The Scalar Curvature of a Riemannian Almost Paracomplex Manifold and Its Conformal Transformations

Abstract

A Riemannian almost paracomplex manifold is a 2n-dimensional Riemannian manifold $(M,g)$, whose structural group $O(2n,\mathbb{R})$ is reduced to the form $O(n,\mathbb{R})\times O(n,\mathbb{R})$. We define the scalar curvature $\pi$ of this manifold and consider relationships between $\pi$ and the scalar curvature s of the metric g and its conformal transformations.

1. Introduction

An almost paracomplex structure on a $2n$-dimensional smooth manifold M is a smooth field J of automorphisms of the tangent spaces, whose square is the identity operator ($J^2={\mathrm{id}}_{TM}$) and two eigenspaces (corresponding to eigenvalues $\pm 1$) have dimension n. In this case, the pair $(M,J)$ is called an almost paracomplex manifold, e.g., [1]. An almost paracomplex structure can alternatively be defined as a G-structure on M that reduces the structural group $GL(2n,\mathbb{R})$ to the form $GL(n,\mathbb{R})\times GL(n,\mathbb{R})$, see [1]. A paracomplex manifold is an almost paracomplex manifold $(M,J)$ such that the G-structure defined by J is integrable. A paracomplex manifold $(M,J)$ is a locally product manifold, i.e., M is locally diffeomorphic to the product $M_1\times M_2$ of two n-dimensional manifolds. An almost paracomplex structure J on a Riemannian manifold $(M,g)$ is said to be orthogonal if two eigenspaces of J are orthogonal. Moreover, every almost paracomplex structure on a Riemannian manifold is always orthogonal with respect to some Riemannian metric, see Section 2.

We can offer an alternative definition of a Riemannian paracomplex manifold. Namely, a $2n$-dimensional Riemannian manifold $(M,g)$ admits an orthogonal almost paracomplex structure if its structure group $O(2n,\mathbb{R})$ can be reduced to the form $O(n,\mathbb{R})\times O(n,\mathbb{R})$. A Riemannian manifold $(M,g)$ with an orthogonal paracomplex structure $(g,J)$ will be called a Riemannian almost paracomplex manifold and denoted by $(M,g,J)$.

The theory of paracomplex structures (e.g., [1,2,3]) has applications (see [4]) to the theory statistical manifolds, see [5]. The long history of the theory of almost paracomplex manifolds and a survey of the results of this theory, as well as examples of almost paracomplex manifolds, can be found in [1,2].

In this article, we define the scalar curvature $\pi$ of a Riemannian almost paracomplex manifold $(M,g,J)$ and consider the relationship between $\pi$ and the scalar curvature s of the metric g and its image under conformal transformations.

2. A Riemannian Orthogonal Paracomplex Manifold and Its Scalar Curvature

Here, we briefly describe the notation and conventions used in this article, see also [1,2]. We will also prove our first results and give illustrative examples.

An almost paracomplex structure on a smooth manifold M is a tensor field $J\in C^{\infty }(T^{*}M\otimes TM)$ such that $J^2=\mathrm{Id}$ and $\mathrm{trace}J=0$, see [2]. As a result, the direct decomposition holds $T_{x}M=H_x\oplus V_x$, where $H_x$ and $V_x$ are horizontal and vertical subspaces of the tangent space $T_{x}M$ at every point $x\in M$. The corresponding distributions $H=\left\{H_x\right\}$ and $V=\left\{V_x\right\}$ on M (i.e., subbundles of $TM$) have equal dimensions and correspond to the eigenvalues $-1$ and $+1$ of the tensor J, respectively. Thus, the dimension of a manifold with almost paracomplex structure is necessarily even. It is known that, for example, a four-dimensional sphere has no globally defined almost paracomplex structures, but there exist a non-integrable almost paracomplex structure on a six-dimensional unit sphere with its standard metric, see [3]. An almost paracomplex structure J on a Riemannian manifold $(M,g)$ is called orthogonal, see [1,2], if

$$g(JX,JY)=g(X,Y),X,Y\in TM,$$

(1)

and it is denoted by $(g,J)$. In this case, the distributions H and V of $(g,J)$ are orthogonal. Note that even if an almost paracomplex structure J is not orthogonal with respect to g, then J is orthogonal with respect to the Riemannian metric $\overline{g}$ defined by

$$\overline{g}(X,Y):=g(X,Y)+g(JX,JY),X,Y\in TM,$$

because $\overline{g}(JX,JY)=\overline{g}(X,Y)$, see (1). The triplet $(M,g,J)$, where $(g,J)$ is an orthogonal almost paracomplex structure on M, is called a Riemannian almost paracomplex manifold.

Remark 1.

An almost paracomplex structure is the antipode of a well-known almost complex structure on a $2n$-dimensional manifold, see [1]. Below, we consider the geometry of Riemannian paracomplex manifolds by analogy with the theory of almost Hermitian manifolds.

The torsion tensor of an almost paracomplex structure J on a smooth manifold M is the (2, 1)-tensor field ${\mathcal{N}}_J$ such that (e.g., [3])

$${\mathcal{N}}_J(X,Y)=[X,Y]+[JX,JY]-J[JX,Y]-J[X,JY],X,Y\in TM,$$

where $[·,·]$ is the Lie bracket of vector fields. The tensor ${\mathcal{N}}_J$ is an analog of the Nijenhuis tensor for an almost complex structure on a smooth manifold of even dimension.

The equality ${\mathcal{N}}_J=0$ holds on M if and only if the distributions H and V are involutive (or integrable, that is the same), see [3] (Theorem 2.4). Then, M is locally the product of two n-dimensional smooth manifolds (e.g., [3]). In this case, the almost paracomplex structure J is called integrable and $(M,J)$ is called a paracomplex manifold. Therefore, an integrable paracomplex structure exists on the product of manifolds of the same dimension, e.g., on the product of n-dimensional unit spheres (see [1]).

Let $(M,g,J)$ be a Riemannian almost paracomplex manifold with the Levi-Civita connection ∇ of the metric g and the Riemannian curvature tensor $R(X,Y)={\nabla }_X{\nabla }_Y-{\nabla }_Y{\nabla }_X-{\nabla }_{[X,Y]}$. Let ${\sigma }_x$ be a plane in $T_{x}M$, i.e., a two-dimensional subspace of $T_{x}M$ at an arbitrary point $x\in M$. Choosing an orthonormal basis $X_x,Y_x$ of ${\sigma }_x$, we define the sectional curvature $sec\left({\sigma }_x\right)$ in direction of ${\sigma }_x$ by

$$sec\left({\sigma }_x\right)=R(X_x,Y_x,X_x,Y_x),$$

where $R(X_x,Y_x,W_x,Z_x)=g(R(X_x,Y_x)Z_x,W_x)$. We shall write also $sec(X_x,Y_x)$ for $sec\left({\sigma }_x\right)$. It is known that $R(X_x,Y_x,X_x,Y_x)$ (the right-hand side) depends only on ${\sigma }_x$, and not on the choice of the orthonormal basis $X_x,Y_x$. The scalar curvature s of the metric g is defined by

$$s={\sum }_{i,j=1}^{2n}sec(e_i,e_j),$$

where $\{e_1,\dots ,e_{2n}\}$ is any orthonormal basis of $T_{x}M$. On the other hand, if $X_x,Y_x$ is an orthonormal basis for ${\sigma }_x$, then $J{X}_x,J{Y}_x$ is an orthonormal basis of another plane ${\sigma }_x^{\prime }$ such that ${\sigma }_x^{\prime }=J{\sigma }_x$. In this case, ${\sigma }_x=J{\sigma }_x^{\prime }=J^2{\sigma }_x$. Therefore, given two J-invariant planes ${\sigma }_x$ and ${\sigma }_x^{\prime }$ in $T_{x}M$, we can define the bisectional curvature $\mathrm{bisec}({\sigma }_x,{\sigma }_x^{\prime })$ by the equality

$$\mathrm{bisec}({\sigma }_x,J{\sigma }_x)=R(X_x,Y_x,J{X}_x,J{Y}_x).$$

One can verify that $R(X_x,Y_x,J{X}_x,J{Y}_x)$ depends only on ${\sigma }_x$ and ${\sigma }_x^{\prime }$. The bisectional curvature is an analog of the holomorphic bisectional curvature of a Kähler manifold, see [6,7] (pp. 303–313). Using the above, we can consider the scalar curvature $\pi$ of an orthogonal paracomplex structure $(g,J)$, or, in other words, of a Riemannian almost paracomplex manifold $(M,g,J)$, defined by the equality

$$\pi ={\sum }_{i,j=1}^{2n}R(e_i,e_j,J{e}_i,J{e}_j)$$

(2)

for a local orthonormal basis $\{e_1,\dots ,e_{2n}\}$ of $TM$. Let $\{e_1,\dots ,e_n\}$ and $\{e_{n+1},\dots ,e_{2n}\}$ be local orthonormal bases of the horizontal distribution H and the vertical distribution V, respectively. Vectors of these bases satisfy the following conditions:

$$J{e}_a=-e_a,J{e}_{\alpha }=e_{\alpha }$$

for $a=1,\dots ,n$ and $\alpha =n+1,\dots ,2n$. Using the above, we can show that

$$\begin{array}{ccc}& & \pi ={\sum }_{i,j=1}^{2n}R(e_i,e_j,J{e}_i,J{e}_j)={\sum }_{a,b=1}^{n}R(e_a,e_b,J{e}_a,J{e}_b)\hfill \\ & & +2{\sum }_{a=1}^n\sum _{\alpha =n+1}^{2n}R(e_a,e_{\alpha },J{e}_a,J{e}_{\alpha })+{\sum }_{\alpha ,\beta =n+1}^{2n}R(e_{\alpha },e_{\beta },J{e}_{\alpha },J{e}_{\beta })\hfill \\ & & ={\sum }_{a,b=1}^{n}R(e_a,e_b,e_a,e_b)-2{\sum }_{a=1}^n{\sum }_{\alpha =n+1}^{2n}R(e_a,e_{\alpha },e_a,e_{\alpha })\hfill \\ & & +{\sum }_{\alpha ,\beta =n+1}^{2n}R(e_{\alpha },e_{\beta },e_{\alpha },e_{\beta })\hfill \\ & & ={\sum }_{i,j=1}^{2n}sec(e_i,e_j)-4{\sum }_{a=1}^n{\sum }_{\alpha =n+1}^{2n}sec(e_a,e_{\alpha })=s-4s_{\mathrm{mix}},\hfill \end{array}$$

where we denoted by

$$s_{\mathrm{mix}}={\sum }_{\alpha =n+1}^{2n}{\sum }_{a=1}^{n}sec(e_a,e_{\alpha })$$

the mixed scalar curvature of an orthogonal paracomplex structure $(g,J)$. The concept of the mixed scalar curvature of a distribution on a Riemannian manifold has a long history and many applications [8,9,10,11]. By the above calculations, we obtain the following.

Theorem 1.

Let $(M,g,J)$ be a Riemannian almost paracomplex manifold. Then,

$$s=\pi +4s_{\mathrm{mix}},$$

(3)

where s is the scalar curvature of the metric g, and π and $s_{\mathrm{mix}}$ are the scalar and mixed scalar curvatures, respectively, of its orthogonal paracomplex structure $(g,J)$.

By (3), if the metric of $(M,g,J)$ has constant sectional curvature 1, then $\pi =-2n$. In contrast, the scalar curvature of such metric g on M is $s=2n(2n-1)$.

We consider three examples with the scalar curvature $\pi$ of a Riemannian almost paracomplex manifold, which is equal to the scalar curvature of its orthogonal paracomplex structure.

Example 1.

Recall that a distribution on a Riemannian manifold is totally geodesic if any geodesic that is tangent to the distribution at one point is tangent to this distribution at all its points. If both structure distributions H and V of a Riemannian paracomplex manifold $(M,g,J)$ are totally geodesic, then $s_{\mathrm{mix}}=(1/8){\parallel \nabla J\parallel }^2$, see [8], and by (3) we obtain

$$\pi =s-{(1/2)\parallel \nabla J\parallel }^2\le s.$$

Example 2.

Recall that a distribution on a Riemannian manifold is minimal (or, harmonic) if its mean curvature vector field (the trace of the second fundamental form) vanishes, see [12] (p. 149). If a minimal distribution is integrable, then its leaves (maximal integral manifolds) are minimal submanifolds, see [12] (p. 151). Let $(M,g,J)$ be a Riemannian paracomplex manifold, then M is locally the product of two n-dimensional manifolds, $M=M_1\times M_2$. If in addition, maximal integral manifolds of H and V are minimal submanifolds of $(M,g,J)$, then $s_{\mathrm{mix}}=-(1/8){\parallel \nabla J\parallel }^2$, see [8], and by (3) we obtain

$$\pi =s+{(1/2)\parallel \nabla J\parallel }^2\ge s.$$

Example 3.

Let $(M,J,g)$ be a 2n-dimensional Riemannian almost paracomplex manifold. Assume that $\nabla J=0$ for the Levi-Civita connection ∇ of the metric g, then both structure distributions H and V are involutive with totally geodesic integral manifolds. In this case, the Riemannian paracomplex manifold $(M,g,J)$ is locally the product of two n-dimensional Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$. The converse is also true. In this case, $s_{\mathrm{mix}}=0$; therefore, $\pi =s$, see also [13]. In particular, the scalar curvature of an orthogonal paracomplex structure of $(S^n\times S^n,g_0\oplus g_0)$ can be expressed in terms of the scalar curvature of $g_0$ via the formula

$$s(g_0\oplus g_0)=s\left(g_0\right)+s\left(g_0\right)=2n(n-1).$$

Therefore, $\pi =s=2n(n-1)$.

Remark 2.

Theorem 1 can be extended for an almost product structure on an m-dimensional Riemannian manifold $(M,g)$. Namely, let $P_i(i=1,2)$ be orthoprojectors on two complementary orthogonal distributions ${\mathcal{D}}_i(i=1,2)$, see [12] (p. 146). Set $G=P_2-P_1$ and define

$$\mathsf{\Pi }={\sum }_{i,j=1}^{m}R(e_i,e_j,G{e}_i,G{e}_j)$$

for a local orthonormal basis $\{e_1,\dots ,e_m\}$ of $TM$, compare with (2). Then, $g(GX,GY)=g(X,Y)$ for $X,Y\in TM$, compare with (1). Now, let $\{e_1,\dots ,e_{n_1}\}$ be a local orthonormal basis of the distribution ${\mathcal{D}}_1$ and $\{e_{n_1+1},\dots ,e_m\}$ be a local orthonormal basis of the distribution ${\mathcal{D}}_2$. Vectors of these bases satisfy the following conditions:

$$G{e}_a=-e_a(a=1,\dots ,n_1),G{e}_{\alpha }=e_{\alpha }(\alpha =n_1+1,\dots ,m).$$

Using the above, we can prove that (see [14])

$$s=\mathsf{\Pi }+4s_{\mathrm{mix}}.$$

In particular, if $(M,g)$ has constant sectional curvature 1, then $s=m(m-1)$ and $s_{\mathrm{mix}}=n_1{n}_2$; hence, $\mathsf{\Pi }=m(m-1)-4n_1{n}_2$.

3. Conformal Transformations of Metrics of Riemannian Almost Paracomplex Manifolds

An identity map $\mathrm{id}:M\to M$ from a differentiable manifold M into itself, also known as an identity transformation, is defined as the map with domain and range M, which satisfies $\mathrm{id}\left(x\right)=x$ for any $x\in M$, and it is the simplest map, which is both continuous and bijective (see [15]). Here, we will consider the conformal geometry of the identity map on a manifold M, and we assume that the domain M and the range M of $\mathrm{id}$ are equipped with metrics g and $\overline{g}$, respectively. The identity map $\mathrm{id}:M\to M$ is called a conformal transformation of the metric g if

$$\overline{g}=e^{2\sigma }g$$

(4)

for some smooth scalar function $\sigma$ on M, e.g., [6] (p. 115) and [7] (p. 269).

In this case, the metric $\overline{g}$ is called a conformal transformation of g; and if $\sigma =\mathrm{const}$, then this transformation is called a homothety. The converse statement (i.e., g is a conformal transformation of $\overline{g}$) is also true, because the equality $g=e^{-2\sigma }\overline{g}$ holds. In addition, the equality ${\overline{g}}^{-1}=e^{-2\sigma }{g}^{-1}$ holds. For such a rescaled metric $\overline{g}$, there is a unique symmetric connection, $\overline{\nabla }$, compatible with $\overline{g}$, i.e., $\overline{\nabla }\overline{g}=0$. Under a conformal transformation (4), the following relation (between two connections) holds, see [6] (p. 115) and [7] (p. 270):

$${\overline{\nabla }}_{X}Y={\nabla }_{X}Y+X\left(\sigma \right)Y+Y\left(\sigma \right)X-g(X,Y)\nabla \sigma ,X,Y\in C^{\infty }\left(TM\right).$$

(5)

We will consider conformal deformations of metrics of a Riemannian almost paracomplex manifold $(M,g,J)$. Obviously,

$$\overline{g}(JX,JY)=e^{2\sigma }g(JX,JY)=e^{2\sigma }g(X,Y)=\overline{g}(X,Y),X,Y\in C^{\infty }\left(TM\right).$$

Hence, a conformal deformation of g preserves the orthogonal decomposition $TM=H\oplus V$ of the tangent bundle of $(M,g,J)$, i.e., it preserves the orthogonal almost paracomplex structure. On the other hand, a diffeomorphism $f:M\to M$ is called a paraholomorhic transformation of $(M,g,J)$, if it preserves the almost paracomplex structure J, see [1]. Therefore, we have the following.

Proposition 1.

Let $(M,g,J)$ be a Riemannian almost paracomplex manifold. Then, the conformal transformation of metric $\mathrm{id}:M\to M$, see (4), represents a paraholomorphic transformation of $(M,g,J)$.

On the contrary, a conformal transformation of the metric of a Riemannian almost paracomplex manifold $(M,g,J)$ does not preserve its scalar curvature $\pi$. Thus, below, we study the relationship between the scalar curvatures $\pi$ and $\overline{\pi }$ of orthogonal paracomplex structures $(g,J)$ and $(\overline{g},J)$, respectively. By the theory of conformal mappings, e.g., [7] (p. 271), the relationship between the curvature tensors (of the Levi-Civita connections ∇ and $\overline{\nabla }$) of the metrics g and $\overline{g}$, respectively, has the following form, e.g., [6] (p. 115) and [7] (p. 271):

$$e^{-2\sigma }{\overline{R}}_{lijk}=R_{lijk}+g_{lk}{\sigma }_{ij}-g_{lj}{\sigma }_{ik}+{\sigma }_{lk}{g}_{ij}-{\sigma }_{lj}{g}_{ik}+(g_{lk}{g}_{ij}-g_{lj}{g}_{ik}){\parallel d\sigma \parallel }^2$$

(6)

with respect to local coordinates $(x^1,\dots ,x^{2n})$, where ${\overline{g}}_{ij}$ and $g_{ij}$ are components of metrics $\overline{g}$ and g. In (6), we denote by ${\overline{R}}_{lijk}$ and $R_{lijk}$, the components of Riemannian curvature tensors $\overline{R}$ and R of metrics $\overline{g}$ and g, respectively. The components ${\sigma }_{ij}$ in (6) are given by

$${\sigma }_{ij}={\nabla }_i{\nabla }_j\sigma -\left({\nabla }_i\sigma \right)\left({\nabla }_j\sigma \right),$$

where ${\nabla }_i={\nabla }_{\partial /\partial x^i}$. From (6), we obtain

$$\begin{array}{ccc}\hfill e^{2\sigma }\overline{\pi }& =& e^{-2\sigma }{\overline{R}}_{lijk}\left(e^{2\sigma }{J}_h^j{\overline{g}}^{hl}\right)\left(e^{2\sigma }{J}_p^i{\overline{g}}^{kp}\right)\hfill \\ \hfill & =& R_{lijk}\left(J_h^j{g}^{hl}\right)\left(J_p^i{g}^{kp}\right)+2g^{ik}{\sigma }_{ik}+2n{\parallel d\sigma \parallel }^2\hfill \\ & =& \pi +2\mathsf{\Delta }\sigma +{2(n-1)\parallel d\sigma \parallel }^2,\hfill \end{array}$$

(7)

where $\left(g^{ij}\right)={\left(g_{ij}\right)}^{-1}$, $\mathsf{\Delta }\sigma =g^{ij}{\nabla }_i{\nabla }_j\sigma$ and $\mathsf{\Delta }=\mathrm{div}\circ \nabla$ is the Laplace–Beltrami operator. We can rewrite (7) as

$$\mathsf{\Delta }\sigma =\frac{1}{2}(e^{2\sigma }\overline{\pi }-\pi )-(n-1){\parallel d\sigma \parallel }^2.$$

(8)

The total scalar curvature $\pi \left(M\right)$ of a compact Riemannian almost paracomplex manifold $(M,g,J)$ is defined by the integral equality

$$\pi \left(M\right)={\int }_M\pi \mathrm{d}{\mathrm{vol}}_g,$$

where $\mathrm{d}{\mathrm{vol}}_g$ is the volume form of the metric g. Note that $\pi \left(M\right)$ is an analog of the total scalar curvature of a compact Riemannian manifold $(M,g)$, see [16] (p. 119) and [9,14],

$$s\left(M\right)={\int }_{M}s\mathrm{d}{\mathrm{vol}}_g.$$

Integrating (8) over M and using the Green’s formula ${\int }_M\mathsf{\Delta }\sigma \mathrm{d}{\mathrm{vol}}_g=0$ yields

$$\pi \left(M\right)={\int }_M(e^{2\sigma }\overline{\pi }-{2(n-1)\parallel d\sigma \parallel }^2)\mathrm{d}{\mathrm{vol}}_g.$$

The above integral equality yields the inequality

$$\pi \left(M\right)\le {\int }_M\left(e^{2\sigma }\overline{\pi }\right)\mathrm{d}{\mathrm{vol}}_g.$$

By the above inequality, if $\overline{\pi }\le 0$ on M and $\pi \left(M\right)\ge 0$, then $\pi \left(M\right)=0$ and $\overline{\pi }\equiv 0$. In this case, $\sigma =\mathrm{const}$.

Theorem 2.

Let $(M,g,J)$ be a compact Riemannian almost paracomplex manifold with nonnegative total scalar curvature, $\pi \left(M\right)\ge 0$, and let $\overline{g}=e^{2\sigma }g$ be another metric conformally related to g for some $\sigma \in C^2\left(M\right)$. If $\overline{\pi }\le 0$ on M, then σ is constant. Thus, the conformal transformation of g to the metric $\overline{g}$ is a homothety; furthermore, $\overline{\pi }=\pi =0$ on M.

Setting $\sigma =\frac{1}{n-1}lnu$ for a positive scalar function $u\in C^2\left(M\right)$, from (4) we obtain $\overline{g}=u^{2/(n-1)}g$ with $u>0$. In this case, (6) can be rewritten as

$$\mathsf{\Delta }u=\frac{n-2}{2}(u^{\frac{n+1}{n-1}}\overline{\pi }-u\pi ).$$

(9)

Integrating (9) over compact manifold M and using the Green’s formula, gives

$${\int }_M{u}^{\frac{n+1}{n-1}}\overline{\pi }\mathrm{d}{\mathrm{vol}}_g={\int }_{M}u\pi \mathrm{d}{\mathrm{vol}}_g.$$

We can formulate the following theorem supplementing Theorem 2.

Theorem 3.

Let $(M,J,g)$ be a compact Riemannian almost paracomplex manifold with scalar curvature $\pi \le 0$ on M, and let a metric $\overline{g}$ be conformally related to g. If $\overline{\pi }\ge 0$ on M, then the conformal deformation of the metric g to $\overline{g}$ is a homothety; furthermore, $\overline{\pi }=\pi =0$ on M.

Corollary 1.

Let $(M,J,g)$ be a compact Riemannian almost paracomplex manifold, and let a metric $\overline{g}$ be conformally related to g. If both orthogonal paracomplex structures $(g,J)$ and $(\overline{g},J)$ have nonvanishing scalar curvatures, i.e., $\pi \ne 0$ and $\overline{\pi }\ne 0$ on M, then these scalar curvatures have the same sign.

If $\overline{\pi }\le 0$ and $\pi \ge 0$, then by (9) we obtain $\mathsf{\Delta }u\le 0$ on M. Thus, from (8), we conclude that $\sigma$ is a superharmonic function. On the other hand, a complete Riemannian manifold $(M,g)$ is called a parabolic manifold if it does not admit a non-constant positive superharmonic function, e.g., [17] (p. 313). For example, a complete Riemannian manifold $(M,g)$ of finite volume is a parabolic manifold because it does not carry non-constant positive superharmonic functions, see [18]. Using the above, we can formulate the following.

Theorem 4.

Let $(M,g,J)$ be a parabolic Riemannian almost paracomplex manifold (in particular, $(M,g,J)$ be a complete manifold of finite volume) with scalar curvature $\pi \ge 0$ on M, and let a metric $\overline{g}$ be conformally related to g. If $\overline{\pi }\le 0$ on M, then the conformal deformation of the metric g to the metric $\overline{g}$ is a homothety; furthermore, $\overline{\pi }=\pi =0$ on M.

If $\overline{\pi }\ge 0$ and $\pi \le 0$ then $\mathsf{\Delta }u\ge 0$ on M, then from (9), we conclude that u is subharmonic function. We recall the following famous theorem by C. Yau: let u be a nonnegative smooth subharmonic function on a complete Riemannian manifold $(M,g)$, then ${\int }_M{u}^p\mathrm{d}{\mathrm{vol}}_g=\infty$ for any $p>1$, unless u is a constant function, see [19] (Theorem 3).

Therefore, we can formulate the following statement on complete Riemannian almost paracomplex manifolds.

Theorem 5.

Let $(M,g,J)$ be a complete Riemannian almost paracomplex manifold with scalar curvature $\pi \le 0$ on M, and let $\overline{g}$ be another metric conformally related to g by the formula $\overline{g}=u^{2/(n-1)}g$ for some positive function $u\in C^2\left(M\right)$. If $\overline{\pi }\ge 0$ on M and $u\in L^p(M,g)$ for some $p>1$, then the conformal deformation of the metric g to the metric $\overline{g}$ is a homothety; furthermore, $\overline{\pi }=\pi =0$ on M.

A Riemannian manifold $(M,g)$ is locally conformally flat if for each point $x\in M$, there exists a neighborhood U of x and a smooth function $\sigma :U\to \mathbb{R}$ such that $(U,e^{2\sigma }g)$ is flat, i.e., the curvature of the metric $e^{2\sigma }g$ vanishes on U. In the case of a Riemannian almost paracomplex manifold $(M,g,J)$, we can formulate the following.

Theorem 6.

Let $(M,g,J)$ be a Riemannian almost paracomplex manifold such that g is a locally conformally flat metric with vanishing scalar curvature s, then its scalar curvature π vanishes on M.

Proof. 

Following [20], denote by $sec\left(D_x\right)$, the sectional curvature of a Riemannian manifold $(M,g)$ associated with an r-plane section $D_x\subset T_{x}M$ for an arbitrary point $x\in M$. Then, for any orthonormal basis $\{e_1,\dots ,e_r\}$ of $D_x$, the scalar curvature $s\left(D_x\right)$ of the r-plane section $D_x$ is defined by, see also [20],

$$s\left(D_x\right)={\sum }_{p,q=1}^{r}sec(e_p,e_q).$$

Now, let $(M,g)$ be a $2r$-dimensional locally conformally flat manifold with vanishing scalar curvature s of the metric g, then $s\left(D_x\right)=-s\left(D_x^{\perp }\right)$, where $D_x^{\perp }$ is the orthogonal complement of $D_x$, see [21]. In the case of a $2n$-dimensional Riemannian almost paracomplex manifold $(M,g,J)$, the scalar curvature s of the metric g can be presented as

$$s=s\left(H\right)+2s_{\mathrm{mix}}+s\left(V\right),$$

(10)

where $s\left(H\right)={\sum }_{a,b=1}^{n}sec(e_a,e_b)$ and $s\left(V\right)={\sum }_{\alpha ,\beta =1}^{n}sec(e_{\alpha },e_{\beta })$ are scalar curvatures of the horizontal and vertical distributions. Moreover, if $(M,g,J)$ is a locally conformally flat manifold with vanishing scalar curvature s of the metric g, then $s\left(H\right)=-s\left(V\right)$. In this case, from (10) we obtain $s_{\mathrm{mix}}=0$. Thus, by our Theorem 1, $\pi =0$. □

For example (see [16]) [p. 61], the product of two Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$, one with sectional curvature 1, and the other with sectional curvature $-1$, is locally conformally flat. In particular, if $dim{M}_1=dim{M}_2=n$, then we have $s=s_1+s_2=n(n-1)-n(n-1)=0$ and $s_{\mathrm{mix}}=0$. Therefore, $\pi =0$.

4. A Riemannian Almost Paracomplex Manifold Conformally Related to the Product of Riemannian Manifolds

Let a 2n-dimensional Riemannian almost paracomplex manifold $(M,J,g)$ satisfy the following conditions: $M=M_1\times M_2$ and $g=e^{2\sigma }(g_1\oplus g_2)$ for some n-dimensional Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$, respectively, and $\sigma \in C^2\left(M\right)$. In this case, the metric of $(M,J,g)$ arises as a result of the conformal transformation of the metric $g_1\oplus g_2$ of the product of Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$. At the same time, there exists a natural integrable orthogonal paracomplex structure J of $(M_1\times M_2,g_1\oplus g_2)$ and the Levi-Civita connection $\overline{\nabla }$ of its metric $\overline{g}=g_1\oplus g_2$ such that $\overline{\nabla }\overline{g}=0$ and $\overline{\nabla }J=0$ (see Example 3). Applying (5), we obtain the following relationship between the covariant derivatives $\overline{\nabla }J$ and $\nabla J$:

$$\begin{array}{ccc}& & g(\left({\overline{\nabla }}_{X}J\right)Y,Z)=g(\left({\nabla }_{X}J\right)Y,Z)-Y\left(\sigma \right)g(JX,Z)-Z\left(\sigma \right)g(JX,Y)\hfill \\ & & +g\left(J\right(\nabla \sigma ),Y)g(X,Z)+g\left(J\right(\nabla \sigma ),X)g(Y,Z),X,Y,Z\in TM.\hfill \end{array}$$

In the case of $\overline{\nabla }J=0$, this formula has the following form:

$$\begin{array}{c}\hfill g(\left({\nabla }_{X}J\right)Y,Z)=Y\left(\sigma \right)g(JX,Z)+Z\left(\sigma \right)g(JX,Y)\\ \hfill -g\left(J\right(\nabla \sigma ),Y)g(X,Z)-g\left(J\right(\nabla \sigma ),X)g(Y,Z),X,Y,Z\in TM.\end{array}$$

(11)

The converse is true only in a local sense. By the above, we can formulate the following.

Theorem 7.

Let a $2n$-dimensional Riemannian almost paracomplex manifold $(M,g,J)$ be conformal to the product of n-dimensional Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$, then the structural tensor J satisfies (11). The converse is true only in a local sense.

Let a $2n$-dimensional Riemannian almost paracomplex manifold $(M,g,J)$ be the product of n-dimensional Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$. In this case, $\nabla J=0$ and $\pi =s$ on $M=M_1\times M_2$. After the conformal deformation $\overline{g}=e^{2\sigma }(g_1\oplus g_2)$ for some $\sigma \in C^2\left(M\right)$ of the metric $g=g_1\oplus g_2$, we obtain the equation, see [7] (p. 271):

$$e^{2\sigma }\overline{s}=s-2(2n-1)\mathsf{\Delta }\sigma -2(n-1)(2n-1){\parallel d\sigma \parallel }^2.$$

(12)

for the scalar curvature $\overline{s}$ of the metric $\overline{g}=e^{2\sigma }(g_1\oplus g_2)$. We rewrite (6) as

$$e^{2\sigma }\overline{\pi }=\pi +2\mathsf{\Delta }\sigma +2(n-1){\parallel d\sigma \parallel }^2.$$

(13)

From (12) and (13), it follows that

$$\mathsf{\Delta }\sigma =\frac{1}{4n}{e}^{2\sigma }(\overline{\pi }-\overline{s})-{\parallel d\sigma \parallel }^2.$$

(14)

Setting $\sigma =lnu$ for a positive scalar function $u\in C^2\left(M\right)$, the equality $\overline{g}=e^{2\sigma }g$ can be rewritten as $\overline{g}=u^{2}g$, $u>0$. In this case, (14) can be rewritten as

$$\mathsf{\Delta }u=\frac{1}{2n}{u}^2(\overline{\pi }-\overline{s}).$$

If $M=M_1\times M_2$ is a compact manifold (in particular, if $M_1$ and $M_2$ are compact manifolds), then from from the above formula we obtain the following integral equation:

$$\frac{1}{2n}{\int }_M{u}^2(\overline{\pi }-\overline{s})\mathrm{d}{\mathrm{vol}}_g=0.$$

(15)

Note that conditions $\overline{\pi }\le \overline{s}$ and $\overline{\pi }<\overline{s}$ (or, $\overline{\pi }\ge \overline{s}$ and $\overline{\pi }>\overline{s}$) for at least one point $x\in M_1\times M_2$ contradict (15). Thus, the following theorem holds.

Theorem 8.

Let $(M,g,J)$ be a $2n$-dimensional Riemannian paracomplex manifold such that $M=M_1\times M_2$ for n-dimensional compact manifolds $M_1$ and $M_2$. If the scalar curvatures π and s satisfy the following condition: $\pi \le s$(resp., $\overline{\pi }\ge \overline{s})$ on M and $\pi <s$ (resp., $\overline{\pi }>\overline{s}$) for at least one point $x\in M$, then M does not admit a metric $\overline{g}=g_1\oplus g_2$ arising as a result of a conformal transformation of g.

Let $(M,J,\overline{g})$ be a $2n$-dimensional integrable Riemannian almost paracomplex manifold with $M=M_1\times M_2$ and

$$g=e^{2{\sigma }_1}{g}_1\oplus e^{2{\sigma }_2}{g}_2$$

(16)

for some scalar functions ${\sigma }_1,{\sigma }_2\in C^2\left(M\right)$. In this case, (16) defines a biconformal deformation (see [22]) of the product metric $\overline{g}=g_1\oplus g_2$ on the product of n-dimensional Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$. At the same time, for a Riemannian manifold $(M,\overline{g})$ such that $M=M_1\times M_2$ and $\overline{g}=g_1\oplus g_2$, there is a unique symmetric connection, $\overline{\nabla }$, compatible with $\overline{g}$ and J, i.e., $\overline{\nabla }\overline{g}=0$ and $\overline{\nabla }J=0$. Applying (5), we can obtain a relationship between the covariant derivatives $\overline{\nabla }J$ and $\nabla J$. In the case of condition $\overline{\nabla }J=0$, this formula has the following form, see [23]:

$$g(\left({\nabla }_{X}J\right)Y,Z)=\phi \left(Y\right)g(JX,Z)+\phi \left(Z\right)g(JX,Y)+\psi \left(Y\right)g(X,Z)+\psi \left(X\right)g(Y,Z)$$

(17)

for all $X,Y,Z\in TM$ and for some nonzero differentiable 1-forms $\varphi$ and $\psi$. The converse is true only in a local sense. Using the above, we can formulate the following.

Theorem 9.

Let a $2n$-dimensional Riemannian almost paracomplex manifold $(M,g,J)$ be biconformal to the product $(M_1\times M_2,g_1\oplus g_2)$ of two n-dimensional Riemannian manifolds. Then, its structural tensor J satisfies (17). The converse is true only in a local sense.

Remark 3.

Formula (17) is similar to (11). In particular, assuming $\phi =d\sigma$ and $\psi =J(\nabla \sigma )$, from (17), we obtain (11).

Recall that a distribution on a Riemannian manifold is totally umbilical if its second fundamental form is proportional to the metric restricted on the distribution, see [12] (p. 151). By the above, an orthogonal almost paracomplex structural $(g,J)$ is integrable and maximal integrable manifolds of its structural distributions H and V are totally umbilical submanifolds of $(M,g,J)$. The converse is also true, see [23].

Using (17), we have proved the integral formula, see [8,24], which for the case $m=2n$ can be rewritten as

$$\pi \left(M\right)=s\left(M\right)-\frac{n-1}{n}{\int }_M{\parallel {\nabla }^{*}J\parallel }^2\mathrm{d}{\mathrm{vol}}_g,$$

(18)

where ${\nabla }^{*}$ is the operator formally adjoint to ∇, and the norm of the tensor field ${\nabla }^{*}J$ is defined using g. From (18), we conclude that $\pi \left(M\right)\le s\left(M\right)$. In addition, for $\pi \left(M\right)=s\left(M\right)$, we obtain from (18) that ${\nabla }^{*}J=0$. In this case, both H and V have totally geodesic maximal integrable manifolds, see [8,24], and the Riemannian almost paracomplex manifold $(M,g,J)$ is locally the product of two n-dimensional Riemannian manifolds.

Using the above, we can formulate the following.

Theorem 10.

Let $(M,g,J)$ be a $2n$-dimensional Riemannian paracomplex manifold such that M is the product of two compact n-dimensional manifolds $M_1$ and $M_2$. If its metric g is obtained from the metric of the product $(M_1\times M_2,g_1\oplus g_2)$ of two n-dimensional Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$ by a biconformal deformation, then $\pi \left(M\right)\le s\left(M\right)$. Moreover, if $\pi \left(M\right)=s\left(M\right)$, then $(M,g)$ is locally isometric to $(M_1\times M_2,g_1\oplus g_2)$.

In [10], we proved a generalization of theorems [25] on two orthogonal complete totally umbilical foliations on a compact and oriented Riemannian manifold. In our case, this result has the following form.

Theorem 11.

Let $(M,g,J)$ be a $2n$-dimensional Riemannian paracomplex manifold such that M is the product of two n-dimensional manifolds $M_1$ and $M_2$, and let g be obtained from the metric of the product $(M_1\times M_2,g_1\oplus g_2)$ of two complete Riemannian manifolds $(M_1,g_1)$ and $(M_2,g_2)$ by the biconformal deformation $g=e^{2{\sigma }_1}{g}_1\oplus e^{2{\sigma }_2}{g}_2$. If $s\le \pi$ and

$$\parallel h_{*}(\nabla {\sigma }_1)+v_{*}(\nabla {\sigma }_2)\parallel \in L^1(M,g),$$

where $h_{*}:TM\to T{M}_1$ and $v_{*}:TM\to T{M}_2$ are natural projections and $\pi \ge s$ on M, then $(M,g)$ is locally isometric to $(M_1\times M_2,g_1\oplus g_2)$.