arXiv subjects

Georges Zafindratafa

Publications and source records attributed to Georges Zafindratafa.

6 recordsLinked to original sources

arXiv2025

Curvature properties of pseudosymmetry type of some 2-quasi-Einstein manifolds

Let (M,g) be a 2-quasi-Einstein non-conformally flat semi-Riemannian manifold of dimension > 3. We prove that if its Riemann-Christoffel curvature tensor R is a linear combination of some Kulkarni-Nomizu tensors formed by the metric tensor g, the Ricci tensor S and its square S^2, then some pseudosymmetry type curvature conditions are satisfied. Certain non-conformally flat warped product manifolds with 2-dimensional base, and in particular some spacetimes, are such 2-quasi Einstein manifolds.

math.DG↗

arXiv2023

On Wintgen ideal submanifolds satisfying some pseudo-symmetry type curvature conditions

Let M be a Wintgen ideal submanifold of dimension n in a real space form R^{n+m}(k) of dimension (n+m) and of constant curvature k, n > 3, m = 1 or m > 1. Let g, R, Ricc, g /\ Ricc and C be the metric tensor, the Riemann-Christoffel curvature tensor, the Ricci tensor, the Kulkarni-Nomizu product of g and Ricc, and the Weyl conformal curvature tensor of M, respectively. In this paper we study Wintgen ideal submanifolds M in real space forms R^{n+m}(k), n > 3, m = 1 or m > 3, satisfying the following pseudo-symmetry type curvature conditions: (i) R.C and Q(g,R) (resp., Q(g,C), Q(g,g/\Ricc), Q(Ricc,R) or Q(Ricc,g/\Ricc)) are linearly dependent; (ii) C.R and Q(g,R) (resp., Q(g,C), Q(g,g/\Ricc), Q(Ricc,R) or Q(Ricc,g/\Ricc)) are linearly dependent; (iii) R.C - C.R and Q(g,R) (resp., Q(g,C), Q(g,g/\Ricc), Q(Ricc,R) or Q(Ricc,g/\Ricc)) are linearly dependent.

math.DG↗

arXiv2023

On semi-Riemannian manifolds satisfying some generalized Einstein metric conditions

The difference tensor R.C-C.R of a semi-Riemannian manifold (M,g), dim M > 3, formed by its Riemannian-Christoffel curvature tensor R and the Weyl conformal curvature tensor C, under some assumptions, can be expressed as a linear combination of (0,6)-Tachibana tensors Q(A,T), where A is a symmetric (0,2)-tensor and T a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions.

math.DG↗

arXiv2023

A note on some generalized curvature tensor

For any semi-Riemannian manifold (M,g) we define some generalized curvature tensor as a linear combination of Kulkarni-Nomizu products formed by the metric tensor, the Ricci tensor and its square of given manifold. That tensor is closely related to quasi-Einstein spaces, Roter spaces and some Roter type spaces.

math.DG↗

arXiv2019

Hypersurfaces in space forms satisfying some generalized Einstein metric condition

The difference tensor C.R - R.C of Einstein manifolds, some quasi-Einstein manifolds and Roter type manifolds, of dimension n > 3, satisfy the following curvature condition: (A) C.R - R.C = Q(S,C) - (k /(n-1)) Q(g,C). We investigate hypersurfaces M in space forms N satisfying (A). The main result states that if the tensor C.R - R.C of a non-quasi-Einstein hypersurface M in N is a linear combination of the tensors Q(g,C) and Q(S,C) then (A) holds on M. In the case when M is a quasi-Einstein hypersurface in N and some additional assumptions are satisfied then (A) also holds on M.

math.DG↗

arXiv2015

Curvature properties of some class of warped product manifolds

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.

math.DG↗