Reduction of Homogeneous Pseudo-Kähler Structures by One-Dimensional Fibers

: 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 C 5 ⊕ C 6 ⊕ C 12 of Chinea-González classiﬁcation.


Introduction
First, Ambrose and Singer [1] gave a tensorial approach to study homogeneous Riemannian manifolds by the so-called homogeneous structure S. Later, Kiričenko [2] 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. [6]). 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 [7], 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 C 5 ⊕ C 6 ⊕ C 7 ⊕ C 8 ⊕ C 9 ⊕ C 10 ⊕ C 12 of Chinea-González classification (cf. [8]).
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 ( [5] 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. [3] Theorem 5.2) in Riemannian manifolds, to other surprising facts in other geometries (see [9] 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 C 5 ⊕ C 6 ⊕ C 12 of Chinea-González classification (cf. [8]). Besides that, if the one-dimensional Lie group is proportional to the sum of the two vectors that defines each projection to the subspaces K 2 ⊕ K 4 of the linear homogeneous structure, then the manifold is Sasakian. Moreover, the reduced manifold is cosymplectic if the sum is zero.

Homogeneous Structures
For a general non-metric framework of homogeneous structures, see [10]. Here, we will focus on the well known notion of pseudo-Riemannian Homogeneous structures that we summarize as follows (see [5] and the references therein for a detailed description). Let (M, g) be a connected pseudo-Riemannian manifold and let ∇ be the Levi-Civita connection. A (1, 2)-tensor field S in M is said to be a pseudo-Riemannian homogeneous structure if and only if it satisfies where ∇ = ∇ − S and R is the curvature form of ∇. A manifold (M, g) is reductive and locally homogeneous if and only if it is endowed with a pseudo-Riemannian homogeneous structure (cf. [11]). It is sometimes more convenient to work with (0, 3)-tensors instead of (1, 2)-tensors with the relation For the sake of simplicity, both tensor fields will be denoted the same. If we fix a point p ∈ M and we consider V = T p M, the condition ∇g = 0 implies that S p 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 S p belongs to the same class for every choice of the point p. If the manifold (M, g) is also equipped with an additional geometric structure defined by a tensor K, then the condition ∇K = 0 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 (M, g, J) be an almost pseudo-Hermitian manifold, that is, a pseudo-Riemannian manifold equipped with a (1, 1)-tensor J that is a point-wise isometry. A pseudo-Hermitian homogeneous structure S is a (1, 2)-tensor satisfying (1) and ∇J = 0. If in addition (M, g, J) is Kähler (that is, ∇J = 0), and we fix a point p ∈ M, V = T p M, the linear space of tensors to be considered is obtained by implementing the condition ∇J = −SJ = 0 to S(V). It is proven in [4,9] that the space K(V) of pseudo-Kähler homogeneous structures decomposes in four mutually orthogonal and irreducible U(p, q)-submodules Their expressions can be found in the Appendix A.
• Let (M, g, φ, ξ, η) be an almost contact metric manifold, that is (for example, see [12]), a pseudo-Riemannian manifold (M, g) equipped with a (1, 1)-tensor φ and a vector field ξ (the 1-form η being its dual with respect to g) such that where ε = g(ξ, ξ). Then, S is a almost contact metric homogeneous structure if and only if (1) is satisfied together with ∇φ = 0. This implies that also ∇ξ = 0 and ∇η = 0. The equation ∇φ = 0 is equivalent to ∇φ = [S, φ], but this condition cannot be easily implemented into the definition of the pointwise space of tensors S(V), V = T p M, 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 U(r, s) × 1. Recall that this group characterizes the canonical almost metric structure of R 2n+1 defined by ξ 0 = e 2n+1 and φ 0 = J 0 0 0 0 , J 0 being the standard complex structure of R 2n . That is, U(r, s) × 1 is the subgroup of O(2r + 1, 2s) or O(2r, 2s + 1) (depending on the value of ε) stabilizing both ξ 0 and φ 0 . Then, S(V) decomposes into two mutually orthogonal U(r, s) × 1 submodules, Additionally, these two submodules decompose in mutually orthogonal and irreducible U(p, q) × 1-submodules where the first classification is given in ( [5] Prop. 4.2.10) and the second in [8]. 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 ∇φ = 0. Hence, a homogeneous structure S belongs to S + (V) if and only if the manifold is cosymplectic.

Reduction of a Homogeneous Structure
Let π :M −→ M be a (left) G-principal bundle, whereM is a pseudo-Riemannian manifold with metricḡ, and the fibers are non-degenerate with respect toḡ. Suppose that G acts onM by isometries.
Givenp, we consider VpM the vertical subspace atp and HpM its orthogonal complement with respect toḡ. As G acts by isometries, the decomposition TpM = VpM ⊕ HpM is a principal G-connection. This connection ω is sometimes called a mechanical connection for its relevant role in some problems in geometric mechanics (see [13]). Furthermore, there is a unique pseudo-Riemannian metric g in M such that the restriction π * : HpM −→ T π(p) M is an isometry for everyp ∈M. Obviously, the metric g satisfies where X H and Y H denote the horizontal lift of X and Y with respect to the mechanical connection.
Let∇ be the Levi-Civita connection andS =∇ − ∇ be a pseudo-Riemannian homogeneous structure onM, invariant under the action of the structure group G. Assume that there is 1-form β taking values in End(h) such that ∇ ω = β · ω.
Then, by ([7] Theorem 3.7), the reduced tensor field S on M defined by is a pseudo-Riemannian homogeneous structure of (M, g).

Fibrations of Pseudo-Hermitian over Almost Contact Metric Manifolds
Let θ be a nowhere-vanishing vector field inM. Around any pointp ∈M there exists a coordinate system (x 1 , ..., x m ), m = dimM, in a neighborhood diffeomorphic to [0, 1] m such that any integral curve of θ is given by x 1 = const, . . . , x m−1 = const. The vector field is said to be regular if the domains can be always chosen such that any orbit of θ intersects them at most once. For regular vector fields, the orbit space M is a smooth manifold and the projection π :M −→ M a submersion (cf. [14]). Furthermore, a regular vector field is said to be strictly regular if all the orbits are diffeomorphic. If θ is a complete, strictly regular vector field, the one-parametric group G generated by θ (G = R or S 1 ) acts freely onM, and π :M −→ M is a G-principal bundle. If we further assume thatM is equipped with a pseudo-Riemannian metricḡ such thatḡ(θ, θ) = ±1 (that is, θ is non-degenerate so that we can normalize it) andḡ is invariant by the group G, then the 1-form where ε = sign(ḡ(θ, θ)) is a G-principal connection form in π :M −→ M, a mechanical connection, as we mentioned above.
Theorem 1. Let (M,ḡ,J) be an almost pseudo-Hermitian manifold and let θ ∈ X(M) be a complete strictly regular unit vector field (ε =ḡ(θ, θ) = ±1). We consider that bothḡ andJ are invariant with respect to the uniparametric group G defined by θ. Then, the orbit space (M, g, φ, ξ, η) is an almost contact metric manifold, with for any X, Y ∈ TM, where X H stands for the horizontal lift with respect to the mechanical connection, and η is the dual form of ξ; that is, η(·) = εg(·, ξ).
With respect to the second equation, given X ∈ TM, we denote by X the orthogonal part of X with respect to ξ. Note that, sinceḡ(J(X H ), θ) = −ḡ(X H ,Jθ) = g(X , ξ) = 0, the vectorJ(X H ) is horizontal. Then and the proof is complete.

Remark 1.
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 (M, g, φ, ξ, η), the canonical 2-form Φ is defined to be Φ(X, Y) = g(φ(X), Y). Since for any metric connection ∇, the manifold M is cosymplectic if and only if ∇Φ = 0. For non-cosymplectic manifolds, if we choose a point p ∈ M, V = T p M, the 3-tensor (∇Φ) p 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 [8]) is equivalent to ∇Φ belonging to different combinations of the irreducible subspaces C 1 , . . . , C 12 in which S − (V) decomposes. These subspaces can be organized in a less coarse classification S −,0 (V) = C 5 ⊕ C 6 ⊕ C 7 ⊕ C 8 ⊕ C 9 ⊕ C 10 ⊕ C 12 , and defined by the expressions In the conditions of Theorem 1, if (M,ḡ,J) is a Kähler manifold, then the quotient space (M, g, φ, ξ, η) is an almost contact manifold such that ∇Φ belongs to the class S −,0 (V) = C 5 ⊕ C 6 ⊕ C 7 ⊕ C 8 ⊕ C 9 ⊕ C 10 ⊕ C 12 .
Proof. For X, Y, Z ∈ X(M) we have Since∇J = 0, the first and third terms of the last step above vanish and we get for any vector field X(M). In particular, Through a comparison with (7) we finally get that is, ∇Φ belongs to S −,0 (V).

Reduction of Homogeneous Structures
We now assume that the pseudo-Hermitian manifold (M,ḡ,J) is equipped with a homogeneous structure tensorS, which in addition is invariant under the action of the group G.

Theorem 2. Let (M,ḡ,J) be an almost pseudo-Kähler manifold equipped with an almost pseudo-Kähler homogeneous structureS that is invariant under the group flow G of a complete strictly regular unit vector field
where ∇ =∇ −S and β is a 1-form on M. Then, the tensor field S on the orbit space M =M/G defined by is a homogeneous, almost contact metric structure on (M, g, φ, ξ, η) belonging to the class S + (V) ⊕ S −,0 (V). Furthermore, the components S + ∈ S + (V) and S −,0 ∈ S −,0 (V) of S are

respectively.
Proof. Since the condition ∇ θ = β ⊗ θ is equivalent to ∇ ω = β · ω, 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 ∇φ = 0, where ∇ = ∇ − S. Let X, Y ∈ X(M) be two vector fields, Following the same steps in Proposition 1, we get which implies that ∇φ = 0, while again taking into consideration the fact that ∇ X H Y H is horizontal. Now, we decompose Y = Y + η(Y)ξ, Z = Z + η(Z)ξ and we get SinceS is a pseudo-Kähler homogeneous structure , which implies that S ∈ S + (V) ⊕ S −,0 (V). Finally, it is a matter of direct checking that S + and S −,0 in the statement satisfy S + ∈ S + (V) and S −,0 ∈ S −,0 (V).

Recall that the Ambrose-Singer condition
If in addition S − belongs to S −,0 (V), then
Finally, taking into account that g(X, Y ) = g(X , Y), g(X, Z ) = g(X , Z), X = X − η(X)ξ, we get the given expression. Proof. This result is a consequence of Theorems 3 and 4.

Conclusions
Since Ambrose-Singer [1] 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. [8]).

•
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 C 5 ⊕ C 6 ⊕ C 7 ⊕ C 8 ⊕ C 9 ⊕ C 10 ⊕ C 12 of the Chinea-González classification (cf. [8]).

•
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 [15] for the real case and [16] 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.
Author Contributions: All authors contributed equally and significantly to writing this article. All authors have read and agreed to the published version of the manuscript.