On The Triviality Of $m$-Modified Conformal Vector Fields

TL;DR

This paper proves that under various geometric conditions, $m$-modified conformal vector fields on Riemannian manifolds are necessarily trivial, extending understanding of symmetries in differential geometry.

Contribution

It establishes the triviality of $m$-modified conformal vector fields on compact and non-compact Riemannian manifolds under specific conditions, providing new rigidity results.

Findings

No non-trivial $m$-modified homothetic vector fields on compact manifolds.

An inequality implying triviality of $m$-modified conformal vector fields.

Triviality of affine Killing $m$-modified conformal vector fields on non-compact manifolds.

Abstract

We prove that a compact Riemannian manifold $M$ does not admit any non-trivial $m$-modified homothetic vector fields. In the corresponding case of an $m$-modified conformal vector field $V$, we establish an inequality that implies the triviality of $V$. Further, we demonstrate that an affine Killing $m$-modified conformal vector field on a non-compact Riemannian manifold $M$ must be trivial. Finally, we show that an $m$-modified gradient conformal vector field is trivial under the assumptions of polynomial volume growth and convergence to zero at infinity.