1. Introduction
An
almost paracomplex structure on a
-dimensional smooth manifold
M is a smooth field
J of automorphisms of the tangent spaces, whose square is the identity operator (
) and two eigenspaces (corresponding to eigenvalues
) have dimension
n. In this case, the pair
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
to the form
, see [
1]. A
paracomplex manifold is an almost paracomplex manifold
such that the
G-structure defined by
J is integrable. A paracomplex manifold
is a
locally product manifold, i.e.,
M is locally diffeomorphic to the product
of two
n-dimensional manifolds. An almost paracomplex structure
J on a Riemannian manifold
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 -dimensional Riemannian manifold admits an orthogonal almost paracomplex structure if its structure group can be reduced to the form . A Riemannian manifold with an orthogonal paracomplex structure will be called a Riemannian almost paracomplex manifold and denoted by .
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 of a Riemannian almost paracomplex manifold and consider the relationship between 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
such that
and
, see [
2]. As a result, the direct decomposition holds
, where
and
are
horizontal and
vertical subspaces of the tangent space
at every point
. The corresponding distributions
and
on
M (i.e., subbundles of
) have equal dimensions and correspond to the eigenvalues
and
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
is called
orthogonal, see [
1,
2], if
and it is denoted by
. In this case, the distributions
H and
V of
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
defined by
because
, see (
1). The triplet
, where
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 -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
such that (e.g., [
3])
where
is the Lie bracket of vector fields. The tensor
is an analog of the
Nijenhuis tensor for an almost complex structure on a smooth manifold of even dimension.
The equality
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
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
be a Riemannian almost paracomplex manifold with the Levi-Civita connection ∇ of the metric g and the Riemannian curvature tensor
. Let
be a plane in
, i.e., a two-dimensional subspace of
at an arbitrary point
. Choosing an orthonormal basis
of
, we define the sectional curvature
in direction of
by
where
. We shall write also
for
. It is known that
(the right-hand side) depends only on
, and not on the choice of the orthonormal basis
. The
scalar curvature s of the metric
g is defined by
where
is any orthonormal basis of
. On the other hand, if
is an orthonormal basis for
, then
is an orthonormal basis of another plane
such that
. In this case,
. Therefore, given two
J-invariant planes
and
in
, we can define the
bisectional curvature by the equality
One can verify that
depends only on
and
. 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
of an orthogonal paracomplex structure
, or, in other words, of a Riemannian almost paracomplex manifold
, defined by the equality
for a local orthonormal basis
of
. Let
and
be local orthonormal bases of the horizontal distribution
H and the vertical distribution
V, respectively. Vectors of these bases satisfy the following conditions:
for
and
. Using the above, we can show that
where we denoted by
the
mixed scalar curvature of an orthogonal paracomplex structure
. 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 be a Riemannian almost paracomplex manifold. Then,where s is the scalar curvature of the metric g, and π and are the scalar and mixed scalar curvatures, respectively, of its orthogonal paracomplex structure . By (
3), if the metric of
has constant sectional curvature 1, then
. In contrast, the scalar curvature of such metric
g on
M is
.
We consider three examples with the scalar curvature 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 are totally geodesic, then , see [8], and by (3) we obtain 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 be a Riemannian paracomplex manifold, then M is locally the product of two n-dimensional manifolds, . If in addition, maximal integral manifolds of H and V are minimal submanifolds of , then , see [8], and by (3) we obtain Example 3. Let be a 2n-dimensional Riemannian almost paracomplex manifold. Assume that 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 is locally the product of two n-dimensional Riemannian manifolds and . The converse is also true. In this case, ; therefore, , see also [13]. In particular, the scalar curvature of an orthogonal paracomplex structure of can be expressed in terms of the scalar curvature of via the formula Therefore, .
Remark 2. Theorem 1 can be extended for an almost product structure on an m-dimensional Riemannian manifold . Namely, let be orthoprojectors on two complementary orthogonal distributions , see [12] (p. 146). Set and definefor a local orthonormal basis of , compare with (2). Then, for , compare with (1). Now, let be a local orthonormal basis of the distribution and be a local orthonormal basis of the distribution . Vectors of these bases satisfy the following conditions: Using the above, we can prove that (see [14]) In particular, if has constant sectional curvature 1, then and ; hence, .
3. Conformal Transformations of Metrics of Riemannian Almost Paracomplex Manifolds
An identity map
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
for any
, 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
are equipped with metrics
g and
, respectively. The identity map
is called a conformal transformation of the metric
g if
for some smooth scalar function
on
M, e.g., [
6] (p. 115) and [
7] (p. 269).
In this case, the metric
is called a
conformal transformation of
g; and if
, then this transformation is called a
homothety. The converse statement (i.e.,
g is a conformal transformation of
) is also true, because the equality
holds. In addition, the equality
holds. For such a rescaled metric
, there is a unique symmetric connection,
, compatible with
, i.e.,
. Under a conformal transformation (
4), the following relation (between two connections) holds, see [
6] (p. 115) and [
7] (p. 270):
We will consider conformal deformations of metrics of a Riemannian almost paracomplex manifold
. Obviously,
Hence, a conformal deformation of
g preserves the orthogonal decomposition
of the tangent bundle of
, i.e., it preserves the orthogonal almost paracomplex structure. On the other hand, a diffeomorphism
is called a
paraholomorhic transformation of
, if it preserves the almost paracomplex structure
J, see [
1]. Therefore, we have the following.
Proposition 1. Let be a Riemannian almost paracomplex manifold. Then, the conformal transformation of metric , see (4), represents a paraholomorphic transformation of . On the contrary, a conformal transformation of the metric of a Riemannian almost paracomplex manifold
does not preserve its scalar curvature
. Thus, below, we study the relationship between the scalar curvatures
and
of orthogonal paracomplex structures
and
, 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
, respectively, has the following form, e.g., [
6] (p. 115) and [
7] (p. 271):
with respect to local coordinates
, where
and
are components of metrics
and
g. In (
6), we denote by
and
, the components of Riemannian curvature tensors
and
R of metrics
and
g, respectively. The components
in (
6) are given by
where
. From (
6), we obtain
where
,
and
is the Laplace–Beltrami operator. We can rewrite (
7) as
The
total scalar curvature of a compact Riemannian almost paracomplex manifold
is defined by the integral equality
where
is the volume form of the metric
g. Note that
is an analog of the
total scalar curvature of a compact Riemannian manifold
, see [
16] (p. 119) and [
9,
14],
Integrating (
8) over
M and using the Green’s formula
yields
The above integral equality yields the inequality
By the above inequality, if on M and , then and . In this case, .
Theorem 2. Let be a compact Riemannian almost paracomplex manifold with nonnegative total scalar curvature, , and let be another metric conformally related to g for some . If on M, then σ is constant. Thus, the conformal transformation of g to the metric is a homothety; furthermore, on M.
Setting
for a positive scalar function
, from (
4) we obtain
with
. In this case, (
6) can be rewritten as
Integrating (
9) over compact manifold
M and using the Green’s formula, gives
We can formulate the following theorem supplementing Theorem 2.
Theorem 3. Let be a compact Riemannian almost paracomplex manifold with scalar curvature on M, and let a metric be conformally related to g. If on M, then the conformal deformation of the metric g to is a homothety; furthermore, on M.
Corollary 1. Let be a compact Riemannian almost paracomplex manifold, and let a metric be conformally related to g. If both orthogonal paracomplex structures and have nonvanishing scalar curvatures, i.e., and on M, then these scalar curvatures have the same sign.
If
and
, then by (
9) we obtain
on
M. Thus, from (
8), we conclude that
is a
superharmonic function. On the other hand, a complete Riemannian manifold
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
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 be a parabolic Riemannian almost paracomplex manifold (in particular, be a complete manifold of finite volume) with scalar curvature on M, and let a metric be conformally related to g. If on M, then the conformal deformation of the metric g to the metric is a homothety; furthermore, on M.
If
and
then
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
, then
for any
, 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 be a complete Riemannian almost paracomplex manifold with scalar curvature on M, and let be another metric conformally related to g by the formula for some positive function . If on M and for some , then the conformal deformation of the metric g to the metric is a homothety; furthermore, on M.
A Riemannian manifold is locally conformally flat if for each point , there exists a neighborhood U of x and a smooth function such that is flat, i.e., the curvature of the metric vanishes on U. In the case of a Riemannian almost paracomplex manifold , we can formulate the following.
Theorem 6. Let 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
, the sectional curvature of a Riemannian manifold
associated with an
r-plane section
for an arbitrary point
. Then, for any orthonormal basis
of
, the scalar curvature
of the
r-plane section
is defined by, see also [
20],
Now, let
be a
-dimensional locally conformally flat manifold with vanishing scalar curvature s of the metric
g, then
, where
is the orthogonal complement of
, see [
21]. In the case of a
-dimensional Riemannian almost paracomplex manifold
, the scalar curvature
s of the metric
g can be presented as
where
and
are scalar curvatures of the horizontal and vertical distributions. Moreover, if
is a locally conformally flat manifold with vanishing scalar curvature
s of the metric
g, then
. In this case, from (
10) we obtain
. Thus, by our Theorem 1,
. □
For example (see [
16]) [p. 61], the product of two Riemannian manifolds
and
, one with sectional curvature 1, and the other with sectional curvature
, is locally conformally flat. In particular, if
, then we have
and
. Therefore,
.
4. A Riemannian Almost Paracomplex Manifold Conformally Related to the Product of Riemannian Manifolds
Let a 2
n-dimensional Riemannian almost paracomplex manifold
satisfy the following conditions:
and
for some
n-dimensional Riemannian manifolds
and
, respectively, and
. In this case, the metric of
arises as a result of the
conformal transformation of the metric
of the product of Riemannian manifolds
and
. At the same time, there exists a natural integrable orthogonal paracomplex structure
J of
and the Levi-Civita connection
of its metric
such that
and
(see Example 3). Applying (
5), we obtain the following relationship between the covariant derivatives
and
:
In the case of
, 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 -dimensional Riemannian almost paracomplex manifold be conformal to the product of n-dimensional Riemannian manifolds and , then the structural tensor J satisfies (11). The converse is true only in a local sense. Let a
-dimensional Riemannian almost paracomplex manifold
be the product of
n-dimensional Riemannian manifolds
and
. In this case,
and
on
. After the conformal deformation
for some
of the metric
, we obtain the equation, see [
7] (p. 271):
for the scalar curvature
of the metric
. We rewrite (
6) as
From (
12) and (
13), it follows that
Setting
for a positive scalar function
, the equality
can be rewritten as
,
. In this case, (
14) can be rewritten as
If
is a compact manifold (in particular, if
and
are compact manifolds), then from from the above formula we obtain the following integral equation:
Note that conditions
and
(or,
and
) for at least one point
contradict (
15). Thus, the following theorem holds.
Theorem 8. Let be a -dimensional Riemannian paracomplex manifold such that for n-dimensional compact manifolds and . If the scalar curvatures π and s satisfy the following condition: (resp., on M and (resp., ) for at least one point , then M does not admit a metric arising as a result of a conformal transformation of g.
Let
be a
-dimensional integrable Riemannian almost paracomplex manifold with
and
for some scalar functions
. In this case, (
16) defines a
biconformal deformation (see [
22]) of the product metric
on the product of
n-dimensional Riemannian manifolds
and
. At the same time, for a Riemannian manifold
such that
and
, there is a unique symmetric connection,
, compatible with
and
J, i.e.,
and
. Applying (
5), we can obtain a relationship between the covariant derivatives
and
. In the case of condition
, this formula has the following form, see [
23]:
for all
and for some nonzero differentiable 1-forms
and
. The converse is true only in a local sense. Using the above, we can formulate the following.
Theorem 9. Let a -dimensional Riemannian almost paracomplex manifold be biconformal to the product 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 and , 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
is integrable and maximal integrable manifolds of its structural distributions
H and
V are
totally umbilical submanifolds of
. The converse is also true, see [
23].
Using (
17), we have proved the integral formula, see [
8,
24], which for the case
can be rewritten as
where
is the operator formally adjoint to ∇, and the norm of the tensor field
is defined using
g. From (
18), we conclude that
. In addition, for
, we obtain from (
18) that
. In this case, both
H and
V have totally geodesic maximal integrable manifolds, see [
8,
24], and the Riemannian almost paracomplex manifold
is locally the product of two
n-dimensional Riemannian manifolds.
Using the above, we can formulate the following.
Theorem 10. Let be a -dimensional Riemannian paracomplex manifold such that M is the product of two compact n-dimensional manifolds and . If its metric g is obtained from the metric of the product of two n-dimensional Riemannian manifolds and by a biconformal deformation, then . Moreover, if , then is locally isometric to .
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 be a -dimensional Riemannian paracomplex manifold such that M is the product of two n-dimensional manifolds and , and let g be obtained from the metric of the product of two complete Riemannian manifolds and by the biconformal deformation . If andwhere and are natural projections and on M, then is locally isometric to .