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Article

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

by
1,
Josef Mikeš
2,* and
Sergey Stepanov
3
1
Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel
2
Department of Algebra and Geometry, Palacky University, 77146 Olomouc, Czech Republic
3
Department of Mathematics, Finance University, 49-55, Leningradsky Prospect, 125468 Moscow, Russia
*
Author to whom correspondence should be addressed.
Mathematics 2021, 9(12), 1379; https://doi.org/10.3390/math9121379
Submission received: 12 May 2021 / Revised: 8 June 2021 / Accepted: 10 June 2021 / Published: 14 June 2021

## Abstract

:
A Riemannian almost paracomplex manifold is a 2n-dimensional Riemannian manifold $( M , g )$, whose structural group $O ( 2 n , R )$ is reduced to the form $O ( n , R ) × O ( n , R )$. We define the scalar curvature $π$ of this manifold and consider relationships between $π$ and the scalar curvature s of the metric g and its conformal transformations.
MSC:
53C15; 53C21

## 1. Introduction

An almost paracomplex structure on a $2 n$-dimensional smooth manifold M is a smooth field J of automorphisms of the tangent spaces, whose square is the identity operator ($J 2 = id T M$) and two eigenspaces (corresponding to eigenvalues $± 1$) 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 $G L ( 2 n , R )$ to the form $G L ( n , R ) × G L ( n , R )$, 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 $M 1 × M 2$ 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 $2 n$-dimensional Riemannian manifold $( M , g )$ admits an orthogonal almost paracomplex structure if its structure group $O ( 2 n , R )$ can be reduced to the form $O ( n , R ) × O ( n , R )$. 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 $π$ of a Riemannian almost paracomplex manifold $( M , g , J )$ and consider the relationship between $π$ 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 $J ∈ C ∞ ( T * M ⊗ T M )$ such that $J 2 = Id$ and $trace J = 0$, see [2]. As a result, the direct decomposition holds $T x M = H x ⊕ V x$, where $H x$ and $V x$ are horizontal and vertical subspaces of the tangent space $T x M$ at every point $x ∈ M$. The corresponding distributions $H = { H x }$ and $V = { V x }$ on M (i.e., subbundles of $T M$) 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
$g ( J X , J Y ) = g ( X , Y ) , X , Y ∈ T M ,$
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 $g ¯$ defined by
$g ¯ ( X , Y ) : = g ( X , Y ) + g ( J X , J Y ) , X , Y ∈ T M ,$
because $g ¯ ( J X , J Y ) = g ¯ ( X , Y )$, 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 $2 n$-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 $N J$ such that (e.g., [3])
$N J ( X , Y ) = [ X , Y ] + [ J X , J Y ] − J [ J X , Y ] − J [ X , J Y ] , X , Y ∈ T M ,$
where $[ · , · ]$ is the Lie bracket of vector fields. The tensor $N J$ is an analog of the Nijenhuis tensor for an almost complex structure on a smooth manifold of even dimension.
The equality $N J = 0$ 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 $R ( X , Y ) = ∇ X ∇ Y − ∇ Y ∇ X − ∇ [ X , Y ]$. Let $σ x$ be a plane in $T x M$, i.e., a two-dimensional subspace of $T x M$ at an arbitrary point $x ∈ M$. Choosing an orthonormal basis $X x , Y x$ of $σ x$, we define the sectional curvature $sec ( σ x )$ in direction of $σ x$ by
$sec ( σ x ) = R ( X x , Y x , X x , Y x ) ,$
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 $sec ( σ x )$. It is known that $R ( X x , Y x , X x , Y x )$ (the right-hand side) depends only on $σ x$, and not on the choice of the orthonormal basis $X x , Y x$. The scalar curvature s of the metric g is defined by
$s = ∑ i , j = 1 2 n sec ( e i , e j ) ,$
where ${ e 1 , … , e 2 n }$ is any orthonormal basis of $T x M$. On the other hand, if $X x , Y x$ is an orthonormal basis for $σ x$, then $J X x , J Y x$ is an orthonormal basis of another plane $σ x ′$ such that $σ x ′ = J σ x$. In this case, $σ x = J σ x ′ = J 2 σ x$. Therefore, given two J-invariant planes $σ x$ and $σ x ′$ in $T x M$, we can define the bisectional curvature $bisec ( σ x , σ x ′ )$ by the equality
$bisec ( σ x , J σ x ) = R ( X x , Y x , J X x , J Y x ) .$
One can verify that $R ( X x , Y x , J X x , J Y x )$ depends only on $σ x$ and $σ x ′$. 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 $π$ of an orthogonal paracomplex structure $( g , J )$, or, in other words, of a Riemannian almost paracomplex manifold $( M , g , J )$, defined by the equality
$π = ∑ i , j = 1 2 n R ( e i , e j , J e i , J e j )$
for a local orthonormal basis ${ e 1 , … , e 2 n }$ of $T M$. Let ${ e 1 , … , e n }$ and ${ e n + 1 , … , e 2 n }$ be local orthonormal bases of the horizontal distribution H and the vertical distribution V, respectively. Vectors of these bases satisfy the following conditions:
$J e a = − e a , J e α = e α$
for $a = 1 , … , n$ and $α = n + 1 , … , 2 n$. Using the above, we can show that
$π = ∑ i , j = 1 2 n R ( e i , e j , J e i , J e j ) = ∑ a , b = 1 n R ( e a , e b , J e a , J e b ) + 2 ∑ a = 1 n ∑ α = n + 1 2 n R ( e a , e α , J e a , J e α ) + ∑ α , β = n + 1 2 n R ( e α , e β , J e α , J e β ) = ∑ a , b = 1 n R ( e a , e b , e a , e b ) − 2 ∑ a = 1 n ∑ α = n + 1 2 n R ( e a , e α , e a , e α ) + ∑ α , β = n + 1 2 n R ( e α , e β , e α , e β ) = ∑ i , j = 1 2 n sec ( e i , e j ) − 4 ∑ a = 1 n ∑ α = n + 1 2 n sec ( e a , e α ) = s − 4 s mix ,$
where we denoted by
$s mix = ∑ α = n + 1 2 n ∑ a = 1 n sec ( e a , e α )$
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,
$s = π + 4 s mix ,$
where s is the scalar curvature of the metric g, and π and $s mix$ 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 $π = − 2 n$. In contrast, the scalar curvature of such metric g on M is $s = 2 n ( 2 n − 1 )$.
We consider three examples with the scalar curvature $π$ 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 $s mix = ( 1 / 8 ) ∥ ∇ J ∥ 2$, see [8], and by (3) we obtain
$π = s − ( 1 / 2 ) ∥ ∇ J ∥ 2 ≤ s .$
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, $M = M 1 × M 2$. If in addition, maximal integral manifolds of H and V are minimal submanifolds of $( M , g , J )$, then $s mix = − ( 1 / 8 ) ∥ ∇ J ∥ 2$, see [8], and by (3) we obtain
$π = s + ( 1 / 2 ) ∥ ∇ J ∥ 2 ≥ s .$
Example 3.
Let $( M , J , g )$ be a 2n-dimensional Riemannian almost paracomplex manifold. Assume that $∇ J = 0$ for the Levi-Civita connectionof 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, $s mix = 0$; therefore, $π = s$, see also [13]. In particular, the scalar curvature of an orthogonal paracomplex structure of $( S n × S n , g 0 ⊕ g 0 )$ can be expressed in terms of the scalar curvature of $g 0$ via the formula
$s ( g 0 ⊕ g 0 ) = s ( g 0 ) + s ( g 0 ) = 2 n ( n − 1 ) .$
Therefore, $π = s = 2 n ( n − 1 )$.
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 $D i ( i = 1 , 2 )$, see [12] (p. 146). Set $G = P 2 − P 1$ and define
$Π = ∑ i , j = 1 m R ( e i , e j , G e i , G e j )$
for a local orthonormal basis ${ e 1 , … , e m }$ of $T M$, compare with (2). Then, $g ( G X , G Y ) = g ( X , Y )$ for $X , Y ∈ T M$, compare with (1). Now, let ${ e 1 , … , e n 1 }$ be a local orthonormal basis of the distribution $D 1$ and ${ e n 1 + 1 , … , e m }$ be a local orthonormal basis of the distribution $D 2$. Vectors of these bases satisfy the following conditions:
$G e a = − e a ( a = 1 , … , n 1 ) , G e α = e α ( α = n 1 + 1 , … , m ) .$
Using the above, we can prove that (see [14])
$s = Π + 4 s mix .$
In particular, if $( M , g )$ has constant sectional curvature 1, then $s = m ( m − 1 )$ and $s mix = n 1 n 2$; hence, $Π = m ( m − 1 ) − 4 n 1 n 2$.

## 3. Conformal Transformations of Metrics of Riemannian Almost Paracomplex Manifolds

An identity map $id : M → M$ 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 $id ( x ) = x$ for any $x ∈ M$, 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 $id$ are equipped with metrics g and $g ¯$, respectively. The identity map $id : M → M$ is called a conformal transformation of the metric g if
$g ¯ = e 2 σ g$
for some smooth scalar function $σ$ on M, e.g., [6] (p. 115) and [7] (p. 269).
In this case, the metric $g ¯$ is called a conformal transformation of g; and if $σ = const$, then this transformation is called a homothety. The converse statement (i.e., g is a conformal transformation of $g ¯$) is also true, because the equality $g = e − 2 σ g ¯$ holds. In addition, the equality $g ¯ − 1 = e − 2 σ g − 1$ holds. For such a rescaled metric $g ¯$, there is a unique symmetric connection, $∇ ¯$, compatible with $g ¯$, i.e., $∇ ¯ g ¯ = 0$. Under a conformal transformation (4), the following relation (between two connections) holds, see [6] (p. 115) and [7] (p. 270):
$∇ ¯ X Y = ∇ X Y + X ( σ ) Y + Y ( σ ) X − g ( X , Y ) ∇ σ , X , Y ∈ C ∞ ( T M ) .$
We will consider conformal deformations of metrics of a Riemannian almost paracomplex manifold $( M , g , J )$. Obviously,
$g ¯ ( J X , J Y ) = e 2 σ g ( J X , J Y ) = e 2 σ g ( X , Y ) = g ¯ ( X , Y ) , X , Y ∈ C ∞ ( T M ) .$
Hence, a conformal deformation of g preserves the orthogonal decomposition $T M = H ⊕ V$ of the tangent bundle of $( M , g , J )$, i.e., it preserves the orthogonal almost paracomplex structure. On the other hand, a diffeomorphism $f : M → M$ 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 $id : M → M$, 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 $π$. Thus, below, we study the relationship between the scalar curvatures $π$ and $π ¯$ of orthogonal paracomplex structures $( g , J )$ and $( g ¯ , J )$, respectively. By the theory of conformal mappings, e.g., [7] (p. 271), the relationship between the curvature tensors (of the Levi-Civita connections ∇ and $∇ ¯$) of the metrics g and $g ¯$, respectively, has the following form, e.g., [6] (p. 115) and [7] (p. 271):
$e − 2 σ R ¯ l i j k = R l i j k + g l k σ i j − g l j σ i k + σ l k g i j − σ l j g i k + ( g l k g i j − g l j g i k ) ∥ d σ ∥ 2$
with respect to local coordinates $( x 1 , … , x 2 n )$, where $g ¯ i j$ and $g i j$ are components of metrics $g ¯$ and g. In (6), we denote by $R ¯ l i j k$ and $R l i j k$, the components of Riemannian curvature tensors $R ¯$ and R of metrics $g ¯$ and g, respectively. The components $σ i j$ in (6) are given by
$σ i j = ∇ i ∇ j σ − ( ∇ i σ ) ( ∇ j σ ) ,$
where $∇ i = ∇ ∂ / ∂ x i$. From (6), we obtain
$e 2 σ π ¯ = e − 2 σ R ¯ l i j k ( e 2 σ J h j g ¯ h l ) ( e 2 σ J p i g ¯ k p ) = R l i j k ( J h j g h l ) ( J p i g k p ) + 2 g i k σ i k + 2 n ∥ d σ ∥ 2 = π + 2 Δ σ + 2 ( n − 1 ) ∥ d σ ∥ 2 ,$
where $( g i j ) = ( g i j ) − 1$, $Δ σ = g i j ∇ i ∇ j σ$ and $Δ = div ∘ ∇$ is the Laplace–Beltrami operator. We can rewrite (7) as
$Δ σ = 1 2 ( e 2 σ π ¯ − π ) − ( n − 1 ) ∥ d σ ∥ 2 .$
The total scalar curvature $π ( M )$ of a compact Riemannian almost paracomplex manifold $( M , g , J )$ is defined by the integral equality
$π ( M ) = ∫ M π d vol g ,$
where $d vol g$ is the volume form of the metric g. Note that $π ( M )$ is an analog of the total scalar curvature of a compact Riemannian manifold $( M , g )$, see [16] (p. 119) and [9,14],
$s ( M ) = ∫ M s d vol g .$
Integrating (8) over M and using the Green’s formula $∫ M Δ σ d vol g = 0$ yields
$π ( M ) = ∫ M ( e 2 σ π ¯ − 2 ( n − 1 ) ∥ d σ ∥ 2 ) d vol g .$
The above integral equality yields the inequality
$π ( M ) ≤ ∫ M ( e 2 σ π ¯ ) d vol g .$
By the above inequality, if $π ¯ ≤ 0$ on M and $π ( M ) ≥ 0$, then $π ( M ) = 0$ and $π ¯ ≡ 0$. In this case, $σ = const$.
Theorem 2.
Let $( M , g , J )$ be a compact Riemannian almost paracomplex manifold with nonnegative total scalar curvature, $π ( M ) ≥ 0$, and let $g ¯ = e 2 σ g$ be another metric conformally related to g for some $σ ∈ C 2 ( M )$. If $π ¯ ≤ 0$ on M, then σ is constant. Thus, the conformal transformation of g to the metric $g ¯$ is a homothety; furthermore, $π ¯ = π = 0$ on M.
Setting $σ = 1 n − 1 ln u$ for a positive scalar function $u ∈ C 2 ( M )$, from (4) we obtain $g ¯ = u 2 / ( n − 1 ) g$ with $u > 0$. In this case, (6) can be rewritten as
$Δ u = n − 2 2 ( u n + 1 n − 1 π ¯ − u π ) .$
Integrating (9) over compact manifold M and using the Green’s formula, gives
$∫ M u n + 1 n − 1 π ¯ d vol g = ∫ M u π d vol g .$
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 $π ≤ 0$ on M, and let a metric $g ¯$ be conformally related to g. If $π ¯ ≥ 0$ on M, then the conformal deformation of the metric g to $g ¯$ is a homothety; furthermore, $π ¯ = π = 0$ on M.
Corollary 1.
Let $( M , J , g )$ be a compact Riemannian almost paracomplex manifold, and let a metric $g ¯$ be conformally related to g. If both orthogonal paracomplex structures $( g , J )$ and $( g ¯ , J )$ have nonvanishing scalar curvatures, i.e., $π ≠ 0$ and $π ¯ ≠ 0$ on M, then these scalar curvatures have the same sign.
If $π ¯ ≤ 0$ and $π ≥ 0$, then by (9) we obtain $Δ u ≤ 0$ on M. Thus, from (8), we conclude that $σ$ 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 $π ≥ 0$ on M, and let a metric $g ¯$ be conformally related to g. If $π ¯ ≤ 0$ on M, then the conformal deformation of the metric g to the metric $g ¯$ is a homothety; furthermore, $π ¯ = π = 0$ on M.
If $π ¯ ≥ 0$ and $π ≤ 0$ then $Δ u ≥ 0$ 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 $∫ M u p d vol g = ∞$ 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 $π ≤ 0$ on M, and let $g ¯$ be another metric conformally related to g by the formula $g ¯ = u 2 / ( n − 1 ) g$ for some positive function $u ∈ C 2 ( M )$. If $π ¯ ≥ 0$ on M and $u ∈ L p ( M , g )$ for some $p > 1$, then the conformal deformation of the metric g to the metric $g ¯$ is a homothety; furthermore, $π ¯ = π = 0$ on M.
A Riemannian manifold $( M , g )$ is locally conformally flat if for each point $x ∈ M$, there exists a neighborhood U of x and a smooth function $σ : U → R$ such that $( U , e 2 σ g )$ is flat, i.e., the curvature of the metric $e 2 σ g$ 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 $sec ( D x )$, the sectional curvature of a Riemannian manifold $( M , g )$ associated with an r-plane section $D x ⊂ T x M$ for an arbitrary point $x ∈ M$. Then, for any orthonormal basis ${ e 1 , … , e r }$ of $D x$, the scalar curvature $s ( D x )$ of the r-plane section $D x$ is defined by, see also [20],
$s ( D x ) = ∑ p , q = 1 r sec ( e p , e q ) .$
Now, let $( M , g )$ be a $2 r$-dimensional locally conformally flat manifold with vanishing scalar curvature s of the metric g, then $s ( D x ) = − s ( D x ⊥ )$, where $D x ⊥$ is the orthogonal complement of $D x$, see [21]. In the case of a $2 n$-dimensional Riemannian almost paracomplex manifold $( M , g , J )$, the scalar curvature s of the metric g can be presented as
$s = s ( H ) + 2 s mix + s ( V ) ,$
where $s ( H ) = ∑ a , b = 1 n sec ( e a , e b )$ and $s ( V ) = ∑ α , β = 1 n sec ( e α , e β )$ 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 $s ( H ) = − s ( V )$. In this case, from (10) we obtain $s mix = 0$. Thus, by our Theorem 1, $π = 0$. □
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 $s mix = 0$. Therefore, $π = 0$.

## 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: $M = M 1 × M 2$ and $g = e 2 σ ( g 1 ⊕ g 2 )$ for some n-dimensional Riemannian manifolds $( M 1 , g 1 )$ and $( M 2 , g 2 )$, respectively, and $σ ∈ C 2 ( M )$. In this case, the metric of $( M , J , g )$ arises as a result of the conformal transformation of the metric $g 1 ⊕ g 2$ 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 $( M 1 × M 2 , g 1 ⊕ g 2 )$ and the Levi-Civita connection $∇ ¯$ of its metric $g ¯ = g 1 ⊕ g 2$ such that $∇ ¯ g ¯ = 0$ and $∇ ¯ J = 0$ (see Example 3). Applying (5), we obtain the following relationship between the covariant derivatives $∇ ¯ J$ and $∇ J$:
$g ( ( ∇ ¯ X J ) Y , Z ) = g ( ( ∇ X J ) Y , Z ) − Y ( σ ) g ( J X , Z ) − Z ( σ ) g ( J X , Y ) + g ( J ( ∇ σ ) , Y ) g ( X , Z ) + g ( J ( ∇ σ ) , X ) g ( Y , Z ) , X , Y , Z ∈ T M .$
In the case of $∇ ¯ J = 0$, this formula has the following form:
$g ( ( ∇ X J ) Y , Z ) = Y ( σ ) g ( J X , Z ) + Z ( σ ) g ( J X , Y ) − g ( J ( ∇ σ ) , Y ) g ( X , Z ) − g ( J ( ∇ σ ) , X ) g ( Y , Z ) , X , Y , Z ∈ T M .$
The converse is true only in a local sense. By the above, we can formulate the following.
Theorem 7.
Let a $2 n$-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 $2 n$-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, $∇ J = 0$ and $π = s$ on $M = M 1 × M 2$. After the conformal deformation $g ¯ = e 2 σ ( g 1 ⊕ g 2 )$ for some $σ ∈ C 2 ( M )$ of the metric $g = g 1 ⊕ g 2$, we obtain the equation, see [7] (p. 271):
$e 2 σ s ¯ = s − 2 ( 2 n − 1 ) Δ σ − 2 ( n − 1 ) ( 2 n − 1 ) ∥ d σ ∥ 2 .$
for the scalar curvature $s ¯$ of the metric $g ¯ = e 2 σ ( g 1 ⊕ g 2 )$. We rewrite (6) as
$e 2 σ π ¯ = π + 2 Δ σ + 2 ( n − 1 ) ∥ d σ ∥ 2 .$
From (12) and (13), it follows that
$Δ σ = 1 4 n e 2 σ ( π ¯ − s ¯ ) − ∥ d σ ∥ 2 .$
Setting $σ = ln u$ for a positive scalar function $u ∈ C 2 ( M )$, the equality $g ¯ = e 2 σ g$ can be rewritten as $g ¯ = u 2 g$, $u > 0$. In this case, (14) can be rewritten as
$Δ u = 1 2 n u 2 ( π ¯ − s ¯ ) .$
If $M = M 1 × M 2$ 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:
$1 2 n ∫ M u 2 ( π ¯ − s ¯ ) d vol g = 0 .$
Note that conditions $π ¯ ≤ s ¯$ and $π ¯ < s ¯$ (or, $π ¯ ≥ s ¯$ and $π ¯ > s ¯$) for at least one point $x ∈ M 1 × M 2$ contradict (15). Thus, the following theorem holds.
Theorem 8.
Let $( M , g , J )$ be a $2 n$-dimensional Riemannian paracomplex manifold such that $M = M 1 × M 2$ for n-dimensional compact manifolds $M 1$ and $M 2$. If the scalar curvatures π and s satisfy the following condition: $π ≤ s$(resp., $π ¯ ≥ s ¯ )$ on M and $π < s$ (resp., $π ¯ > s ¯$) for at least one point $x ∈ M$, then M does not admit a metric $g ¯ = g 1 ⊕ g 2$ arising as a result of a conformal transformation of g.
Let $( M , J , g ¯ )$ be a $2 n$-dimensional integrable Riemannian almost paracomplex manifold with $M = M 1 × M 2$ and
$g = e 2 σ 1 g 1 ⊕ e 2 σ 2 g 2$
for some scalar functions $σ 1 , σ 2 ∈ C 2 ( M )$. In this case, (16) defines a biconformal deformation (see [22]) of the product metric $g ¯ = g 1 ⊕ g 2$ 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 $( M , g ¯ )$ such that $M = M 1 × M 2$ and $g ¯ = g 1 ⊕ g 2$, there is a unique symmetric connection, $∇ ¯$, compatible with $g ¯$ and J, i.e., $∇ ¯ g ¯ = 0$ and $∇ ¯ J = 0$. Applying (5), we can obtain a relationship between the covariant derivatives $∇ ¯ J$ and $∇ J$. In the case of condition $∇ ¯ J = 0$, this formula has the following form, see [23]:
$g ( ( ∇ X J ) Y , Z ) = φ ( Y ) g ( J X , Z ) + φ ( Z ) g ( J X , Y ) + ψ ( Y ) g ( X , Z ) + ψ ( X ) g ( Y , Z )$
for all $X , Y , Z ∈ T M$ and for some nonzero differentiable 1-forms $ϕ$ and $ψ$. The converse is true only in a local sense. Using the above, we can formulate the following.
Theorem 9.
Let a $2 n$-dimensional Riemannian almost paracomplex manifold $( M , g , J )$ be biconformal to the product $( M 1 × M 2 , g 1 ⊕ g 2 )$ 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 $φ = d σ$ and $ψ = J ( ∇ σ )$, 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 $m = 2 n$ can be rewritten as
$π ( M ) = s ( M ) − n − 1 n ∫ M ∥ ∇ * J ∥ 2 d vol g ,$
where $∇ *$ is the operator formally adjoint to ∇, and the norm of the tensor field $∇ * J$ is defined using g. From (18), we conclude that $π ( M ) ≤ s ( M )$. In addition, for $π ( M ) = s ( M )$, we obtain from (18) that $∇ * J = 0$. 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 $2 n$-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 $( M 1 × M 2 , g 1 ⊕ g 2 )$ of two n-dimensional Riemannian manifolds $( M 1 , g 1 )$ and $( M 2 , g 2 )$ by a biconformal deformation, then $π ( M ) ≤ s ( M )$. Moreover, if $π ( M ) = s ( M )$, then $( M , g )$ is locally isometric to $( M 1 × M 2 , g 1 ⊕ g 2 )$.
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 $2 n$-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 $( M 1 × M 2 , g 1 ⊕ g 2 )$ of two complete Riemannian manifolds $( M 1 , g 1 )$ and $( M 2 , g 2 )$ by the biconformal deformation $g = e 2 σ 1 g 1 ⊕ e 2 σ 2 g 2$. If $s ≤ π$ and
$∥ h * ( ∇ σ 1 ) + v * ( ∇ σ 2 ) ∥ ∈ L 1 ( M , g ) ,$
where $h * : T M → T M 1$ and $v * : T M → T M 2$ are natural projections and $π ≥ s$ on M, then $( M , g )$ is locally isometric to $( M 1 × M 2 , g 1 ⊕ g 2 )$.

## Author Contributions

Methodology, J.M., V.R. and S.S.; investigation, writing—review and editing. All authors have read and agreed to the published version of the manuscript.

## Funding

For the second author, this research was funded by the grant IGA PrF 2021030 at Palacky University in Olomouc.

Not applicable.

Not applicable.

Not applicable.

## Conflicts of Interest

The authors declare no conflict of interest.

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Rovenski, V.; Mikeš, J.; Stepanov, S. The Scalar Curvature of a Riemannian Almost Paracomplex Manifold and Its Conformal Transformations. Mathematics 2021, 9, 1379. https://doi.org/10.3390/math9121379

AMA Style

Rovenski V, Mikeš J, Stepanov S. The Scalar Curvature of a Riemannian Almost Paracomplex Manifold and Its Conformal Transformations. Mathematics. 2021; 9(12):1379. https://doi.org/10.3390/math9121379

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Rovenski, Vladimir, Josef Mikeš, and Sergey Stepanov. 2021. "The Scalar Curvature of a Riemannian Almost Paracomplex Manifold and Its Conformal Transformations" Mathematics 9, no. 12: 1379. https://doi.org/10.3390/math9121379

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