right))\ni f_{\ast }X$. This yields $W\in E_H\left(f\right)$.    □

Let

$${\tan }_x:T_{f\left(x\right)}\left(A\right)\to T_x\left(M\right),{\mathrm{nor}}_x:T_{f\left(x\right)}\left(A\right)\to E{\left(f\right)}_x,$$

$${\tan }_x^{ϵ}:T_{f\left(x\right)}\left(A\right)\to T_x\left(M\right),{\mathrm{nor}}_x^{ϵ}:T_{f\left(x\right)}\left(A\right)\to E_{ϵ}{\left(f\right)}_x,$$

be the projections associated with the decompositions (79) and (80). Next, let us set

$$T_A^{\perp }=\mathrm{nor}\left(T_A\right)\in C^{\infty }\left(E\left(f\right)\right)$$

(82)

so that

$$T_A^f=f_{\ast }\tan \left(T_A\right)+T_A^{\perp }.$$

(83)

Lemma 9.

Let $f:M\to A$ be a CR immersion of strictly pseudoconvex CR manifolds, and let $\theta \in {\mathcal{P}}_{+}\left(M\right)$ and $\mathsf{\Theta }\in {\mathcal{P}}_{+}\left(A\right)$ be positively oriented contact forms on M and A. Then

$${\mathsf{\Lambda }}^2\tan \left(T_A\right)=\left\{1-{\mathsf{\Theta }}^f\left(T_A^{\perp }\right)\right\}\left\{\mathsf{\Lambda }T-J{\nabla }^H\mathsf{\Lambda }\right\}.$$

(84)

In particular, if f is isopseudohermitian, then

$$\tan \left(T_A\right)=\left\{1-{\mathsf{\Theta }}^f\left(T_A^{\perp }\right)\right\}T.$$

(85)

Proof. 

As $\mathsf{\Theta }\left(T_A\right)=1$

$$1=\mathsf{\Theta }{\left(T_A\right)}^f={\mathsf{\Theta }}^f\left(T_A^f\right)$$

by (83)

$$={\mathsf{\Theta }}^f\left(f_{\ast }\tan \left(T_A\right)\right)+{\mathsf{\Theta }}^f\left(T_A^{\perp }\right)$$

$$=\left(f^{\ast }\mathsf{\Theta }\right)\tan \left(T_A\right)+{\mathsf{\Theta }}^f\left(T_A^{\perp }\right)=\mathsf{\Lambda }\theta \left[\tan \left(T_A\right)\right]+{\mathsf{\Theta }}^f\left(T_A^{\perp }\right)$$

so that

$$\mathsf{\Lambda }\theta \left[\tan \left(T_A\right)\right]=1-{\mathsf{\Theta }}^f\left(T_A^{\perp }\right).$$

(86)

Moreover, for any $X\in H\left(M\right)$,

$$\mathsf{\Lambda }{g}_{\theta }(X,\tan \left(T_A\right))=\mathsf{\Lambda }{G}_{\theta }(X,{\mathsf{\Pi }}_H\tan \left(T_A\right))$$

$$=\left(f^{\ast }{G}_{\mathsf{\Theta }}\right)(X,{\mathsf{\Pi }}_H\tan \left(T_A\right))=G_{\mathsf{\Theta }}^f\left(f_{\ast }X,f_{\ast }{\mathsf{\Pi }}_H\tan \left(T_A\right)\right)$$

$$=g_{\mathsf{\Theta }}^f\left(f_{\ast }X,f_{\ast }{\mathsf{\Pi }}_H\tan \left(T_A\right)\right)$$

$$=g_{\mathsf{\Theta }}^f\left(f_{\ast }X,f_{\ast }\left\{\tan \left(T_A\right)-\theta \left(\tan \left(T_A\right)\right)T\right\}\right)$$

$$=g_{\mathsf{\Theta }}^f\left(f_{\ast }X,f_{\ast }\tan \left(T_A\right)\right)-\theta \left(\tan \left(T_A\right)\right)g_{\mathsf{\Theta }}^f\left(f_{\ast }X,f_{\ast }T\right).$$

As $f_{\ast }X$ is tangential, one has $g_{\mathsf{\Theta }}^f(f_{\ast }X,T_A^{\perp })=0$, hence, by (83),

$$g_{\mathsf{\Theta }}^f\left(f_{\ast }X,f_{\ast }\tan \left(T_A\right)\right)=g_{\mathsf{\Theta }}^f\left(f_{\ast }X,T_A^f\right)=0$$

because $f_{\ast }X\in C^{\infty }\left(f^{-1}H\left(A\right)\right)$ and $H\left(A\right)\perp T_A$ with respect to $g_{\mathsf{\Theta }}$. We may conclude that

$$\mathsf{\Lambda }{g}_{\theta }(X,\tan \left(T_A\right))=-\theta \left(\tan \left(T_A\right)\right)\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left(X,T\right).$$

(87)

On the other hand,

$$\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left(X,T\right)=g_{\mathsf{\Theta }}^f\left(f_{\ast }X,f_{\ast }T\right)$$

$$=g_{\mathsf{\Theta }}^f\left(f_{\ast }X,f_{\ast }T-\mathsf{\Lambda }{T}_A^f\right)+\mathsf{\Lambda }{g}_{\mathsf{\Theta }}^f\left(f_{\ast }X,T_A^f\right)$$

$$=g_{\mathsf{\Theta }}^f\left(f_{\ast }X,{\mathfrak{X}}_{\mathsf{\Theta }}\right)+\mathsf{\Lambda }{\mathsf{\Theta }}^f\left(f_{\ast }X\right)=G_{\mathsf{\Theta }}^f\left(f_{\ast }X,{\mathfrak{X}}_{\mathsf{\Theta }}\right)+\mathsf{\Lambda }\left(f^{\ast }\mathsf{\Theta }\right)X$$

$$=G_{\mathsf{\Theta }}^f\left(f_{\ast }X,f_{\ast }{\tan }_H{\mathfrak{X}}_{\mathsf{\Theta }}\right)+{\mathsf{\Lambda }}^2\theta \left(X\right)$$

as $X\in H\left(M\right)=\mathrm{Ker}\left(\theta \right)$

$$=\left(f^{\ast }{G}_{\mathsf{\Theta }}\right)\left(X,{\tan }_H{\mathfrak{X}}_{\mathsf{\Theta }}\right)=\mathsf{\Lambda }{G}_{\theta }\left(X,{\tan }_H{\mathfrak{X}}_{\mathsf{\Theta }}\right)=-\left(JX\right)\left(\mathsf{\Lambda }\right)$$

by (70). Then, by (87),

$$\mathsf{\Lambda }{\mathsf{\Pi }}_H\tan T_A=-\theta \left(\tan T_A\right)J{\nabla }^H\mathsf{\Lambda }.$$

(88)

Finally,

$$\mathsf{\Lambda }\tan \left(T_A\right)=\mathsf{\Lambda }\left\{{\mathsf{\Pi }}_H\tan \left(T_A\right)+\theta \left(\tan T_A\right)T\right\}$$

by (88)

$$=\theta \left(\tan \left(T_A\right)\right)\left\{-J{\nabla }^H\mathsf{\Lambda }+\mathsf{\Lambda }T\right\}$$

that is, by (86)

$${\mathsf{\Lambda }}^2\tan \left(T_A\right)=\left\{1-{\mathsf{\Theta }}^f\left(T_A^{\perp }\right)\right\}\left\{\mathsf{\Lambda }T-J{\nabla }^H\mathsf{\Lambda }\right\},$$

and Lemma 9 is proved.    □

4. Pseudohermitian Immersions

Definition 37.

Let $(M,T_{1,0}\left(M\right))$ and $(A,T_{1,0}\left(A\right))$ be strictly pseudoconvex CR manifolds of CR dimensions n and $N=n+k$, $k\ge 1$. Let $\theta \in {\mathcal{P}}_{+}\left(M\right)$ and $\mathsf{\Theta }\in {\mathcal{P}}_{+}\left(A\right)$. A CR immersion $f:M\to A$ is said to be a pseudohermitian immersion of $\left(M\theta \right)$ into $(A,\mathsf{\Theta })$ if

(i)

f is isopseudohermitian, i.e., $f^{\ast }\mathsf{\Theta }=\theta$,

(ii)

$T_A^{\perp }=0$.

Proposition 4.

Let $f:M\to A$ be an isopseudohermitian immersion. The following statements are equivalent:

(i)

f is a pseudohermitian immersion.

(ii)

$f^{\ast }{g}_{\mathsf{\Theta }}=g_{\theta }$.

(iii)

${\mathfrak{X}}_{\mathsf{\Theta }}=0$.

(iv)

$\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T, T)=1$.

Proof. 

(i) ⟹ (ii). Let f be a pseudohermitian immersion. Then, for any $X,Y\in H\left(M\right)$,

$$\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(X,Y)=g_{\mathsf{\Theta }}^f\left(f_{\ast }X,f_{\ast }Y\right)=G_{\theta }^f\left(f_{\ast }X,f_{\ast }Y\right)$$

$$=\left(f^{\ast }{G}_{\mathsf{\Theta }}\right)(X,Y)=G_{\theta }(X,Y)=g_{\theta }(X,Y).$$

Hence, $f^{\ast }{g}_{\mathsf{\Theta }}=g_{\theta }$ on $H\left(M\right)\otimes H\left(M\right)$. Next, $T_A^{\perp }=0$ together with Lemma 9 yields

$$\tan \left(T_A\right)=T.$$

(89)

Then,

$$\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(X, T)=g_{\mathsf{\Theta }}^f\left(f_{\ast }X,f_{\ast }T\right)$$

by (89)

$$=g_{\mathsf{\Theta }}^f\left(f_{\ast }X,f_{\ast }\tan \left(T_A\right)\right)$$

by $T_A^{\perp }=0$

$$=g_{\mathsf{\Theta }}^f\left(f_{\ast }X,T_A^f\right)=0=g_{\theta }(X,T).$$

Hence, $f^{\ast }{g}_{\mathsf{\Theta }}=g_{\theta }$ on $H\left(M\right)\otimes \mathbb{R}T$. It remains necessary that we check (ii) on $\mathbb{R}T\otimes \mathbb{R}T$. Indeed,

$$\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T, T)=g_{\mathsf{\Theta }}^f\left(f_{\ast }T,f_{\ast }T\right)$$

by (89)

$$=g_{\mathsf{\Theta }}^f\left(f_{\ast }\tan \left(T_A\right),f_{\ast }\tan \left(T_A\right)\right)$$

by $T_A^{\perp }=0$

$$=g_{\mathsf{\Theta }}^f\left(T_A^f,T_A^f\right)=g_{\mathsf{\Theta }}(T_A,T_A{)}^f=1=g_{\theta }(T, T)$$

by the very definition of the Webster metrics. Q.E.D.

(ii) ⟹ (iii). Let $f:M\to A$ be a CR immersion such that (ii) holds, i.e., $f^{\ast }{g}_{\mathsf{\Theta }}=g_{\theta }$. As f is a CR map, the assumption (ii) implies $f^{\ast }{G}_{\mathsf{\Theta }}=G_{\theta }$ and then, by (43), $\mathsf{\Lambda }\equiv 1$. Moreover,

$${\parallel {\mathfrak{X}}_{\mathsf{\Theta }}\parallel }_{\mathsf{\Theta }}^2=g_{\mathsf{\Theta }}^f\left({\mathfrak{X}}_{\mathsf{\Theta }},{\mathfrak{X}}_{\mathsf{\Theta }}\right)=g_{\mathsf{\Theta }}^f\left(f_{\ast }T-T_A^f,f_{\ast }T-T_A^f\right)$$

$$=\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T, T)+g_{\mathsf{\Theta }}{\left(T_A,T_A\right)}^f-2g_{\mathsf{\Theta }}^f\left(f_{\ast }T,T_A^f\right)$$

by (67)

$$=g_{\theta }(T,T)-1=0$$

so that ${\mathfrak{X}}_{\mathsf{\Theta }}=0$. Q.E.D.

(iii) ⟹ (iv). Let $f:M\to A$ be an isopseudohermitian immersion such that ${\mathfrak{X}}_{\mathsf{\Theta }}=0$. Then,

$${\parallel {\mathfrak{X}}_{\mathsf{\Theta }}\parallel }_{\mathsf{\Theta }}^2=\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T,T)-1$$

(90)

yields (iv). Q.E.D.

(iv) ⟹ (i). Let $f:M\to A$ be an isopseudohermitian immersion such that

$$\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T,T)=1.$$

Then, by (90), ${\mathfrak{X}}_{\mathsf{\Theta }}=0$ i.e.,

$$T_A^f=f_{\ast }T\in f_{\ast }\mathfrak{X}\left(M\right)⟹T_A^{\perp }=0.$$

□

5. Gauss and Weingarten Formulas

Let $f:M\to A$ be a CR immersion of strictly pseudoconvex CR manifolds, and let $\theta \in {\mathcal{P}}_{+}\left(M\right)$ and $\mathsf{\Theta }\in {\mathcal{P}}_{+}\left(A\right)$. We adopt the following notations for the various linear connections we shall work with:

$$D^{ϵ}\mathrm{Levi}–\mathrm{Civita}\mathrm{connection}\mathrm{of}\left(A,g_{ϵ}^A\right),$$

$$D\mathrm{Tanaka}–\mathrm{Webster}\mathrm{connection}\mathrm{of}(A,\mathsf{\Theta }),$$

$${\nabla }^{f,ϵ}\mathrm{Levi}–\mathrm{Civita}\mathrm{connection}\mathrm{of}\left(M,g_{ϵ}\left(f\right)\right),$$

$${\nabla }^{ϵ}\mathrm{Levi}–\mathrm{Civita}\mathrm{connection}\mathrm{of}\left(M,g_{ϵ}^M\right),$$

$$\nabla \mathrm{Tanaka}–\mathrm{Webster}\mathrm{connection}\mathrm{of}(M,\theta ).$$

The Gauss and Weingarten formulas for the isometric immersion $f:\left(M,g_{ϵ}\left(f\right)\right)\to \left(A,g_{ϵ}^A\right)$ are:

$${\left(D^{ϵ}\right)}_X^f{f}_{\ast }Y=f_{\ast }{\nabla }_X^{f,ϵ}Y+B_{ϵ}\left(f\right)(X,Y),$$

(91)

$${\left(D^{ϵ}\right)}_X^f\xi =-f_{\ast }{a}_{\xi }^{ϵ}X+D_X^{{\perp }_{ϵ}}\xi ,$$

(92)

for any $X,Y\in \mathfrak{X}\left(M\right)$ and any $\xi \in C^{\infty }\left(E_{ϵ}\left(f\right)\right)$. Here, $B_{ϵ}\left(f\right)$, $a_{\xi }^{ϵ}$ and $D^{{\perp }_{ϵ}}$ are, respectively, the second fundamental form, the Weingarten operator (associated with the normal vector field $\xi$), and the normal connection, a connection in the vector bundle $E_{ϵ}\left(f\right)\to M$, of the given isometric immersion. The symbol ${\left(D^{ϵ}\right)}^f$ in (91) and (92) denotes the connection induced by the Levi–Civita connection of $(A,g_{ϵ}^A)$ in the pullback bundle $f^{-1}T\left(A\right)\to M$; i.e., ${\left(D^{ϵ}\right)}^f$ is the pullback connection $f^{-1}{D}_{ϵ}$. One has

$${\nabla }_X^{f,ϵ}Y={\tan }_{ϵ}\left\{{\left(D^{ϵ}\right)}_X^f{f}_{\ast }Y\right\},B_{ϵ}\left(f\right)={\mathrm{nor}}_{ϵ}\left\{{\left(D^{ϵ}\right)}_X^f{f}_{\ast }Y\right\},$$

$$a_{\xi }^{ϵ}X=-{\tan }_{ϵ}\left\{{\left(D^{ϵ}\right)}_X^f\xi \right\},D_X^{{\perp }_{ϵ}}\xi =\mathrm{nor}\left\{{\left(D^{ϵ}\right)}_X^f\xi \right\}.$$

The second fundamental form $B_{ϵ}\left(f\right)$ and Weingarten operator $a_{\xi }^{ϵ}$ are related by

$${\left(g_{ϵ}^A\right)}^f\left(B_{ϵ}\left(f\right)(X,Y),\xi \right)=g_{ϵ}\left(f\right)\left(a_{\xi }^{ϵ}X,Y\right).$$

(93)

The second fundamental form $B_{ϵ}\left(f\right)$ is symmetric and, merely as a consequence of (93), the Weingarten operator $a_{\xi }^{ϵ}$ is self-adjoint with respect to $g_{ϵ}\left(f\right)$.

6. Gauss–Ricci–Codazzi Equations

Let $\mathbf{E}\to M$ be a vector bundle and $\mathbf{D}\in \mathcal{C}\left(\mathbf{E}\right)$ a connection. The curvature form $R^{\mathbf{D}}=R\left(\mathbf{D}\right)$ is

$$R^{\mathbf{D}}(X,Y)=\left[{\mathbf{D}}_X,{\mathbf{D}}_Y\right]-{\mathbf{D}}_{[X,Y]},X,Y\in \mathfrak{X}\left(M\right).$$

The curvature forms of the connections in the Gauss and Weingarten formulas are

$$\begin{array}{|ccc|}\hline \mathbf{D}& \mathbf{E}& R\left(\mathbf{D}\right)\\ {\left(D^{ϵ}\right)}^f& f^{-1}T\left(A\right)\to M& R\left({\left(D^{ϵ}\right)}^f\right)\\ {\nabla }^{f,ϵ}& T\left(M\right)\to M& R\left({\nabla }^{f,ϵ}\right)\\ D^{{\perp }_{ϵ}}& E_{ϵ}\left(f\right)\to M& R\left(D^{{\perp }_{ϵ}}\right)\\ \hline\end{array}$$

The Gauss–Codazzi equation for the isometric immersion $f:(M,g_{ϵ}\left(f\right))\to (A,g_{ϵ}^A)$ is (see, e.g., [20]):

$$\begin{array}{c}R\left({\left(D^{ϵ}\right)}^f\right)(X,Y)f_{\ast }Z=f_{\ast }R\left({\nabla }^{f,ϵ}\right)(X,Y)Z\\ -f_{\ast }{a}_{B_{ϵ}\left(f\right)(Y,Z)}^{ϵ}X+f_{\ast }{a}_{B_{ϵ}\left(f\right)(X,Z)}^{ϵ}Y+\left(D_X^{ϵ}{B}_{ϵ}\left(f\right)\right)(Y,Z)-(D_Y^{ϵ}{B}_{ϵ}\left(f\right))(X,Z)\end{array}$$

(94)

for any $X,Y,Z\in \mathfrak{X}\left(M\right)$. Here, $D_X^{ϵ}{B}_{ϵ}\left(f\right)$ is the Van der Waerden–Bortolotti covariant derivative (of the second fundamental form), i.e.,

$$\left(D_X^{ϵ}{B}_{ϵ}\left(f\right)\right)(Y,Z)=D_X^{{\perp }_{ϵ}}{B}_{ϵ}\left(f\right)(Y,Z)-B_{ϵ}\left(f\right)\left({\nabla }_X^{f,ϵ}Y,Z\right)-B_{ϵ}\left(f\right)\left(Y,{\nabla }_X^{f,ϵ}Z\right).$$

The Codazzi equation is obtained by identifying the $E_{ϵ}\left(f\right)$ components of the Gauss–Codazzi Equation (94)

$$\begin{array}{c}{\mathrm{nor}}_{ϵ}\left\{R\left({\left(D^{ϵ}\right)}^f\right)(X,Y)f_{\ast }Z\right\}=\\ \left(D_X^{ϵ}{B}_{ϵ}\left(f\right)\right)(Y,Z)-(D_Y^{ϵ}{B}_{ϵ}\left(f\right))(X,Z).\end{array}$$

(95)

Let us take the inner product of (94) with $W\in \mathfrak{X}\left(M\right)$ in order to identity the tangential components of (94)

$${\left(g_{ϵ}^A\right)}^f\left(R\left({\left(D^{ϵ}\right)}^f\right)(X,Y)f_{\ast }Z,f_{\ast }W\right)=g_{ϵ}\left(f\right)\left(R\left({\nabla }^{f,ϵ}\right)(X,Y)Z,W\right)$$

$$-g_{ϵ}\left(f\right)\left(a_{B_{ϵ}\left(f\right)(Y,Z)}^{ϵ}X,W\right)+g_{ϵ}\left(f\right)\left(a_{B_{ϵ}\left(f\right)(X,Z)}^{ϵ}Y,W\right)$$

and let us substitute from (93) so as to obtain (the Gauss equation of the given isometric immersion)

$$\begin{array}{c}{\left(g_{ϵ}^A\right)}^f\left(R\left({\left(D^{ϵ}\right)}^f\right)(X,Y)f_{\ast }Z,f_{\ast }W\right)\\ =g_{ϵ}\left(f\right)\left(R\left({\nabla }^{f,ϵ}\right)(X,Y)Z,W\right)\\ -{\left(g_{ϵ}^A\right)}^f\left(B_{ϵ}\left(f\right)(X,W),B_{ϵ}\left(f\right)(Y,Z)\right)+{\left(g_{ϵ}^A\right)}^f\left(B_{ϵ}\left(f\right)(Y,W),B_{ϵ}\left(f\right)(X,Z)\right)\end{array}$$

(96)

for any $X,Y,Z,W\in \mathfrak{X}\left(M\right)$. For any $X,Y\in \mathfrak{X}\left(M\right)$ and any $\xi \in C^{\infty }\left(E_{ϵ}\left(f\right)\right)$ as a consequence of the Gauss and Weingarten formulas (91) and (92),

$$R\left({\left(D^{ϵ}\right)}^f\right)(X,Y)\xi =R\left(D^{{\perp }_{ϵ}}\right)(X,Y)\xi$$

$$+{\nabla }_Y^{f,ϵ}{a}_{\xi }^{ϵ}X-{\nabla }_X^{f,ϵ}{a}_{\xi }^{ϵ}Y+a_{\xi }^{ϵ}[X,Y]-a_{D_X^{{\perp }_{ϵ}}\xi }^{ϵ}Y+a_{D_Y^{{\perp }_{ϵ}}\xi }^{ϵ}X$$

$$+B_{ϵ}\left(f\right)\left(a_{\xi }^{ϵ}X,Y\right)-B_{ϵ}\left(f\right)\left(X,a_{\xi }^{ϵ}Y\right)$$

and, taking the inner product with $\eta \in C^{\infty }\left(E_{ϵ}\left(f\right)\right)$, gives

$$\begin{array}{c}{\left(g_{ϵ}^A\right)}^f\left(R\left({\left(D^{ϵ}\right)}^f\right)(X,Y)\xi ,\eta \right)={\left(g_{ϵ}^A\right)}^f\left(R\left(D^{{\perp }_{ϵ}}\right)(X,Y)\xi ,\eta \right)\\ +{\left(g_{ϵ}^A\right)}^f\left(B_{ϵ}\left(f\right)\left(a_{\xi }^{ϵ}X,Y\right),\eta \right)-{\left(g_{ϵ}^A\right)}^f\left(B_{ϵ}\left(f\right)\left(X,a_{\xi }^{ϵ}Y\right),\eta \right)\end{array}$$

(97)

or, by applying (93) to modify the last two terms in (97),

$$\begin{array}{c}{\left(g_{ϵ}^A\right)}^f\left(R\left({\left(D^{ϵ}\right)}^f\right)(X,Y)\xi ,\eta \right)={\left(g_{ϵ}^A\right)}^f\left(R\left(D^{{\perp }_{ϵ}}\right)(X,Y)\xi ,\eta \right)\\ -g_{ϵ}\left(f\right)\left(\left[a_{\xi }^{ϵ},a_{\eta }^{ϵ}\right]X,Y\right)\end{array}$$

(98)

(the Ricci equation for the given isometric immersion).

7. The Projections ${\tan }_{\mathbf{ϵ}}$ and ${\mathrm{nor}}_{\mathbf{ϵ}}$

Our main purpose in the present section is to compute the projection ${\tan }_{ϵ}:f^{-1}T\left(A\right)\to T\left(M\right)$ in terms of pseudohermitian invariants. One has (at every point of M)

$$E_{ϵ}\left(f\right)=\left\{\xi \in f^{-1}T\left(A\right):{\left(g_{ϵ}^A\right)}^f(\xi ,f_{\ast }V)=0,\forall V\in T\left(M\right)\right\}.$$

Also, for every $V\in T\left(M\right)$ by (48),

$${\left(g_{ϵ}^A\right)}^f(\xi ,f_{\ast }V)=g_{\mathsf{\Theta }}^f(\xi ,f_{\ast }V)+\mathsf{\Lambda }\left(\frac{1}{{ϵ}^2}-1\right){\mathsf{\Theta }}^f\left(\xi \right)\theta \left(V\right).$$

Therefore, if we set (again pointwise)

$$S_{ϵ}\left(f\right)=\left\{f_{\ast }V+\mathsf{\Lambda }\left(\frac{1}{{ϵ}^2}-1\right)\theta \left(V\right)T_A^f:V\in T\left(M\right)\right\},$$

then

$$E_{ϵ}\left(f\right)=S_{ϵ}{\left(f\right)}^{\perp }.$$

(99)

As to the notation adopted in (99), if $S_x\subset T_{f\left(x\right)}\left(A\right)$ is a linear subspace, then $S_x^{\perp }$ denotes the orthogonal complement of $S_x$ in $T_{f\left(x\right)}\left(A\right)$ with respect to the inner product $g_{\mathsf{\Theta },f\left(x\right)}$. We shall need the linear operator $L_{ϵ}:\mathfrak{X}\left(M\right)\to C^{\infty }\left(f^{-1}T\left(A\right)\right)$ given by

$$L_{ϵ}V\equiv f_{\ast }V+\mathsf{\Lambda }\left(\frac{1}{{ϵ}^2}-1\right)\theta \left(V\right)T_A^f,V\in \mathfrak{X}\left(M\right).$$

(100)

The relation (100) also defines a vector bundle morphism $L_{ϵ}:T\left(M\right)\to f^{-1}T\left(A\right)$, denoted by the same symbol. Then,

$$E_{ϵ}\left(f\right)=\left\{\xi \in f^{-1}T\left(A\right):g_{\mathsf{\Theta }}^f\left(L_{ϵ}V,\xi \right)=0,\forall V\in T\left(M\right)\right\}.$$

For arbitrary $W\in f^{-1}T\left(A\right)$, we take the inner product of

$$W=f_{\ast }{\tan }_{ϵ}W+{\mathrm{nor}}_{ϵ}W$$

with $f_{\ast }V$, $V\in T\left(M\right)$, with respect to the inner product ${\left(g_{ϵ}^A\right)}^f$, so as to obtain

$${\left(g_{ϵ}^A\right)}^f(W,f_{\ast }V)=g_{ϵ}\left(f\right)({\tan }_{ϵ}W,V).$$

(101)

Lemma 10.

The function

$$\lambda \equiv \left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T,T)\in C^{\infty }\left(M\right)$$

(102)

is strictly positive; i.e., $\lambda \left(x\right)>0$ for any $x\in M$.

Proof. 

One has

$$\lambda \left(x\right)=g_{\mathsf{\Theta },f\left(x\right)}\left(\left(d_{x}f\right)T_x,\left(d_{x}f\right)T_x\right)\ge 0.$$

If there is $x_0\in M$ such that $\lambda \left(x_0\right)=0$, then $\left(d_{x_0}f\right)T_{x_0}=0$, yielding $T_{x_0}=0$ (as f is an immersion), a contradiction.    □

At this point, we employ the relations (see (48) and (75))

$$g_{ϵ}^A=g_{\mathsf{\Theta }}+\left(\frac{1}{{ϵ}^2}-1\right)\mathsf{\Theta }\otimes \mathsf{\Theta },$$

$$g_{ϵ}\left(f\right)=\mathsf{\Lambda }{g}_{\theta }+\left\{{\left(\frac{\mathsf{\Lambda }}{ϵ}\right)}^2-\mathsf{\Lambda }(\mathsf{\Lambda }+1)+\lambda \right\}\theta \otimes \theta +2\theta \odot g_{\theta }\left(J{\nabla }^H\mathsf{\Lambda },·\right),$$

and modify (101) accordingly. We obtain

$$\begin{gathered} g_{\mathsf{\Theta }}^f\left(W,f_{\ast }V\right)+\mathsf{\Lambda }\left(\frac{1}{{ϵ}^2}-1\right){\mathsf{\Theta }}^f\left(W\right)\theta \left(V\right) \\ =\mathsf{\Lambda }{g}_{\theta }\left(V,{\tan }_{ϵ}W\right)+\left\{{\left(\frac{\mathsf{\Lambda }}{ϵ}\right)}^2-\mathsf{\Lambda }(\mathsf{\Lambda }+1)+\lambda \right\}\theta \left(V\right)\theta \left({\tan }_{ϵ}W\right)+ \\ +\theta \left(V\right)g_{\theta }\left(J{\nabla }^H\mathsf{\Lambda },{\tan }_{ϵ}W\right)+\theta \left({\tan }_{ϵ}W\right)g_{\theta }\left(J{\nabla }^H\mathsf{\Lambda },V\right). \end{gathered}$$

(103)

We ought to examine a few consequences of (103). First, let us use (103) for $V=X\in H\left(M\right)$, i.e., as $\theta \left(X\right)=0$,

$$g_{\mathsf{\Theta }}^f\left(W,f_{\ast }X\right)=\mathsf{\Lambda }{g}_{\theta }\left(X,{\tan }_{ϵ}W\right)+\theta \left({\tan }_{ϵ}W\right)g_{\theta }\left(J{\nabla }^H\mathsf{\Lambda },X\right).$$

(104)

Let $\{X_a:1\le a\le 2n\}$ be a local $G_{\theta }$-orthonormal [i.e., $G_{\theta }(X_a,X_b)={\delta }_{ab}$, $1\le a,b\le 2n$] frame of $H\left(M\right)$, defined on the open set $U\subset M$. Then,

$$\mathsf{\Lambda }{\mathsf{\Pi }}_H{\tan }_{ϵ}W=\mathsf{\Lambda }\sum _{a=1}^{2n}{g}_{\theta }\left(X_a,{\tan }_{ϵ}W\right)X_a$$

by (104)

$$=\sum _{a=1}^{2n}\left\{g_{\mathsf{\Theta }}^f\left(W,f_{\ast }{X}_a\right)-g_{\theta }\left(J{\nabla }^H\mathsf{\Lambda },X_a\right)\right\}{X}_a$$

or

$$\mathsf{\Lambda }{\mathsf{\Pi }}_H{\tan }_{ϵ}W=\sum _{a=1}^{2n}{g}_{\mathsf{\Theta }}^f\left(W,f_{\ast }{X}_a\right)X_a-J{\nabla }^H\mathsf{\Lambda }$$

(105)

everywhere in U. Second, let us use (103) for $V=T$; i.e., as $\theta \left(T\right)=1$ and $g_{\theta }(T,Y)=\theta \left(Y\right)$ for any $Y\in \mathfrak{X}\left(M\right)$,

$$\begin{array}{c}{g}_{\mathsf{\Theta }}^f\left(W,f_{\ast }T\right)+\mathsf{\Lambda }\left(\frac{1}{{ϵ}^2}-1\right){\mathsf{\Theta }}^f\left(W\right)\\ \left\{{\mathsf{\Lambda }}^2\left(\frac{1}{{ϵ}^2}-1\right)+\lambda \right\}\theta \left({\tan }_{ϵ}W\right)+g_{\theta }\left(J{\nabla }^H\mathsf{\Lambda },{\tan }_{ϵ}W\right).\end{array}$$

(106)

We shall conduct an asymptotic analysis of our equations as $ϵ\to 0^{+}$, so we consider $0<ϵ<1$ to start with. Consequently, by Lemma 10,

$$\mathsf{\Lambda }{\left(x\right)}^2\left(\frac{1}{{ϵ}^2}-1\right)+\lambda \left(x\right)>0,x\in M.$$

For simplicity, we set

$$u_{ϵ}=\theta \left({\tan }_{ϵ}W\right)\in C^{\infty }\left(M\right),X_{ϵ}={\mathsf{\Pi }}_{H\left(M\right)}{\tan }_{ϵ}W\in C^{\infty }\left(H\left(M\right)\right),$$

so that ${\tan }_{ϵ}W=X_{ϵ}+u_{ϵ}T$ and Equations (105) and (106) read:

$$\mathsf{\Lambda }{X}_{ϵ}=\sum _{a=1}^{2n}{g}_{\mathsf{\Theta }}^f\left(W,f_{\ast }{X}_a\right)X_a-J{\nabla }^H\mathsf{\Lambda },$$

(107)

$$\begin{array}{c}{g}_{\mathsf{\Theta }}^f\left(W,f_{\ast }T\right)+\mathsf{\Lambda }\left(\frac{1}{{ϵ}^2}-1\right){\mathsf{\Theta }}^f\left(W\right)\\ =\left\{{\mathsf{\Lambda }}^2\left(\frac{1}{{ϵ}^2}-1\right)+\lambda \right\}{u}_{ϵ}+g_{\theta }\left(J{\nabla }^H\mathsf{\Lambda },X_{ϵ}\right).\end{array}$$

(108)

Let us multiply (108) by $\mathsf{\Lambda }$ and substitute $\mathsf{\Lambda }{X}_{ϵ}$ from (107) into the resulting equation. We obtain

$$\begin{array}{c}{g}_{\mathsf{\Theta }}^f\left(W,\mathsf{\Lambda }{f}_{\ast }T\right)+{\mathsf{\Lambda }}^2\left(\frac{1}{{ϵ}^2}-1\right){\mathsf{\Theta }}^f\left(W\right)\\ =\mathsf{\Lambda }\left\{{\mathsf{\Lambda }}^2\left(\frac{1}{{ϵ}^2}-1\right)+\lambda \right\}{u}_{ϵ}+g_{\mathsf{\Theta }}^f\left(W,J{\nabla }^H\mathsf{\Lambda }\right)-{\parallel {\nabla }^H\mathsf{\Lambda }\parallel }_{\theta }^2.\end{array}$$

(109)

Let $Z\left(\mathsf{\Lambda }\right)=\{x\in M:\mathsf{\Lambda }(x)=0\}$ be the zero set of $\mathsf{\Lambda }$. Note that (107) and (109) determine $X_{ϵ}$ and $u_{ϵ}$ on the open set $M\backslash Z\left(\mathsf{\Lambda }\right)$.

From now on, we confine our calculations to isopseudohermitian (i.e., $\mathsf{\Lambda }\equiv 1$) CR immersions $f:M\to A$. Then, (107) and (109) read

$$X_{ϵ}=\sum _{a=1}^{2n}{g}_{\mathsf{\Theta }}^f\left(W,f_{\ast }{X}_a\right)X_a,$$

(110)

$$g_{\mathsf{\Theta }}^f\left(W,f_{\ast }T\right)+\left(\frac{1}{{ϵ}^2}-1\right){\mathsf{\Theta }}^f\left(W\right)=\left(\frac{1}{{ϵ}^2}-1+\lambda \right)u_{ϵ}.$$

(111)

Summing up, we have proved:

Lemma 11.

For every $W\in f^{-1}T\left(A\right)$

$$\begin{array}{c}{\tan }_{ϵ}W=\sum _{a=1}^{2n}{g}_{\mathsf{\Theta }}^f\left(W,f_{\ast }{X}_a\right)X_a\\ +\frac{1}{1+{ϵ}^2(\lambda -1)}\left\{{\mathsf{\Theta }}^f\left(W\right)+{ϵ}^2{g}_{\mathsf{\Theta }}^f(W,{\mathfrak{X}}_{\mathsf{\Theta }})\right\}T\end{array}$$

(112)

everywhere in U, where ${\mathfrak{X}}_{\mathsf{\Theta }}=f_{\ast }T-T_A^f$.

Finally, for the calculation of the projection ${\mathrm{nor}}_{ϵ}$, we shall use

$${\mathrm{nor}}_{ϵ}W=W-f_{\ast }{\tan }_{ϵ}W$$

together with (112).

8. Gauss Formula for $\mathit{f}:(\mathit{M},{\mathit{g}}_{\mathit{ϵ}}\left(\mathit{f}\right))\to (\mathit{A},{\mathit{g}}_{\mathit{ϵ}}^{\mathit{A}})$

The purpose of the present section is to give an explicit form of the Gauss formula

$${\left(D^{ϵ}\right)}_V^f{f}_{\ast }W=f_{\ast }{\nabla }_V^{f,ϵ}W+B_{ϵ}\left(f\right)(V,W),V,W\in \mathfrak{X}\left(M\right).$$

(113)

To this end, we shall compute

$${\nabla }_V^{f,ϵ}W={\tan }_{ϵ}\left\{{\left(D^{ϵ}\right)}_V^f{f}_{\ast }W\right\},B_{ϵ}\left(f\right)(V,W)={\mathrm{nor}}_{ϵ}\left\{{\left(D^{ϵ}\right)}_V^f{f}_{\ast }W\right\},$$

by essentially using (112) in Lemma 11. Calculations are considerably simplified by exploiting the decomposition $T\left(M\right)=H\left(M\right)\oplus \mathbb{R}T$. Let us set $V=X$ and $W=Y$ with $X,Y\in H\left(M\right)$ in the Gauss formula (113), i.e.,

$${\left(D^{ϵ}\right)}_X^f{f}_{\ast }Y=f_{\ast }{\nabla }_X^{f,ϵ}Y+B_{ϵ}\left(f\right)(X,Y).$$

(114)

On the other hand, by (12)–(15), with ${\nabla }^{ϵ}$ replaced by $D^{ϵ}$,

$$D_{\mathbf{X}}^{ϵ}\mathbf{Y}=D_{\mathbf{X}}\mathbf{Y}+\left\{g_{\mathsf{\Theta }}(\mathbf{X},J_A\mathbf{Y})-{ϵ}^2{g}_{\mathsf{\Theta }}(\mathbf{X},{\tau }_A\mathbf{Y})\right\}{T}_A,$$

(115)

$$D_{\mathbf{X}}^{ϵ}{T}_A={\tau }_A\mathbf{X}+\frac{1}{{ϵ}^2}{J}_A\mathbf{X},$$

(116)

$$D_{T_A}^{ϵ}\mathbf{X}=D_{T_A}\mathbf{X}+\frac{1}{{ϵ}^2}{J}_A\mathbf{X},$$

(117)

$$D_{T_A}^{ϵ}{T}_A=0,$$

(118)

for any $\mathbf{X},\mathbf{Y}\in H\left(A\right)$. We systematically apply our findings in Section 2 to the pseudohermitian manifold $(A,\mathsf{\Theta })$ and to the Riemannian metric $g_{ϵ}^A$ (the $ϵ$-contraction of the Levi form $G_{\mathsf{\Theta }}$). By (115),

$$\begin{array}{c}{\left(D^{ϵ}\right)}_X^f{f}_{\ast }Y=D_X^f{f}_{\ast }Y\\ +\left\{g_{\mathsf{\Theta }}^f(f_{\ast }X,J_A^f{f}_{\ast }Y)-{ϵ}^2{g}_{\mathsf{\Theta }}^f(f_{\ast }X,{\tau }_A^f{f}_{\ast }Y)\right\}{T}_A^f.\end{array}$$

(119)

Here, $D^f=f^{-1}D$ is the pullback of the Tanaka–Webster connection D–a connection in the pullback bundle $f^{-1}T\left(A\right)\to M$. We shall substitute from (119) into the left-hand side of (114). Our ultimate goal is to relate the pseudohermitian geometry of the ambient space $(A, \mathsf{\Theta })$ to that of the submanifold $(M, \theta )$. Therefore, to compute the right-hand side of (114), one needs a lemma relating the induced connection ${\nabla }^{f,ϵ}$, associated with the isometric immersion $f:(M,g_{ϵ}\left(f\right))\to (A,g_{ϵ}^A)$, to the Tanaka–Webster connection ∇ of $(M, \theta )$.

Lemma 12.

Let $f:M\to A$ be an isopseudohermitian (i.e., $f^{\ast }\mathsf{\Theta }=\theta$) CR immersion. The Levi–Civita connection ${\nabla }^{f,ϵ}$ of $(M,g_{ϵ}\left(f\right))$ and the Tanaka–Webster connection ∇ of $(M, \theta )$ are related by

$${\nabla }_X^{f,ϵ}Y={\nabla }_{X}Y+\left\{\mathsf{\Omega }(X,Y)-\frac{1}{{\mu }_{ϵ}}A(X,Y)\right\}T,$$

(120)

$${\nabla }_X^{f,ϵ}T=\tau X+{\mu }_{ϵ}JX+\frac{1}{2{\mu }_{ϵ}}X\left(\lambda \right)T,$$

(121)

$${\nabla }_T^{f,ϵ}X={\nabla }_{T}X+{\mu }_{ϵ}JX+\frac{1}{2{\mu }_{ϵ}}X\left(\lambda \right)T,$$

(122)

$${\nabla }_T^{f,ϵ}T=-\frac{1}{2}{\nabla }^H\lambda +\frac{1}{2{\mu }_{ϵ}}T\left(\lambda \right)T,$$

(123)

for any $X,Y\in H\left(M\right)$. Here ${\mu }_{ϵ}={\mu }_{ϵ}(f;\theta ,\theta )\in C^{\infty }\left(M\right)$ is given by

$${\mu }_{ϵ}=\frac{1}{{ϵ}^2}-1+\lambda ,\lambda =\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(T, T).$$

(124)

Proof. 

We start from the well-known (see, e.g., Proposition 2.3 in [10], Volume I, p. 160) expression of the Levi–Civita connection ${\nabla }^{f,ϵ}$ in terms of the Riemannian metric $g_{ϵ}\left(f\right)$

$$2g_{ϵ}\left(f\right)\left({\nabla }_U^{f,ϵ}V,W\right)$$

$$=U\left(g_{ϵ}\left(f\right)(V,W)\right)+V\left(g_{ϵ}\left(f\right)(W,V)\right)-W\left(g_{ϵ}\left(f\right)(U,V)\right)$$

$$+g_{ϵ}\left(f\right)\left([U,V],W\right)+g_{ϵ}\left(f\right)\left([W,U],V\right)+g_{ϵ}\left(f\right)\left(U,[W,V]\right)$$

for any $U,V,W\in \mathfrak{X}\left(M\right)$. We adopt the notations in Section 2; i.e., we set

$$C_g(U,V,W)$$

$$=U\left(g(V,W)\right)+V\left(g(W,V)\right)-W\left(g(U,V)\right)$$

$$+g\left([U,V],W\right)+g\left([W,U],V\right)+g\left(U,[W,V]\right)$$

so that

$$2g_{ϵ}\left(f\right)\left({\nabla }_U^{f,ϵ}V,W\right)=C_{g_{ϵ}\left(f\right)}(U,V,W).$$

(125)

By (78) in Corollary 2 (to Proposition 3),

$$g_{ϵ}\left(f\right)=g_{ϵ}^M+(\lambda -1)\theta \otimes \theta ,$$

(126)

where $\lambda \in C^{\infty }\left(M\right)$ is given by (102).    □

Lemma 13.

The Christoffel mappings $C_{g_{ϵ}\left(f\right)}$ and $C_{g_{ϵ}^M}$ are related by

$$\begin{array}{c}{C}_{g_{ϵ}\left(f\right)}(U,V,W)=C_{g_{ϵ}^M}(U,V,W)\\ +U\left(\lambda \right)\theta \left(V\right)\theta \left(W\right)+V\left(\lambda \right)\theta \left(U\right)\theta \left(W\right)-W\left(\lambda \right)\theta \left(U\right)\theta \left(V\right)\\ +2(\lambda -1)[\theta \left(W\right)\theta \left({\nabla }_{U}V\right)\\ +\mathsf{\Omega }(U,V)\theta \left(W\right)+\mathsf{\Omega }(W,U)\theta \left(V\right)+\mathsf{\Omega }(W,V)\theta \left(U\right)]\end{array}$$

(127)

for any $U,V,W\in \mathfrak{X}\left(M\right)$.

Proof. 

(127) is a straightforward (yet rather involved) consequence of (126). We give a few details, for didactic reasons, as follows. Let us substitute from (126) into $C_{g_{ϵ}\left(f\right)}$ and recognize the term $C_{g_{ϵ}^M}$. To bring into the picture the Tanaka–Webster connection, we substitute the remaining Lie products from

$$[V,W]={\nabla }_{V}W-{\nabla }_{W}V-2(\theta \wedge \tau )(V,W)+2\mathsf{\Omega }(V,W)T$$

(128)

and use $\nabla \theta =0$. At its turn, (128) is a mere consequence of

$$T_{\nabla }=2\left\{\theta \wedge \tau -\mathsf{\Omega }\otimes T\right\}.$$

Let ${\nabla }^{ϵ}$ be the Levi–Civita connection of $(M,g_{ϵ}^M)$. Similar to (125),

$$2g_{ϵ}^M\left({\nabla }_U^{ϵ}V,W\right)=C_{g_{ϵ}^M}(U,V,W).$$

(129)

Then, (125)–(129) yield

$$\begin{array}{c}2g_{ϵ}\left({\nabla }_U^{f,ϵ}V,W\right)+2(\lambda -1)\theta \left({\nabla }_U^{f,ϵ}V\right)\theta \left(W\right)=2g_{ϵ}\left({\nabla }_U^{ϵ}V,W\right)\\ +U\left(\lambda \right)\theta \left(V\right)\theta \left(W\right)+V\left(\lambda \right)\theta \left(U\right)\theta \left(W\right)-W\left(\lambda \right)\theta \left(U\right)\theta \left(V\right)\\ +2(\lambda -1)[\theta \left(W\right)\theta \left({\nabla }_{U}V\right)\\ +\mathsf{\Omega }(U,V)\theta \left(W\right)+\mathsf{\Omega }(W,U)\theta \left(V\right)+\mathsf{\Omega }(W,V)\theta \left(U\right)].\end{array}$$

(130)

Let us substitute from $g_{ϵ}=g_{\theta }+({ϵ}^{-2}-1)\theta \otimes \theta$ into (130) and use nondegeneracy of $g_{\theta }$ to “simplify” W. We obtain

$$\begin{array}{c}2{\nabla }_U^{f,ϵ}V+2\left(\frac{1}{{ϵ}^2}+\lambda -2\right)\theta \left({\nabla }_U^{f,ϵ}V\right)T\\ =2{\nabla }_U^{ϵ}V+2\left(\frac{1}{{ϵ}^2}-1\right)\theta \left({\nabla }_U^{ϵ}V\right)T\\ +\left[U\left(\lambda \right)\theta \left(V\right)+V\left(\lambda \right)\theta \left(U\right)\right]T-\theta \left(U\right)\theta \left(V\right)\nabla \lambda \\ +2(\lambda -1)\left\{\left[\theta \left({\nabla }_{U}V\right)+\mathsf{\Omega }(U,V)\right]T+\theta \left(V\right)JU+\theta \left(U\right)JV\right\}.\end{array}$$

(131)

Next, let us apply $\theta$ to both sides of (131) and use $\theta \left(T\right)=1$ and $\theta \circ J=0$ in order to yield

$$\begin{array}{c}2\left(\frac{1}{{ϵ}^2}+\lambda -1\right)\theta \left({\nabla }_U^{f,ϵ}V\right)=\frac{2}{{ϵ}^2}\theta \left({\nabla }_U^{ϵ}V\right)\\ +\left[U\left(\lambda \right)\theta \left(V\right)+V\left(\lambda \right)\theta \left(U\right)\right]-\theta \left(U\right)\theta \left(V\right)T\left(\lambda \right)\\ +2(\lambda -1)\left\{\left[\theta \left({\nabla }_{U}V\right)+\mathsf{\Omega }(U,V)\right]\right\}.\end{array}$$

(132)

In particular, for $U=X$ and $V=Y$ with $X,Y\in H\left(M\right)$, the formulas (131) and (132) become

$$\begin{array}{c}{\nabla }_X^{f,ϵ}Y+\left(\frac{1}{{ϵ}^2}+\lambda -2\right)\theta \left({\nabla }_X^{f,ϵ}Y\right)T\\ ={\nabla }_X^{ϵ}Y+\left\{\left(\frac{1}{{ϵ}^2}-1\right)\theta \left({\nabla }_X^{ϵ}Y\right)+(\lambda -1)\mathsf{\Omega }(X,Y)\right\}T,\end{array}$$

(133)

$$\left(\frac{1}{{ϵ}^2}+\lambda -1\right)\theta \left({\nabla }_X^{f,ϵ}Y\right)=\frac{1}{{ϵ}^2}\theta \left({\nabla }_X^{ϵ}Y\right)+(\lambda -1)\mathsf{\Omega }(X,Y).$$

(134)

Here, one also uses $\theta \left({\nabla }_{X}Y\right)=0$ because ∇ parallelizes $H\left(M\right)$. Let us multiply (134) by T and subtract the resulting equation from (133). We obtain

$${\nabla }_X^{f,ϵ}Y={\nabla }_X^{ϵ}Y+\left\{\theta \left({\nabla }_X^{f,ϵ}Y\right)-\theta \left({\nabla }_X^{ϵ}Y\right)\right\}T,$$

(135)

a simplified form of (133) equivalent to

$${\mathsf{\Pi }}_{H\left(M\right)}{\nabla }_X^{f,ϵ}Y={\mathsf{\Pi }}_{H\left(M\right)}{\nabla }_X^{ϵ}Y.$$

(136)

At this point, we exploit the relationship between the Levi–Civita connection ${\nabla }^{ϵ}$ and the Tanaka–Webster connection ∇ as established in Lemma 2. Ssee formulas (12)–(15). For instance, by (12) and (136),

$${\nabla }_X^{ϵ}Y={\nabla }_{X}Y+\left\{\mathsf{\Omega }(X,Y)-{ϵ}^{2}A(X,Y)\right\}T⟹$$

$${\mathsf{\Pi }}_{H\left(M\right)}{\nabla }_X^{ϵ}Y={\nabla }_{X}Y⟹$$

$${\mathsf{\Pi }}_{H\left(M\right)}{\nabla }_X^{f,ϵ}Y={\nabla }_{X}Y$$

(137)

for any $X,Y\in H\left(M\right)$. Also, by applying $\theta$ to both sides of (12),

$$\theta \left({\nabla }_X^{ϵ}Y\right)=\mathsf{\Omega }(X,Y)-{ϵ}^{2}A(X,Y)$$

(138)

and substitution into (134) furnishes

$$\theta \left({\nabla }_X^{f,ϵ}Y\right)=\mathsf{\Omega }(X,Y)-\frac{1}{{\mu }_{ϵ}}A(X,Y).$$

(139)

Finally, by (137) and (138),

$${\nabla }_X^{f,ϵ}Y={\nabla }_{X}Y+\left\{\mathsf{\Omega }(X,Y)-\frac{1}{{\mu }_{ϵ}}A(X,Y)\right\}T,$$

which is (120). Q.E.D.

Similarly, the formulas (131) and (132) for $U=X$ and $V=T$ become, by (13), i.e., ${\nabla }_X^{ϵ}T=\tau X+{ϵ}^{-2}JX\in H\left(M\right)$,

$${\mathsf{\Pi }}_H{\nabla }_X^{f,ϵ}T=\tau X+{\mu }_{ϵ}JX,\theta \left({\nabla }_X^{f,ϵ}T\right)=\frac{1}{2{\mu }_{ϵ}}X\left(\lambda \right),$$

yielding (121). Q.E.D. The proof of the remaining relations (122) and (123) is similar.    □

Let us go back to (119). As $J_A^f\circ f_{\ast }=f_{\ast }\circ J$ formula (119) reads

$$\begin{array}{c}{\left(D^{ϵ}\right)}_X^f{f}_{\ast }Y=D_X^f{f}_{\ast }Y\\ +\left\{\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(X,JY)-{ϵ}^2{g}_{\mathsf{\Theta }}^f(f_{\ast }X,{\tau }_A^f{f}_{\ast }Y)\right\}{T}_A^f.\end{array}$$

(140)

Let us substitute from (140) and (120) into the Gauss formula (114) in order to obtain:

Proposition 5.

Let M and A be strictly pseudoconvex CR manifolds, and let $\theta \in {\mathcal{P}}_{+}\left(M\right)$ and $\mathsf{\Theta }\in {\mathcal{P}}_{+}\left(A\right)$. Let $f:M\to A$ be an isopseudohermitian immersion of $(M, \theta )$ into $(A,\mathsf{\Theta })$. Let $g_{ϵ}^A$, $0<ϵ<1$, be the ϵ-contraction of the Levi form $G_{\mathsf{\Theta }}$, and let $g_{ϵ}\left(f\right)=f^{\ast }{g}_{ϵ}^A$. Then

$$\begin{array}{c}{D}_X^f{f}_{\ast }Y+\left\{\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(X,JY)-{ϵ}^2{g}_{\mathsf{\Theta }}^f(f_{\ast }X,{\tau }_A^f{f}_{\ast }Y)\right\}{T}_A^f\\ =f_{\ast }{\nabla }_{X}Y+\left\{\mathsf{\Omega }(X,Y)-\frac{1}{{\mu }_{ϵ}}A(X,Y)\right\}{f}_{\ast }T+B_{ϵ}\left(f\right)(X,Y),\\ X,Y\in H\left(M\right),\end{array}$$

(141)

is the Gauss formula for the isometric immersion $f:(M,g_{ϵ}\left(f\right))\to (A,g_{ϵ}^A)$ along $H\left(M\right)\otimes H\left(M\right)$.

It should be noted that all terms in the Gauss formula (113), except for the second fundamental form $B_{ϵ}\left(f\right)$, were expressed in terms of pseudohermitian invariants of $(M, \theta )$ and $(A,\mathsf{\Theta })$. The remaining components of (113), respectively, along $H\left(M\right)\otimes \mathbb{R}T$, $\mathbb{R}T\otimes H\left(M\right)$, and $\mathbb{R}T\otimes \mathbb{R}T$, can be derived by setting

$$(U,V)\in \left\{(X,T),(T,X),(T,T)\right\},X\in H\left(M\right),$$

(142)

into (113). We relegate the derivation of the components (142) to further work. For the time being, we seek to further split (141) into tangential and normal parts, with respect to the direct sum decomposition (80). This amounts to decomposing $D_X^f{f}_{\ast }Y$ and the Reeb vector field $T_A^f$ with respect to (80).

We start with the decomposition of $T_A^f$. Formula (112) for $W=T_A^f$ gives, as ${\mathfrak{X}}_{\mathsf{\Theta }}\in f^{-1}H\left(A\right)$, so that ${\mathfrak{X}}_{\mathsf{\Theta }}$ and $T_A^f$ are $g_{\mathsf{\Theta }}^f$-orthogonal,

$${\tan }_{ϵ}\left(T_A\right)=\sum _{a=1}^{2n}{g}_{\mathsf{\Theta }}^f\left(f_{\ast }{X}_a,T_A^f\right)X_a+\frac{1}{{ϵ}^2{\mu }_{ϵ}}T$$

and

$$g_{\mathsf{\Theta }}^f\left(f_{\ast }{X}_a,T_A^f\right)={\mathsf{\Theta }}^f\left(f_{\ast }{X}_a\right)=\theta \left(X_a\right)=0$$

so that:

Lemma 14.

Let $f:M\to A$ be an isopseudohermitian immersion of $(M, \theta )$ into $(A,\mathsf{\Theta })$. The tangential and normal components of the Reeb vector field $T_A^f$, with respect to the decomposition (80), are

$${\tan }_{ϵ}(T_A^f)=\frac{1}{{ϵ}^2{\mu }_{ϵ}}T,{\mathrm{nor}}_{ϵ}(T_A^f)=T_A^f-\frac{1}{{ϵ}^2{\mu }_{ϵ}}{f}_{\ast }T.$$

(143)

Next, we attack the decomposition of $D_X^f{f}_{\ast }Y$ with respect to (80). To this end, we need to introduce pseudohermitian analogs to familiar objects in the theory of isometric immersions between Riemannian manifolds, such as the induced and normal connections, the second fundamental form, and the Weingarten operator. For any $V,W\in \mathfrak{X}\left(M\right)$ and any $\xi \in C^{\infty }\left(E\left(f\right)\right)$, we set by definition

$${\nabla }_V^{\top }W=\tan \left\{D_V^f{f}_{\ast }W\right\},B\left(f\right)(V,W)=\mathrm{nor}\left\{D_V^f{f}_{\ast }W\right\},$$

$$a_{\xi }V=-\tan \left\{D_V^f\xi \right\},{\nabla }_V^{\perp }\xi =\mathrm{nor}\left\{D_V^f\xi \right\},$$

where

$$\tan :f^{-1}T\left(A\right)\to T\left(M\right),\mathrm{nor}:f^{-1}T\left(A\right)\to E\left(f\right),$$

are the natural projections.

Theorem 2.

(i) ${\nabla }^{\top }$ is a linear connection on M.

(ii)

$B\left(f\right)$ is $C^{\infty }(M,\mathbb{R})$-bilinear.

(iii)

a is $C^{\infty }(M,\mathbb{R})$-bilinear.

(iv)

${\nabla }^{\perp }$ is a connection in the vector bundle $E\left(f\right)\to M$.

(v)

For any $V,W\in \mathfrak{X}\left(M\right)$ and any $\xi \in C^{\infty }\left(E\left(f\right)\right)$.

$$D_V^f{f}_{\ast }W=f_{\ast }{\nabla }_V^{\top }W+B\left(f\right)(V,W),$$

(144)

$$D_V^f\xi =-f_{\ast }{a}_{\xi }V+{\nabla }_V^{\perp }\xi .$$

(145)

The proof of Theorem 2 is straightforward. We adopt the following pseudohermitian analog to the ordinary terminology in use within the theory of isometric immersions between Riemannian manifolds.

Definition 38.

${\nabla }^{\top }$ is the induced connection (the connection induced by D via f). ${\nabla }^{\perp }$ is the normal Tanaka–Webster connection. $B\left(f\right)$ and $a_{\xi }$ are, respectively, the pseudohermitian second fundamental form and the pseudohermitian Weingarten operator (associated with the normal vector field $\xi$) of the CR immersion $f:M\to (A,\mathsf{\Theta })$. (144) is the pseudohermitan Gauss formula. (145) is the pseudohermitian Weingarten formula.

The induced connection ${\nabla }^{\top }$ and the (intrinsic) Tanaka–Webster connection ∇ of $(M, \theta )$ do not coincide, in general, unless, e.g., $f:(M,\theta )\to (A,\mathsf{\Theta })$ is a pseudohermitian immersion. The ambient connection–the Tanaka–Webster connection of $(A,\mathsf{\Theta })$–has torsion so that $B\left(f\right)$, unlike its Riemannian counterpart, it is never symmetric. We expect that $B\left(f\right)$ is the second fundamental form of f as introduced by P. Ebenfelt, X. Huang and D. Zaitsev (see formula 2.3 in [12], p. 636) by making use of B. Lamel’s spaces $E_k\left(p\right)$ (actually of $E_1\left(p\right)$; see Definition 1 in [13], p. 1). The main properties of ${\nabla }^{\top }$, $B\left(f\right)$, and $a_{\xi }$ are collected in the following.

Theorem 3.

Let $f:M\to A$ be a CR immersion, and let $\theta \in {\mathcal{P}}_{+}\left(M\right)$ and $\mathsf{\Theta }\in {\mathcal{P}}_{+}\left(A\right)$.

(i) The induced connection ${\nabla }^T$ has torsion, i.e.,

$$\begin{array}{c}{\mathrm{Tor}}_{{\nabla }^{\top }}(V,W)=-2\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(V,JW)T_A^{\top }\\ +\mathsf{\Lambda }\left\{\theta \left(V\right)\tan \left[{\tau }_A\left(f_{\ast }W\right)\right]-\theta \left(W\right)\tan \left[{\tau }_A\left(f_{\ast }V\right)\right]\right\}.\end{array}$$

(146)

In particular, if $g_{\mathsf{\Theta }}$ is Sasakian, then ${\nabla }^{\top }$ is symmetric ⟺ the Reeb vector field of $(A,\mathsf{\Theta })$ if $g_{\mathsf{\Theta }}$-orthogonal to $f\left(M\right)$.

(ii) The pseudohermitian second fundamental form $B\left(f\right)$ is not symmetric, in general, i.e.,

$$\begin{array}{c}B\left(f\right)(V,W)-B\left(f\right)(W,V)=-2\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(V,JW)T_A^{\perp }\\ +\mathsf{\Lambda }\left\{\theta \left(V\right)\mathrm{nor}\left[{\tau }_A\left(f_{\ast }W\right)\right]-\theta \left(W\right)\mathrm{nor}\left[{\tau }_A\left(f_{\ast }V\right)\right]\right\}.\end{array}$$

(147)

In particular, if $g_{\mathsf{\Theta }}$ is Sasakian, then $B\left(f\right)$ is symmetric ⟺$T_A$ is tangent to $f\left(M\right)$.

(iii) The metric $f^{\ast }{g}_{\mathsf{\Theta }}$ is parallel with respect to ${\nabla }^{\top }$ i.e., ${\nabla }^{\top }{f}^{\ast }{g}_{\mathsf{\Theta }}=0$.

(iv) For any $V,W\in \mathfrak{X}\left(M\right)$ and $\xi \in C^{\infty }\left(E\left(f\right)\right)$

$$g_{\mathsf{\Theta }}^f\left(B\left(f\right)(V,W),\xi \right)=\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(a_{\xi }V,W).$$

(148)

(v) For any $V,W\in \mathfrak{X}\left(M\right)$

$$\left({\nabla }_V^{\top }J\right)W=\tan \left\{J_A^{f}B\left(f\right)(V,W)\right\},$$

(149)

$$B\left(f\right)(V,JW)=\mathrm{nor}\left\{J_A^{f}B\left(f\right)(V,W)\right\}.$$

(150)

(vi) For any $V\in \mathfrak{X}\left(M\right)$

$${\nabla }_V^{\top }{T}_A^{\top }-a_{T_A^{\perp }}V=0,B\left(f\right)(V,T_A^{\top })+{\nabla }_V^{\perp }{T}_A^{\perp }=0.$$

(151)

Proof. 

(i)–(ii) The torsion of D is

$${\mathrm{Tor}}_D=2\left\{\mathsf{\Theta }\wedge {\tau }_A-{\mathsf{\Omega }}_A\otimes T_A\right\}$$

where ${\mathsf{\Omega }}_A(\mathbf{V},\mathbf{W})=g_{\mathsf{\Theta }}(\mathbf{V},J_A\mathbf{W})$ for any $\mathbf{V},\mathbf{W}\in \mathfrak{X}\left(A\right)$ see e.g., (17). Then, for any $V,W\in \mathfrak{X}\left(M\right)$

$$\mathsf{\Lambda }\left\{\theta \left(V\right){\tau }_A\left(f_{\ast }W\right)-\theta \left(W\right){\tau }_A\left(f_{\ast }V\right)\right\}-2\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(V,JW)T_A^f$$

$$=D_V^f{f}_{\ast }W-D_W^f{f}_{\ast }W-f_{\ast }[V,W]=$$

by the pseudohermitian Gauss formula (144)

$$=f_{\ast }{\mathrm{Tor}}_{{\nabla }^{\top }}(V,W)+B(V,W)-B(W,V)$$

yielding, by $T_A^f=f_{\ast }{T}_A^{\top }+T_A^{\perp }$, (146) and (147). Q.E.D.

(iii) For any $U,V,W\in \mathfrak{X}\left(M\right)$

$$0=\left(D_U^f{g}_{\mathsf{\Theta }}^f\right)\left(f_{\ast }V,f_{\ast }W\right)$$

$$=U\left(\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(V,W)\right)-g_{\mathsf{\Theta }}^f\left(D_U^f{f}_{\ast }V,f_{\ast }W\right)-g_{\mathsf{\Theta }}^f\left(f_{\ast }V,D_U^f{f}_{\ast }W\right)$$

again by (144)

$$=\left({\nabla }_U^{\top }\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\right)(V,W).$$

Q.E.D.

(iv) By (144)

$$g_{\mathsf{\Theta }}^f\left(B\left(f\right)(V,W),\xi \right)=g_{\mathsf{\Theta }}^f\left(D_V^f{f}_{\ast }W,\xi \right)$$

by $D^f{g}_{\mathsf{\Theta }}^f=0$ and $f_{\ast }W\perp \xi$ together with the pseudohermitian Weingarten formula (145)

$$=-g_{\mathsf{\Theta }}^f\left(f_{\ast }W,D_V^f\xi \right)=\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(a_{\xi }V,W).$$

Q.E.D.

(v) By (144)

$$f_{\ast }{\nabla }_V^{\top }JW+B\left(f\right)(V,JW)=D_V^f{f}_{\ast }JW$$

as f is a CR map, and by $D^f{J}_A^f=0$ and again (144)

$$=J_A^f{D}_V^f{f}_{\ast }W=f_{\ast }J{\nabla }_V^{\top }W+J_A^{f}B\left(f\right)(V,W).$$

Q.E.D.

(vi) Follows from $D^f{J}_A^f=0$, by (144) and (145).    □

Formula (112) for $W=D_X^f{f}_{\ast }Y$ gives

$$\begin{array}{c}{\tan }_{ϵ}\left(D_X{f}_{\ast }Y\right)=\sum _{a=1}^{2n}{g}_{\mathsf{\Theta }}^f\left(D_X{f}_{\ast }Y,f_{\ast }{X}_a\right)X_a\\ +\frac{1}{{ϵ}^2{\mu }_{ϵ}}\left\{{\mathsf{\Theta }}^f\left(D_X^f{f}_{\ast }Y\right)+{ϵ}^2{g}_{\mathsf{\Theta }}^f\left(D_X{f}_{\ast }Y,{\mathfrak{X}}_{\mathsf{\Theta }}\right)\right\}T.\end{array}$$

(152)

On the other hand, by the pseudohermitian Gauss formula (144) for $(V,W)=(X,Y)$ with $X,Y\in H\left(M\right)$,

$$g_{\mathsf{\Theta }}^f\left(D_X^f{f}_{\ast }Y,f_{\ast }{X}_a\right)=\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,X_a\right).$$

(153)

Let us set

$$T_A^{\top }=\tan \left(T_A^f\right),T_A^{\perp }=\mathrm{nor}\left(T_A^f\right),$$

so that

$$T_A^f=f_{\ast }{T}_A^{\top }+T_A^{\perp }$$

(154)

everywhere on M. Then, by (154), the pseudohermitian Gauss formula (144), and ${\mathfrak{X}}_{\mathsf{\Theta }}=f_{\ast }T-T_A^f$, the functions

$${\mathsf{\Theta }}^f(D_X^f{f}_{\ast }Y),g_{\mathsf{\Theta }}^f\left(D_X^f{f}_{\ast }Y,{\mathfrak{X}}_{\mathsf{\Theta }}\right)\in C^{\infty }(M,\mathbb{R})$$

can be written as

$${\mathsf{\Theta }}^f\left(D_X^f{f}_{\ast }Y\right)=\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,T_A^{\top }\right)+g_{\mathsf{\Theta }}^f\left(B\left(f\right)(X,Y),T_A^{\perp }\right),$$

(155)

$$\begin{array}{c}{g}_{\mathsf{\Theta }}^f\left(D_X^f{f}_{\ast }Y,{\mathfrak{X}}_{\mathsf{\Theta }}\right)=\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,T-T_A^{\top }\right)\\ -g_{\mathsf{\Theta }}^f\left(B\left(f\right)(X,Y),T_A^{\perp }\right).\end{array}$$

(156)

Finally, let us substitute from (153) and from (155) and (156) into (152). We obtain:

Lemma 15.

Let $f:M\to A$ be an isopseudohermitian immersion. The tangential component of $D_X^f{f}_{\ast }Y$ with respect to (80) is

$$\begin{array}{c}{\tan }_{ϵ}\left(D_X{f}_{\ast }Y\right)=\sum _{a=1}^{2n}\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,X_a\right)X_a\\ +\frac{1}{{ϵ}^2{\mu }_{ϵ}}\{\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,T_A^{\top }+{ϵ}^2\left(T-T_A^{\top }\right)\right)\\ +(1-{ϵ}^2)g_{\mathsf{\Theta }}^f\left(B\left(f\right)(X,Y),T_A^{\perp }\right)\}T,\end{array}$$

(157)

for any $X,Y\in H\left(M\right)$.

At this point, we may go back to (141), the Gauss formula for the isometric immersion $f:(M,g_{ϵ}\left(f\right))\to (A,g_{ϵ}^A)$ along $H\left(M\right)\otimes H\left(M\right)$, and apply the projections ${\tan }_{ϵ}$ and ${\mathrm{nor}}_{ϵ}$ to both sides. We obtain, by $\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)(X,JY)=\mathsf{\Omega }(X,Y)$,

$$\begin{array}{c}{\tan }_{ϵ}\left(D_X^f{f}_{\ast }Y\right)+\left\{\mathsf{\Omega }(X,Y)-{ϵ}^2{g}_{\mathsf{\Theta }}^f\left(f_{\ast }X,{\tau }_A^f{f}_{\ast }Y\right)\right\}{\tan }_{ϵ}(T_A^f)\\ ={\nabla }_{X}Y+\left\{\mathsf{\Omega }(X,Y)-\frac{1}{{\mu }_{ϵ}}A(X,Y)\right\}T,\end{array}$$

(158)

$$\begin{array}{c}{\mathrm{nor}}_{ϵ}\left(D_X^f{f}_{\ast }Y\right)+\left\{\mathsf{\Omega }(X,Y)-{ϵ}^2{g}_{\mathsf{\Theta }}^f\left(f_{\ast }X,{\tau }_A^f{f}_{\ast }Y\right)\right\}{\mathrm{nor}}_{ϵ}(T_A^f)\\ =B_{ϵ}\left(f\right)(X,Y).\end{array}$$

(159)

Let us substitute from (143) and (157) into (158). We obtain

$$\begin{array}{c}\sum _a\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{T}Y,X_a\right)X_a\\ +\frac{1}{{ϵ}^2{\mu }_{ϵ}}\left\{\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,{ϵ}^{2}T+(1-{ϵ}^2)T_A^{\top }\right)+g_{\mathsf{\Theta }}^f\left(B\left(f\right)(X,Y),(1-{ϵ}^2)T_A^{\perp }\right)\right\}T\\ +\frac{1}{{ϵ}^2{\mu }_{ϵ}}\left\{\mathsf{\Omega }(X,Y)-{ϵ}^2{g}_{\mathsf{\Theta }}^f\left(f_{\ast }X,{\tau }_A^f{f}_{\ast }Y\right)\right\}T={\nabla }_{X}Y+\left\{\mathsf{\Omega }(X,Y)-\frac{1}{{\mu }_{ϵ}}A(X,Y)\right\}T.\end{array}$$

(160)

Moreover, by $D_X^f{f}_{\ast }Y\in f^{-1}H\left(A\right)=\mathrm{Ker}\left({\mathsf{\Theta }}^f\right)$,

$$0={\mathsf{\Theta }}^f\left(D_X^f{f}_{\ast }Y\right)=g_{\mathsf{\Theta }}^f\left(D_X^f{f}_{\ast }Y,T_A^f\right)$$

by the pseudohermitian Gauss formula (144)

$$=g_{\mathsf{\Theta }}^f\left(f_{\ast }{\nabla }_X^{T}Y,T_A^f\right)+g_{\mathsf{\Theta }}^f\left(B\left(f\right)(X,Y),T_A^f\right)$$

by (154)

$$=\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{T}Y,T_A^{\top }\right)+g_{\mathsf{\Theta }}^f\left(B\left(f\right)(X,Y),T_A^{\perp }\right).$$

Summing up, we have proved the identity

$$\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{T}Y,T_A^{\top }\right)+g_{\mathsf{\Theta }}^f\left(B\left(f\right)(X,Y),T_A^{\perp }\right)=0.$$

(161)

As a consequence of (161), Equation (160) simplifies to

$$\begin{array}{c}\sum _a\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{T}Y,X_a\right)X_a\\ +\frac{1}{{\mu }_{ϵ}}\left\{\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,T\right)-g_{\mathsf{\Theta }}^f\left(f_{\ast }X,{\tau }_A^f{f}_{\ast }Y\right)\right\}T\\ ={\nabla }_{X}Y+\frac{1}{{\mu }_{ϵ}}\left\{(\lambda -1)\mathsf{\Omega }(X,Y)-A(X,Y)\right\}T.\end{array}$$

(162)

Identifying the $H\left(M\right)$ and $\mathbb{R}T$ in (162) leads to:

Proposition 6.

Let $f:M\to A$ be an isopseudohermitian immersion of $(M, \theta )$ and $(A,\mathsf{\Theta })$. The equations

$$\sum _a\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{T}Y,X_a\right)X_a={\nabla }_{X}Y,$$

(163)

$$\begin{array}{c}\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,T\right)+A(X,Y)\\ =(\lambda -1)\mathsf{\Omega }(X,Y)+g_{\mathsf{\Theta }}^f\left(f_{\ast }X,{\tau }_A^f{f}_{\ast }Y\right),\end{array}$$

(164)

are equivalent to the tangential part (158) of the Gauss formula (141) along $H\left(M\right)\otimes H\left(M\right)$ for the isometric immersion $f:(M,g_{ϵ}\left(f\right))\to (A,g_{ϵ}^A)$.

A remark is in order. Let us apply ${\mathsf{\Theta }}^f$ to both sides of the pseudohermitian Gauss formula (144) for $W=X\in C^{\infty }\left(H\left(M\right)\right)$, i.e.,

$$D_V^f{f}_{\ast }X=f_{\ast }{\nabla }_V^{\top }X+B\left(f\right)(V,X),V\in \mathfrak{X}\left(M\right).$$

As D parallelizes $H\left(A\right)$ and $f^{\ast }\mathsf{\Theta }=\theta$

$$0={\mathsf{\Theta }}^f\left(D_V^f{f}_{\ast }X\right)=\theta \left({\nabla }_V^{\top }X\right)+{\mathsf{\Theta }}^f\left(B\left(f\right)(V,X)\right)$$

showing that in general ${\nabla }^{\top }$ does not parallelize $H\left(M\right)$ (unlike the Tanaka–Webster connection ∇).

Let us go back to the Gauss formula (159). We need to compute ${\mathrm{nor}}_{ϵ}(D_X^f{f}_{\ast }Y)$. To this end, let us simplify (157) according to our successive finding (161), i.e.,

$$\begin{array}{c}{\tan }_{ϵ}\left(D_X^f{f}_{\ast }Y\right)=\sum _a\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,X_a\right)X_a\\ +\frac{1}{{\mu }_{ϵ}}\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,T\right)T.\end{array}$$

(165)

Then

$${\mathrm{nor}}_{ϵ}\left(D_X^f{f}_{\ast }Y\right)=D_X^f{f}_{\ast }Y-f_{\ast }{\tan }_{ϵ}\left(D_X^f{f}_{\ast }Y\right)$$

(166)

by (165)

$$=D_X^f{f}_{\ast }Y-\sum _a\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,X_a\right)f_{\ast }{X}_a-\frac{1}{{\mu }_{ϵ}}\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,T\right)f_{\ast }T.$$

Also, let us recall that, by (143),

$${\mathrm{nor}}_{ϵ}\left(T_A^f\right)=T_A^f-\frac{1}{{ϵ}^2{\mu }_{ϵ}}{f}_{\ast }T.$$

(167)

Next, let us modify the Gauss formula (159) by substitution from (166) and (167). We obtain

$$D_X^f{f}_{\ast }Y-\sum _a\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,X_a\right)f_{\ast }{X}_a-\frac{1}{{\mu }_{ϵ}}\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,T\right)f_{\ast }T$$

$$+\left\{\mathsf{\Omega }(X,Y)-{ϵ}^2{g}_{\mathsf{\Theta }}^f\left(f_{\ast }X,{\tau }_A^f{f}_{\ast }Y\right)\right\}\left\{T_A^f-\frac{1}{{ϵ}^2{\mu }_{ϵ}}{f}_{\ast }T\right\}=B_{ϵ}\left(f\right)(X,Y)$$

or, by (163):

Proposition 7.

Let $f:M\to A$ be an isopseudohermitian immersion of $(M, \theta )$ into $(A,\mathsf{\Theta })$. The normal part (159) of the Gauss formula (141) for the isometric immersion $f:(M,g_{ϵ}\left(f\right))\to (A,g_{ϵ}^A)$ along $H\left(M\right)\otimes H\left(M\right)$, is equivalent to

$$\begin{array}{c}{D}_X^f{f}_{\ast }Y-\frac{1}{{\mu }_{ϵ}}\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,T\right)f_{\ast }T\\ +\left\{\mathsf{\Omega }(X,Y)-{ϵ}^2{g}_{\mathsf{\Theta }}^f\left(f_{\ast }X,{\tau }_A^f{f}_{\ast }Y\right)\right\}\left\{T_A^f-\frac{1}{{ϵ}^2{\mu }_{ϵ}}{f}_{\ast }T\right\}\\ =f_{\ast }{\nabla }_{X}Y+B_{ϵ}\left(f\right)(X,Y)\end{array}$$

(168)

for any $X,Y\in H\left(M\right)$.

Another useful form of (168) is obtained by the substitution $f_{\ast }T={\mathfrak{X}}_{\mathsf{\Theta }}+T_A^f$; i.e.,

$$\begin{array}{c}{D}_X^f{f}_{\ast }Y-\frac{1}{{\mu }_{ϵ}}\left(f^{\ast }{g}_{\mathsf{\Theta }}\right)\left({\nabla }_X^{\top }Y,T\right)\left({\mathfrak{X}}_{\mathsf{\Theta }}+T_A^f\right)\\ +\left\{\mathsf{\Omega }(X,Y)-{ϵ}^2{g}_{\mathsf{\Theta }}^f\left(f_{\ast }X,{\tau }_A^f\right)\right\}\end{array}$$