We study the reduction procedure applied to pseudo-Kähler manifolds by a one dimensional Lie group acting by isometries and preserving the complex tensor. We endow the quotient manifold with an almost contact metric structure. We use this fact to connect pseudo-Kähler homogeneous structures with almost contact metric homogeneous structures. This relation will have consequences in the class of the almost contact manifold. Indeed, if we choose a pseudo-Kähler homogeneous structure of linear type, then the reduced, almost contact homogeneous structure is of linear type and the reduced manifold is of type of Chinea-González classification.
First, Ambrose and Singer  gave a tensorial approach to study homogeneous Riemannian manifolds by the so-called homogeneous structure S. Later, Kiričenko  extended this approach to homogeneous Riemannian manifolds with additional geometric structures. Furthermore, references [3,4,5] decomposed the space of tensors S in cases of the additional geometry being purely Riemannian, Kähler or almost contact metric, respectively.
Nowadays, the application of homogeneous structures is a main tool in the investigation of homogeneous manifolds; see [3,5], among others. Moreover, homogeneous manifolds are a central object for many mathematical models of physical theories (for example, linear degenerate homogeneous structures are related to homogeneous plane waves; cf. ). This is specially relevant when the space is also equipped with additional geometry, such as contact or Kähler. Nevertheless, the scare knowledge about the relationships between homogeneous structures when there is a map between homogeneous manifolds is remarkable. An example of this is the reduction procedure of homogeneous structures, which was first introduced in , wherein, in particular, the authors reduced pseudo-Riemannian almost contact homogeneous structures to pseudo-Riemannian, almost-Hermitian homogeneous structures.
In this paper, we examine the reduction procedure of a pseudo-Kähler homogeneous manifold to almost contact metric homogeneous manifolds by one dimensional fibers. We show that the almost contact metric manifold is of type of Chinea-González classification (cf. ).
Some of the most important cases of homogeneous structures are those ones of linear type, in which the dimension of the class grows linearly with the dimension of the manifold; see ( Chapter 5). Linear classes always provide, in the different geometries with which they have been studied, interesting results, from the characterization of negative constant curvature (cf.  Theorem 5.2) in Riemannian manifolds, to other surprising facts in other geometries (see  for a survey). In our work, we show that if the Kähler manifold has a homogeneous structure of linear type, then the reduced homogeneous structure is of linear type. Furthermore, as the homogeneous structures of almost contact metric manifolds are related with the covariant derivative of the fundamental 2-form associated to it, we prove that the reduced manifold by a homogeneous linear structure is of type of Chinea-González classification (cf. ). Besides that, if the one-dimensional Lie group is proportional to the sum of the two vectors that defines each projection to the subspaces of the linear homogeneous structure, then the manifold is Sasakian. Moreover, the reduced manifold is cosymplectic if the sum is zero.
2.1. Homogeneous Structures
For a general non-metric framework of homogeneous structures, see . Here, we will focus on the well known notion of pseudo-Riemannian Homogeneous structures that we summarize as follows (see  and the references therein for a detailed description). Let be a connected pseudo-Riemannian manifold and let ∇ be the Levi–Civita connection. A -tensor field S in M is said to be a pseudo-Riemannian homogeneous structure if and only if it satisfies
where and R is the curvature form of ∇. A manifold is reductive and locally homogeneous if and only if it is endowed with a pseudo-Riemannian homogeneous structure (cf. ). It is sometimes more convenient to work with -tensors instead of -tensors with the relation
For the sake of simplicity, both tensor fields will be denoted the same. If we fix a point and we consider , the condition implies that belongs to the space
The decomposition of this space into irreducible components under the action of the orthogonal group provides a set of classes such that belongs to the same class for every choice of the point p. If the manifold is also equipped with an additional geometric structure defined by a tensor K, then the condition must be included in (1) to characterize the reductive local homogeneity of the manifold under (local) transformations preserving both g and K. We apply this situation to two instances:
Let be an almost pseudo-Hermitian manifold, that is, a pseudo-Riemannian manifold equipped with a -tensor J that is a point-wise isometry. A pseudo-Hermitian homogeneous structure S is a -tensor satisfying (1) and . If in addition is Kähler (that is, ), and we fix a point , , the linear space of tensors to be considered is
obtained by implementing the condition to . It is proven in [4,9] that the space of pseudo-Kähler homogeneous structures decomposes in four mutually orthogonal and irreducible -submodules
Let be an almost contact metric manifold, that is (for example, see ), a pseudo-Riemannian manifold equipped with a -tensor and a vector field (the 1-form being its dual with respect to g) such that
where . Then, S is a almost contact metric homogeneous structure if and only if (1) is satisfied together with . This implies that also and . The equation is equivalent to , but this condition cannot be easily implemented into the definition of the pointwise space of tensors , , since the Levi–Civita connection depends on the metric and the first derivatives of the metric. However, we can still split this space of linear tensors under the group . Recall that this group characterizes the canonical almost metric structure of defined by and
being the standard complex structure of . That is, is the subgroup of or (depending on the value of ) stabilizing both and . Then, decomposes into two mutually orthogonal submodules,
Additionally, these two submodules decompose in mutually orthogonal and irreducible -submodules
where the first classification is given in ( Prop. 4.2.10) and the second in . See the Appendix A for their expressions. Cosymplectic manifolds are an important subcase of almost contact metric manifolds. They are characterized by the additional condition . Hence, a homogeneous structure S belongs to if and only if the manifold is cosymplectic.
2.2. Reduction of a Homogeneous Structure
Let be a (left) G-principal bundle, where is a pseudo-Riemannian manifold with metric , and the fibers are non-degenerate with respect to . Suppose that G acts on by isometries.
Given , we consider the vertical subspace at and its orthogonal complement with respect to . As G acts by isometries, the decomposition
is a principal G-connection. This connection is sometimes called a mechanical connection for its relevant role in some problems in geometric mechanics (see ). Furthermore, there is a unique pseudo-Riemannian metric g in M such that the restriction is an isometry for every . Obviously, the metric g satisfies
where and denote the horizontal lift of X and Y with respect to the mechanical connection.
Let be the Levi–Civita connection and be a pseudo-Riemannian homogeneous structure on , invariant under the action of the structure group G. Assume that there is 1-form taking values in such that
Then, by ( Theorem 3.7), the reduced tensor field S on M defined by
is a pseudo-Riemannian homogeneous structure of .
3. Fibrations of Pseudo-Hermitian over Almost Contact Metric Manifolds
Let be a nowhere-vanishing vector field in . Around any point there exists a coordinate system , , in a neighborhood diffeomorphic to such that any integral curve of is given by . The vector field is said to be regular if the domains can be always chosen such that any orbit of intersects them at most once. For regular vector fields, the orbit space M is a smooth manifold and the projection a submersion (cf. ). Furthermore, a regular vector field is said to be strictly regular if all the orbits are diffeomorphic. If is a complete, strictly regular vector field, the one-parametric group G generated by ( or ) acts freely on , and is a G-principal bundle. If we further assume that is equipped with a pseudo-Riemannian metric such that (that is, is non-degenerate so that we can normalize it) and is invariant by the group G, then the 1-form
where is a G-principal connection form in , a mechanical connection, as we mentioned above.
Let be an almost pseudo-Hermitian manifold and let be a complete strictly regular unit vector field (). We consider that both and are invariant with respect to the uniparametric group G defined by θ. Then, the orbit space is an almost contact metric manifold, with
for any , where stands for the horizontal lift with respect to the mechanical connection, and η is the dual form of ξ; that is, .
As acts by preserving the metric and the complex structure tensor, we have that the tensors given in (5) are well defined. In addition, , being orthogonal to , is horizontal with respect to the mechanical connection and . We have to check that
If , then , which means that is an horizontal vector. Then,
With respect to the second equation, given , we denote by the orthogonal part of X with respect to . Note that, since , the vector is horizontal. Then
and the proof is complete. □
On top of the structure on the reduced manifold provided in the previous result, it is easy to check that the Levi–Civita connection on M associated with g is characterized by the condition
Associated with an almost contact metric manifold , the canonical 2-form is defined to be . Since
for any metric connection , the manifold M is cosymplectic if and only if . For non-cosymplectic manifolds, if we choose a point , , the 3-tensor belongs to the space
defined above. The classification of almost contact metric manifold in a category other than cosymplectic (Sasaki, trans-Sasaki, Kenmotsu, etc.; see ) is equivalent to belonging to different combinations of the irreducible subspaces in which decomposes. These subspaces can be organized in a less coarse classification
and defined by the expressions
In the conditions of Theorem 1, if is a Kähler manifold, then the quotient space is an almost contact manifold such that belongs to the class .
For we have
Since , the first and third terms of the last step above vanish and we get
We now assume that the pseudo-Hermitian manifold is equipped with a homogeneous structure tensor , which in addition is invariant under the action of the group G.
Let be an almost pseudo-Kähler manifold equipped with an almost pseudo-Kähler homogeneous structure that is invariant under the group flow G of a complete strictly regular unit vector field θ. Suppose that
where and β is a 1-form on M. Then, the tensor field S on the orbit space defined by
is a homogeneous, almost contact metric structure on belonging to the class . Furthermore, the components and of S are
Since the condition is equivalent to , we are working in the conditions explained in Section 2.2 so that S is a pseudo-Riemannian homogeneous structure. To show that S is an almost contact metric homogeneous structure, we have to prove that , where . Let be two vector fields,
so that is horizontal. Then projects to . Following the same steps in Proposition 1, we get
which implies that , while again taking into consideration the fact that is horizontal.
Now, we decompose , and we get
Since is a pseudo-Kähler homogeneous structure,
which implies that . Finally, it is a matter of direct checking that and in the statement satisfy and . □
Let be a pseudo-Kähler manifold equipped with a pseudo-Kähler homogeneous structure invariant under the flow group G of a complete strictly regular unit vector field θ. Assume that belongs to the class , parametrized by G-invariant vector fields and . Then, the component of the reduced homogeneous almost-contact metric structure S of the almost contact metric manifold belongs to with projections
for , where .
In the expression
of the component of S, we apply that , defined by vector fields and (see Theorem A1), and we have
where , . As we get
One can easily check from expression given in the Theorem A3 that first, second and third lines of the last equality belong to , and respectively. □
Recall that the Ambrose–Singer condition is equivalent to . Hence
If in addition belongs to , then
Under the conditions of Theorem 3, the covariant derivative of the fundamental form of the almost-contact metric manifold belongs to with components
for , where .
Moreover, the manifold is cosymplectic if and only if .
From Theorem 2, the part of the reduced homogeneous structure S in belongs to and we can apply (8). By means of a straightforward computation, one can show that the components , and provide the expressions , and in the statement, which belong to , and respectively.
Finally, the three components of vanish if and only if and , that is, . □
Let be a pseudo- Kähler manifold equipped with a pseudo-Kähler homogeneous structure of the class defined by vector fields and . Suppose that is a complete strictly regular vector field, and let G be its flow group. Then the orbit manifold is Sasakian.
The vector fields and satisfy (cf. ) so that satisfies the conditions of Theorem 2. From Proposition 2, since , we have that belongs to the class , which is equivalent to being Sasakian (). □
Let be a pseudo-Kähler manifold equipped with a pseudo-Kähler homogeneous structure invariant under the flow group G of a complete strictly regular unit vector field θ. Assume that belongs to the class , parametrized by G-invariant vector fields and . Then, the component of the reduced homogeneous almost-contact metric structure S of the almost contact metric manifold belongs to , and its expression is
for , where , and , .
In the expression,
of the component of S, we apply the fact that , defined by vector fields and (see Theorem A1), and we have
We use the fact that , , and we get
Finally, taking into account that , , , we get the given expression. □
Let be a pseudo-Kähler manifold equipped with a pseudo-Kähler homogeneous structure invariant under the flow group G of a complete strictly regular unit vector field θ. Assume that belongs to the class . Then the reduced homogeneous almost-contact metric structure S of the almost contact metric manifold is of linear type.
This result is a consequence of Theorems 3 and 4. □
Since Ambrose–Singer  to the present, homogeneous structure tensors have been a main object in the study of the homogeneous manifolds with very interesting applications to other fields in differential geometry and theoretical physics, for example, those derived from linear structures. This paper is a contribution in those two lines of work:
We showed a reduction procedure by one dimensional fibers between almost pseudo-Hermitian manifolds to almost contact metric manifolds of general type in the sense of Chinea-González (cf. ).
We applied this fibration result to the case of homogeneous structures. In this context we got a reduction result between pseudo-Kähler homogeneous structures to almost contact metric homogeneous structures. Moreover, the reduced manifold lies in of the Chinea-González classification (cf. ).
We proved that the reduction procedure sends pseudo-Kähler homogeneous structures of linear type to almost contact metric homogeneous structures of linear type. Indeed, we showed the explicit expressions of the reduced homogeneous structure and gave a characterization of the reduced manifold being cosymplectic and Sasakian.
The study of homogeneous structures of linear type is connected with models of singular plane waves in general relativity (see  for the real case and  for the pseudo-Kähler setting explored in this work). The models associated with the particular instances that arose from our results, along with some other examples, will be the topic of future research.
Furthermore, we can conclude once more that homogeneous structures have an important and restrictive influence on the geometric structures of the manifold (for example, being xosymplectic or Sasakian). In our opinion, that makes homogeneous structures a fruitful main tool in the study of homogeneous manifolds.
All authors contributed equally and significantly to writing this article. All authors have read and agreed to the published version of the manuscript.
Both authors have been partially supported by MICINN (Spain) under grant number PGC2018-098321-B-I00. M.C.L. has been partially supported by Junta de Castilla y León (Spain) under grant number SA090G19. J.L.C.J. acknowledge financial support from “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554) and from the Spanish National Research Council, through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205- CEX001).
Conflicts of Interest
The authors declare no conflict of interest.
Appendix A. Expressions of Homogeneous Structures
We now give the explicit expressions of the classes given in (2)–(4) and their corresponding classification theorems.
().If , the space is decomposed into mutually orthogonal and irreducible -submodules as
for an orthonormal basis .
().If , the space decomposes into irreducible and mutually orthogonal -submodules as
for the orthogonal complement to ξ, an orthonormal basis of and .
().If , the space decomposes into irreducible and mutually orthogonal -submodules as
for the orthogonal complement to ξ, an orthonormal basis of and .
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