Hypersurfaces in space forms satisfying some curvature conditions

Deszcz, Ryszard; Głogowska, Małgorzata; Hotloś, Marian; Zafindratafa, Georges

doi:10.1016/j.geomphys.2015.10.010

Cited by 39 publications

(82 citation statements)

References 24 publications

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“…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 .…”

Section: Preliminariesmentioning

confidence: 99%

“…For an (0, k)-tensor T , k ≥ 1 and a symmetric (0, 2)-tensor E we define the (0, k)-tensor E · T and the (0, k + 2)-tensors R · T , C · T and Q(E, T ) by [19,27,34,49]…”

Section: Preliminariesmentioning

confidence: 99%

“…The tensor Q(E, T ) is called the Tachibana tensor of the tensors E and T , or the Tachibana tensor for short [34]. We note that (cf., [21, Section 4.4])…”

Section: Preliminariesmentioning

confidence: 99%

“…A semi-Riemannian manifold (M, g), n ≥ 4, is said to be a manifold with pseudosymmetric Weyl conformal curvature tensor [22,36] (see also [13,26,46]) if the tensor C · C and the Tachibana tensor Q(g, C) are linearly dependent at every point of M , that is…”

Section: Preliminariesmentioning

confidence: 99%

“…An important subclass of Ricci-generalized pseudosymmetric manifolds is formed by the manifolds realizing the condition [17,36] R · R = Q(S, R).…”

Section: Preliminariesmentioning

confidence: 99%

See 4 more Smart Citations

Curvature properties of some 4-dimensional semi-Riemannian metrics

A.¹,

Matsuyama²

2017

Kragujevac J Math

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Section: Preliminariesmentioning

confidence: 99%

“…For an (0, k)-tensor T , k ≥ 1 and a symmetric (0, 2)-tensor E we define the (0, k)-tensor E · T and the (0, k + 2)-tensors R · T , C · T and Q(E, T ) by [19,27,34,49]…”

Section: Preliminariesmentioning

confidence: 99%

“…The tensor Q(E, T ) is called the Tachibana tensor of the tensors E and T , or the Tachibana tensor for short [34]. We note that (cf., [21, Section 4.4])…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…An important subclass of Ricci-generalized pseudosymmetric manifolds is formed by the manifolds realizing the condition [17,36] R · R = Q(S, R).…”

Section: Preliminariesmentioning

confidence: 99%

See 3 more Smart Citations

Curvature properties of some 4-dimensional semi-Riemannian metrics

A.¹,

Matsuyama²

2017

Kragujevac J Math

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Curvature properties of Robinson–Trautman metric

2018

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The main objective of the present paper is to investigate the curvature properties of generalized pp-wave metric. It is shown that generalized pp-wave spacetime is Ricci generalized pseudosymmetric, 2-quasi-Einstein and generalized quasi-Einstein in the sense of Chaki. As a special case it is shown that pp-wave spacetime is semisymmetric, semisymmetric due to conformal and projective curvature tensors, R-space by Venzi and satisfies the pseudosymmetric type condition P · P = − 1 3 Q(S, P ). Again we investigate the sufficient condition for which a generalized pp-wave spacetime turns into pp-wave spacetime, pure radiation spacetime, locally symmetric and recurrent.Finally, it is shown that the energy-momentum tensor of pp-wave spacetime is parallel if and only if it is cyclic parallel. And the energy momentum tensor is Codazzi type if it is cyclic parallel but the converse is not true as shown by an example. Finally we make a comparison between the curvature properties of the Robinson-Trautman metric and generalized pp-wave metric.

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On some curvature conditions of pseudosymmetry type

et al. 2015

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It is known that the difference tensor R•C −C • R and the Tachibana tensor Q(S, C) of any semi-Riemannian Einstein manifold (M, g) of dimension n ≥ 4 are linearly dependent at every point of M. More precisely R • C − C • R = (1/(n − 1)) Q(S, C) holds on M. In the paper we show that there are quasi-Einstein, as well as non-quasi-Einstein semi-Riemannian manifolds for which the above mentioned tensors are linearly dependent. For instance, we prove that every non-locally symmetric and non-conformally flat manifold with parallel Weyl tensor (essentially conformally symmetric manifold) satisfies R • C = C • R = Q(S, C) = 0. Manifolds with parallel Weyl tensor having Ricci tensor of rank two form a subclass of the class of Roter type manifolds. Therefore we also investigate Roter type manifolds for which the tensors R • C − C • R and Q(S, C) are linearly dependent. We determine necessary and sufficient conditions for a Roter type manifold to be a manifold having that property.

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Hypersurfaces in space forms satisfying some curvature conditions

Cited by 39 publications

References 24 publications

Curvature properties of some 4-dimensional semi-Riemannian metrics

Curvature properties of some 4-dimensional semi-Riemannian metrics

Curvature properties of Robinson–Trautman metric

On some curvature conditions of pseudosymmetry type

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