On Semi-Riemannian Manifolds Satisfying Some Generalized Einstein Metric Conditions

Year 2023, Volume: 16 Issue: 2, 539 - 576, 29.10.2023

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

https://doi.org/10.36890/iejg.1323352

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

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