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

: A Riemannian almost paracomplex manifold is a 2 n -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.


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 = id TM ) 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 GL(2n, R) to the form GL(n, R) × GL(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 2n-dimensional Riemannian manifold (M, g) admits an orthogonal almost paracomplex structure if its structure group O(2n, 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.

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 ⊗ TM) 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 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 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 metricḡ defined bȳ 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 N J such that (e.g., [3]) 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, 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 . 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 where {e 1 , . . . , e 2n } 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 JX x , JY x is an orthonormal basis of another plane σ x such that σ x = Jσ x . In this case, σ x = Jσ 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 One can verify that R(X x , Y x , JX x , JY 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 for a local orthonormal basis {e 1 , . . . , e 2n } of TM. Let {e 1 , . . . , e n } and {e n+1 , . . . , 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: for a = 1, . . . , n and α = n + 1, . . . , 2n. Using the above, we can show that where we denoted by 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.
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).
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 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 Example 3. Let (M, J, g) be a 2n-dimensional Riemannian almost paracomplex manifold. Assume that ∇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 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 Therefore, π = s = 2n(n − 1).

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 andḡ, respectively. The identity map id : M → M is called a conformal transformation of the metric g ifḡ = e 2σ g for some smooth scalar function σ on M, e.g., [6] (p. 115) and [7] (p. 269).
Hence, a conformal deformation of g preserves the orthogonal decomposition TM = 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. 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 andπ of orthogonal paracomplex structures (g, J) and (ḡ, 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 andḡ, respectively, has the following form, e.g., [6] (p. 115) and [7] (p. 271): e −2σR lijk = R lijk + g lk σ ij − g lj σ ik + σ lk g ij − σ lj g ik + (g lk g ij − g lj g ik ) dσ 2 (6) with respect to local coordinates (x 1 , . . . , x 2n ), whereḡ ij and g ij are components of metrics g and g. In (6), we denote byR lijk and R lijk , the components of Riemannian curvature tensorsR and R of metricsḡ and g, respectively. The components σ ij in (6) are given by where (g ij ) = (g ij ) −1 , ∆ σ = g ij ∇ i ∇ j σ and ∆ = div • ∇ is the Laplace-Beltrami operator. We can rewrite (7) as The total scalar curvature π(M) of a compact Riemannian almost paracomplex manifold (M, g, J) is defined by the integral equality 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], By the above inequality, ifπ ≤ 0 on M and π(M) ≥ 0, then π(M) = 0 andπ ≡ 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 letḡ = e 2σ g be another metric conformally related to g for some σ ∈ C 2 (M). Ifπ ≤ 0 on M, then σ is constant. Thus, the conformal transformation of g to the metricḡ is a homothety; furthermore,π = π = 0 on M.
Theorem 3. Let (M, J, g) be a compact Riemannian almost paracomplex manifold with scalar curvature π ≤ 0 on M, and let a metricḡ be conformally related to g. Ifπ ≥ 0 on M, then the conformal deformation of the metric g toḡ 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 metricḡ be conformally related to g. If both orthogonal paracomplex structures (g, J) and (ḡ, J) have nonvanishing scalar curvatures, i.e., π = 0 andπ = 0 on M, then these scalar curvatures have the same sign.
Ifπ ≤ 0 and π ≥ 0, then by (9) we obtain ∆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 metricḡ be conformally related to g. Ifπ ≤ 0 on M, then the conformal deformation of the metric g to the metricḡ is a homothety; furthermore,π = π = 0 on M.
Ifπ ≥ 0 and π ≤ 0 then ∆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 letḡ be another metric conformally related to g by the formulā g = u 2/(n−1) g for some positive function u ∈ C 2 (M). Ifπ ≥ 0 on M and u ∈ L p (M, g) for some p > 1, then the conformal deformation of the metric g to the metricḡ 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 2σ g) is flat, i.e., the curvature of the metric e 2σ 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 ) = ∑ r p,q=1 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 where s(H) = ∑ n a,b=1 sec(e a , e b ) and s(V) = ∑ n α,β=1 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.

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 2σ (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 metricḡ = g 1 ⊕ g 2 such that∇ḡ = 0 and∇J = 0 (see Example 3). Applying (5), we obtain the following relationship between the covariant derivatives∇J and ∇J: In the case of∇J = 0, this formula has the following form: 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, ∇J = 0 and π = s on M = M 1 × M 2 . After the conformal deformationḡ = e 2σ (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): for the scalar curvatures of the metricḡ = e 2σ (g 1 ⊕ g 2 ). We rewrite (6) as From (12) and (13), it follows that Setting σ = ln u for a positive scalar function u ∈ C 2 (M), the equalityḡ = e 2σ g can be rewritten asḡ = u 2 g, u > 0. In this case, (14) can be rewritten as 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: Note that conditionsπ ≤s andπ <s (or,π ≥s andπ >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 2n-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 metricḡ = g 1 ⊕ g 2 arising as a result of a conformal transformation of g.
Let (M, J,ḡ) be a 2n-dimensional integrable Riemannian almost paracomplex manifold with M = M 1 × M 2 and g = e 2σ 1 g 1 ⊕ e 2σ 2 g 2 (16) for some scalar functions σ 1 , σ 2 ∈ C 2 (M). In this case, (16) defines a biconformal deformation (see [22]) of the product metricḡ = 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,ḡ) such that M = M 1 × M 2 andḡ = g 1 ⊕ g 2 , there is a unique symmetric connection,∇, compatible withḡ and J, i.e.,∇ḡ = 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]: for all X, Y, Z ∈ TM 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.

Remark 3. Formula
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 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 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 × 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.