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Recurrent curvature over four-dimensional homogeneous manifolds

Recurrent curvature properties are considered on four-dimensional pseudo-Riemannian homogeneous manifolds with non-trivial isotropy, and also on some geometric manifolds.


Introduction
It is well known that the most important geometric object living on a manifold is the curvature tensor. Many different conditions may be applied on the curvature tensor, and each of them is the geometric interpretation of an outer property. Being parallel is a famous condition on the curvature tensor, which determines the (locally) symmetric spaces. Locally symmetric manifolds have an important application in various fields of sciences like applied physics. In this way, manifolds with recurrent curvature, as a generalization of locally symmetric spaces, will find a special place. Many authors focused their studies on spaces with recurrent curvature. We present a brief survey about them.
Spaces with recurrent curvature were firstly introduced and characterized by H. S. Ruse [10]. He investigated these spaces and showed that the necessary and sufficient condition for a three-dimensional Riemannian manifold to have recurrent curvature is to accept a parallel vector field. Also in dimension 2, he showed that every Riemannian manifold has recurrent curvature.
A. G. Walker [13] made more extensive studies and proved several interesting results; probably the most important was the characterization of three-spaces with recurrent curvature, spaces known as strictly Walker manifolds.
In recent years, for the Lorentzian setting, several authors have studied threemanifolds with recurrent curvature; for example, E. García-Río et al. obtained a complete description of all locally homogeneous Lorentzian manifolds with recurrent curvature [5]. Using this classification, G. Calvaruso and A. Zaeim [1] investigated symmetries on this space and computed Ricci and curvature collineations 244 MILAD BASTAMI, ALI HAJI-BADALI, AND AMIRHESAM ZAEIM on Lorentzian three-manifolds with recurrent curvature. A. Haji-Badali [7] investigated these spaces and showed that a Lorentzian three-manifold with recurrent curvature does not accept non-trivial proper gradient Ricci solitons; he also obtained a classification for Ricci almost solitons on these spaces.
Due to the previous research, we concentrate our work on the study of the recurrent curvature property of four-dimensional homogeneous pseudo-Riemannian manifolds. These examples contain important classes of pseudo-Riemannian manifolds and it is worthwhile to study different geometric properties. Our study is based on the classification of four-dimensional homogeneous manifolds with nontrivial isotropy, which was given by Komrakov in [9]. We complete this study by checking the recurrent condition for these spaces, obtaining a new classification for homogeneous four-dimensional pseudo-Riemannian manifolds with recurrent curvature.
This paper is organized in the following way. The second section contains preliminaries and some basic facts necessary for our study. Section 3 is devoted to the main theorem of this study, i.e., the classification of homogeneous four-spaces with recurrent curvature, and then we consider the geometry of each class in Section 4. In the last section, the recurrent curvature property is considered on some geometric manifolds with physical applications, e.g. Walker spaces.

Preliminaries
A pseudo-Riemannian manifold (M, g) is called homogeneous if for any pair of points p, q ∈ M , there exists an isometry ϕ on M such that ϕ(p) = q. Homogeneous spaces with many interesting geometric properties are studied by several authors [2,3,5,9]. It is well known that a homogeneous space may be realized and studied as a quotient space G/H, where G is the group of isometries, which acts transitively on (M, g), and H is the isotropy subgroup.
We denote the Lie algebra of G and its subgroup H by g and h, respectively, and by m we denote the subspace of g complementary to h. One of the most important properties of homogeneous spaces is the existence of a one-to-one correspondence between invariant metrics on M = G/H and non-degenerate symmetric bilinear forms on m. The pair (g, h) uniquely defines the following map: By using this map, we can determine bilinear forms g on m with respect to the basis {e 1 , . . . , e r , u 1 , . . . , u n }, where {e j } r j=1 and {u i } n i=1 are bases for h and m, respectively. Now, the necessary and sufficient condition for the bilinear form g to be invariant is The Levi-Civita connection is computed by Then, the curvature tensor is determined by and the Ricci tensor ρ of the metric g with respect to the basis We recall the following definition.
Generally, for any tensor field T on (M, g) we can consider the spaces with recurrent tensor field T by studying the existence of a 1-form ω such that ∇T = ω⊗ T . Trivially, a locally symmetric manifold, i.e., ∇R = 0, has recurrent curvature.

Four-dimensional homogeneous manifolds with non-trivial isotropy
Now, we start with the classification of four-dimensional homogeneous manifolds with non-trivial isotropy, which is given by Komrakov in [9], and from now on, in this section and the following one, we will assume that (M = G/H, g) is an arbitrary pseudo-Riemannian four-dimensional homogeneous manifold with nontrivial isotropy, equipped with an invariant metric g.  Table 1 and ω i are the components of the oneform ω with respect to the basis Proof. We start by the case-by-case study of homogeneous spaces with non-trivial isotropy in [9]. Referring to [3], where a complete list of locally symmetric examples is presented, we restrict our study to the non-locally symmetric cases. We show the details for the case 1.1 1 .01, and the other cases may be treated in a similar way. As stated in the preliminaries, every invariant metric on the homogeneous space M = G/H is in one-to-one correspondence with an invariant inner product on m. For this case, there exists a basis {e 1 , u 1 , . . . , u 4 } of g; the nonzero brackets are invariant metric nonzero ω i and conditions and the isotropy subalgebra is generated by h = span{e 1 }. If we take m = span{u 1 , . . . , u 4 }, then the matrix related to ψ(e 1 ), which was defined in the previous section, and the invariant metrics with respect to and for the metric g to be nondegenerate it must satisfy a 2 (c 2 − bd) = 0. Therefore, using equation (2.1), we have the following components of the Levi-Civita connection: So, by a direct computation over (2.2), the components of curvature tensor are obtained as MILAD BASTAMI, ALI HAJI-BADALI, AND AMIRHESAM ZAEIM . Now, let ω = ω 1 θ 1 +ω 2 θ 2 +ω 3 θ 3 +ω 4 θ 4 ; a direct calculation using (2.4) gives (∇R − ω ⊗ R) 3142i = 1 2 bω i . Since ω = 0, we immediately get b = 0 (the case ω = 0 means that (M, g) is locally symmetric). Applying b = 0 for the remaining terms of (∇R − ω ⊗ R) ijklm gives ω i c 2 (a 2 − c 2 ) = 0, i = 1, . . . , 3; (ω 4 + 2)(a 2 − c 2 ) = 0.

Geometry of Theorem 3.1
An Einstein-like metric and a commutative curvature operator over the homogenous four-manifold (M, g) were presented in [14] and [8], respectively. Here, we introduce a large class of some geometric examples in our classification of Theorem 3.1. The Einstein manifold is one of the most important manifolds in geometry and physics. It is well known that the manifold (M, g) is called Einstein if ρ = ηg for a real constant η. Also, it is obvious that the manifold is flat if the curvature tensor vanishes identically.
The Weyl conformal tensor W is a (0, 4) tensor field on (M, g) which is completely determined by its components in the following way: where Sc is the scalar curvature. A four-dimensional pseudo-Riemannian manifold is called conformally flat if its Weyl conformal tensor W vanishes identically.  Table 2.   Proof. We proceed similarly to the proof of Theorem 3.1; so, we present the details for the cases 1.
and from the nondegeneracy of the metric g we have a 4 = 0. So, the Levi-Civita connection (2.1) and the curvature operator are as follows: and by using equation (2.2), Thus, by direct calculations, this case will be flat if the following equations are satisfied: Therefore, this case can be flat just if b = 0 and λ = 1 as it appeared in Table 2. According to equation (2.3), the corresponding Ricci tensor is So, this case is Ricci-flat if λ = 1, but if also b = 0 then the corresponding space is flat, and also by taking direct covariant derivative over ρ, this case is Ricci parallel if λ = 0, −1 and b = 0. Then, this case is Ricci parallel non-locally symmetric. Now, the Einstein condition yields and for the metric g to be nondegenerate we must have a 4 = 0. Using equation (2.1) gives and also by (2.2) we have the curvature operators It is obvious that if r 2 + p + s = 0 and r 2 + p − s = 0 then the metric is flat. The corresponding Ricci tensor is therefore, this case is Ricci-flat if r 2 + p = 0, and to have non-flat examples we must have s = 0. The same computation shows that this condition also holds for the Ricci parallel non-flat case. The Einstein condition in this case translates to aη − 2r 2 + 2p = 0, aη = 0, bη = 0, so, as a = 0, we must have r 2 + p = 0; so the metric will be flat. The conformal flatness in this case means that as = 0; since a = 0, this case is conformally flat if s = 0.
If in (2.4), instead of the curvature tensor we use the Ricci tensor, the space is called Ricci recurrent. Then, we consider the recurrent curvature condition for a strict Walker metric. Let Now, by applying the recurrent curvature condition (2.4), we have Since A = 0 (the case A = 0 implies the trivial flat condition), satisfying the recurrent curvature condition is equivalent to establishing the following: So, we have proved the next theorem. In the same coordinate system, the functions a and c can be further specialized to satisfy a(x 3 , x 4 ) ≡ 0, c(x 3 , x 4 ) ≡ 0 (cf. [11]). So, A reduces to A = ∂ 2 b ∂x3 2 . Therefore, for a special non-flat example, we present the following one.
Example 5.2. The geodesically complete strict Walker metric in R 4 , with vanishing a and c and b as b = f (x 4 )x 2 3 + g(x 4 )x 3 + h(x 4 ), for arbitrary smooth functions f = 0, g, and h, has recurrent curvature (∇R = ω ⊗ R) with the following 1-form ω on R 4 :

Conclusion
Homogeneous examples of pseudo-Riemannian manifolds which have recurrent curvature without Riemannian counterpart have been presented. Especially, there are some open questions like the existence of some common underlying structure (presumably the existence of a parallel field of degenerate lines/planes). Here just for an example we investigate the existence of recurrent manifolds within the framework of strictly Walker metrics in dimension four. This allows us the construction of simple examples which are not locally homogeneous in signature (2, 2), but a full description is still an open question.