Curvature properties of some class of warped product manifolds

Abstract

Warped product manifolds with p-dimensional base, p=1,2, satisfy some curvature conditions of pseudosymmetry type. These conditions are formed from the metric tensor g, the Riemann-Christoffel curvature tensor R, the Ricci tensor S and the Weyl conformal curvature C of the considered manifolds. The main result of the paper states that if p=2 and the fibre is a semi-Riemannian space of constant curvature, if n is greater or equal to 4, then the (0,6)-tensors R.R - Q(S,R) and C.C of such warped products are proportional to the (0,6)-tensor Q(g,C) and the tensor C is expressed by a linear combination of some Kulkarni-Nomizu products formed from the tensors g and S. Thus these curvature conditions satisfy non-conformally flat non-Einstein warped product spacetimes (p=2, n=4). We also investigate curvature properties of pseudosymmetry type of quasi-Einstein manifolds. In particular, we obtain some curvature property of the Goedel spacetime

Curvature properties of some 4-dimensional semi-Riemannian metrics

“…M is called locally symmetric due to Cartan if ∇R = 0, which is equivalent to the fact that the local geodesic symmetry at each point of M is an isometry. During the last eight decades the notion of locally symmetric spaces have been generalized by many authors in different ways and several steps such as κ-space by Ruse [73][74][75] (which is called recurrent space by Walker in 1950 [114]), conformally recurrent manifolds by Adati and Miyazawa [2], projectively recurrent manifolds by Adati and Miyazawa [3], 2-recurrent manifolds by Lichnerowicz [57], generalized recurrent manifolds by Dubey [47], quasi-generalized recurrent manifolds by Shaikh and Roy [101], hyper generalized recurrent manifolds by Shaikh and Patra [100], weakly generalized recurrent manifolds by Shaikh and Roy [102], semisymmetric manifolds by Cartan [9] (which were classified by Szabó [107][108][109], in the Riemannian case), pseudosymmetric manifolds by Deszcz [23,35,83], pseudosymmetric manifolds by Chaki [10], weakly symmetric manifolds by Selberg [78], weakly symmetric manifolds by Tamássy and Binh [110]. It may be mentioned that the notion of weakly symmetric manifold by Selberg is different from that by Tamássy and Binh [83], and pseudosymmetric manifold by Chaki is also different from pseudosymmetric manifold by Deszcz [83].…”

“…It is clear that the 1-form A as well as the function β are non-zero at every point on U S . The manifold is said to be a 2-quasi-Einstein [29,31,33] manifold if rank(S − αg) ≤ 2 and rank(S − αg) = 2 on some open non-empty subset of U S , where α is some function on U S .…”

“…A semi-Riemannian manifold (M, g), n ≥ 3, is said to be pseudosymmetric by Deszcz [23,35] (see also [13,113]) (resp. Ricci-pseudosymmetric [21,37]) if on U R = {x ∈ M : R − r n(n−1) G x = 0} (resp.…”

“…We note that U C ∪ U S = U R (see, e.g., [31]). A semi-Riemannian manifold (M, g), n ≥ 3, is said to be Ricci-generalized pseudosymmetric [17,18] if at every point of M , the tensor R · R and the Tachibana tensor Q(S, R) are linearly dependent.…”

“…We note that any Roter type manifold is a generalized Roter type but not conversely. Very recently generalized Roter type manifolds were investigated in [31,97]. In [71, Theorem 1] it was stated that some 2-recurrent manifolds satisfy R = 1 r S ∧ S. Curvature properties of pseudosymmetry type of manifolds satisfying the condition R = φS ∧ S, where φ is some function, were obtained in [54].…”