On Semi-Riemannian Manifolds Satisfying Some Generalized Einstein Metric Conditions

Authors : Ryszard Deszcz, Małgorzata Głogowska, Marian Hotloś, Miroslava Petrović-torgašev, Georges Zafindratafa

Pages : 539-576

Doi:10.36890/iejg.1323352

View : 77 | Download : 151

Publication Date : 2023-10-29

Article Type : Research Paper

Abstract :The derivation-commutator $R \\cdot C - C \\cdot R$ of a semi-Riemannian manifold $(M,g)$, $\\dim M \\geq 4$, formed by its Riemann-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.
Keywords : Warped product manifold, Einstein, quasi Einstein, 2 quasi Einstein and partially Einstein manifold, generalized Einstein metric condition, pseudosymmetry type curvature condition, hypersurface, Chen ideal submanifold