Critical point equation on almost f-cosymplectic manifolds

H Aruna Kumara,Devaraja Mallesha Naik,V Venkatesha

doi:10.1108/ajms-10-2020-0094

H Aruna Kumara, Devaraja Mallesha Naik + Show 1 more

Open Access

https://doi.org/10.1108/ajms-10-2020-0094

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Abstract

PurposeBesse first conjectured that the solution of the critical point equation (CPE) must be Einstein. The CPE conjecture on some other types of Riemannian manifolds, for instance, odd-dimensional Riemannian manifolds has considered by many geometers. Hence, it deserves special attention to consider the CPE on a certain class of almost contact metric manifolds. In this direction, the authors considered CPE on almost f-cosymplectic manifolds.Design/methodology/approachThe paper opted the tensor calculus on manifolds to find the solution of the CPE.FindingsIn this paper, in particular, the authors obtained that a connected f-cosymplectic manifold satisfying CPE with \lambda=\tilde{f} is Einstein. Next, the authors find that a three dimensional almost f-cosymplectic manifold satisfying the CPE is either Einstein or its scalar curvature vanishes identically if its Ricci tensor is pseudo anti‐commuting.Originality/valueThe paper proved that the CPE conjecture is true for almost f-cosymplectic manifolds.

Highlights

One of the natural ways of finding canonical Riemannian metric, that is, Riemannian metrics with constant curvature in various form on a smooth manifold is to look for metrics which are critical points of a natural functional on the space of all metrics on a given manifold
It is very interesting to investigate the critical points of total scalar curvature functional S : M → R given by
Defined on a compact orientable Riemannian n-manifold (M, g), where M denotes set of all Riemannian metrics on (M, g) of unit volume, rg is the scalar curvature and dvg is the volume form

Summary

Introduction

One of the natural ways of finding canonical Riemannian metric, that is, Riemannian metrics with constant curvature in various form on a smooth manifold is to look for metrics which are critical points of a natural functional on the space of all metrics on a given manifold. In this context, it is very interesting to investigate the critical points of total scalar curvature functional S : M → R given by. Venkatesha are thankful to Department of Science and Technology, New Delhi for financial assistance to the Department of Mathematics, Kuvempu University under the FISt program (Ref. No SR/FST/MS-I/2018-23(C))

Arab Journal of Mathematical Sciences

Δg λ þ n rλ À

CPE on almost

Suppose that f

Next suppose

1Þa means λ is constant