On an Anti-Torqued Vector Field on Riemannian Manifolds

Sharief Deshmukh, Ibrahim Al-Dayel and Devaraja Mallesha Naik
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Sharief Deshmukh: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Ibrahim Al-Dayel: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, P.O. Box 65892, Riyadh 11566, Saudi Arabia
Devaraja Mallesha Naik: Department of Mathematics, CHRIST (Deemed to Be University), Bengaluru 560029, India

Mathematics, 2021, vol. 9, issue 18, 1-12

Abstract: A torqued vector field ? 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.

Keywords: torse-forming vector fields; concircular vector fields; torqued vector fields; Einstein manifolds; scalar curvature; Fischer–Marsden equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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