On some curvature conditions of pseudosymmetry type

Ryszard Deszcz (1) , Małgorzata Głogowska (1) , Marian Hotloś (2) , Georges Zafindratafa (3)

1 Wroclaw University of Environmental and Life Sciences
2 Wroclaw University of Science and Technology
3 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015

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.

Mots clés: Einstein manifold; Quasi-Einstein manifold; Manifold with parallel Weyl tensor; Roter type manifold; Pseudosymmetric manifold; Generalized Einstein metric condition; Tachibana tensor

Domaines: Mathématiques [math]

Ryszard Deszcz, Małgorzata Głogowska, Marian Hotloś, Georges Zafindratafa. On some curvature conditions of pseudosymmetry type. Periodica Mathematica Hungarica, 2015, 70 (2), pp.153-170. ⟨10.1007/s10998-014-0081-9⟩. ⟨hal-03146519⟩