On an Anti-Torqued Vector Field on Riemannian Manifolds

Sharief Deshmukh,Ibrahim Al-Dayel,Devaraja Mallesha Naik

doi:10.3390/math9182201

Sharief Deshmukh, Ibrahim Al-Dayel + Show 1 more

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https://doi.org/10.3390/math9182201

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Abstract

A torqued vector field ξ is a torse-forming vector field on a Riemannian manifold that is orthogonal to the dual vector field of the 1-form in the definition of torse-forming vector field. In this paper, we introduce an anti-torqued vector field which is opposite to torqued vector field in the sense it is parallel to the dual vector field to the 1-form in the definition of torse-forming vector fields. It is interesting to note that anti-torqued vector fields do not reduce to concircular vector fields nor to Killing vector fields and thus, give a unique class among the classes of special vector fields on Riemannian manifolds. These vector fields do not exist on compact and simply connected Riemannian manifolds. We use anti-torqued vector fields to find two characterizations of Euclidean spaces. Furthermore, a characterization of an Einstein manifold is obtained using the combination of a torqued vector field and Fischer–Marsden equation. We also find a condition under which the scalar curvature of a compact Riemannian manifold admitting an anti-torqued vector field is strictly negative.

Highlights

A concircular vector field w on a Riemannian manifold ( M, g) is defined by the equationAccepted: 25 August 2021∇ E w = f E, E ∈ X( M), Published: 8 September 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. (1)where ∇ denotes the Riemannian connection of ( M, g), f is a smooth function, and X( M )is the Lie algebra of smooth vector fields on M
We find a condition under which the scalar curvature of a compact Riemannian manifold admitting an anti-torqued vector field is strictly negative
Concircular vector fields are well known for their applications in physics

Summary

Introduction

A concircular vector field w on a Riemannian manifold ( M, g) is defined by the equation. It is interesting to note that the twisted product I × f N of an interval I and N an (m − 1)-dimensional Riemannian manifold has a torqued vector field it is not a concircular vector field (cf [20]). We study a torse-forming vector field on a Riemannian manifold ( M, g) for which the vector field v dual to ω (ω ( E) = g(v, E)) is parallel to w We are interested in a unit torse-forming w on ( M, g) with dual 1-form η satisfying f η ( E) = −ω ( E), that is, v = − f w (w is parallel to v ) and call this torse-forming vector field an anti-torqued vector field on the Riemannian manifold ( M, g). We find a condition under which the scalar curvature of a compact Riemannian manifold admitting an anti-torqued vector field is strictly negative (see Theorem 4)

Preliminaries

Characterizing Euclidean Spaces via Anti-Torqued Vector Fields

A Characterization of Einstein Manifolds