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Some curvature restricted geometric structures for projective curvature tensors

Absos Ali Shaikh

http://orcid.org/0000-0001-6312-2564

Department of Mathematics, The University of Burdwan, Golapbag, Burdwan 713104, West Bengal, India

E-mail Address: aask2003@yahoo.co.in

E-mail Address: aashaikh@math.buruniv.ac.in

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and

Haradhan Kundu

Department of Mathematics, The University of Burdwan, Golapbag, Burdwan 713104, West Bengal, India

E-mail Address: kundu.haradhan@gmail.com

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https://doi.org/10.1142/S0219887818501578Cited by:17 (Source: Crossref)

Abstract

The projective curvature tensor is an invariant under geodesic preserving transformations on semi-Riemannian manifolds. It possesses different geometric properties than other generalized curvature tensors. The main object of the present paper is to study some semisymmetric type and pseudosymmetric type curvature restricted geometric structures due to projective curvature tensor. The reduced pseudosymmetric type structures for various Walker type conditions are deduced and the existence of Venzi space is ensured. It is shown that the geometric structures formed by imposing projective operator on a (0,4)-tensor is different from that for the corresponding (1,3)-tensor. Characterization of various semisymmetric type and pseudosymmetric type curvature restricted geometric structures due to projective curvature tensor are obtained on semi-Riemannian manifolds, and it is shown that some of them reduce to Einstein manifolds for the Riemannian case. Finally, to support our theorems, four suitable examples are presented.

Keywords:

AMSC: Primary: 53B20, Primary: 53B30, Secondary: 53C25, Secondary: 53C50

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