A geometric heat flow for vector fields

TL;DR

This paper introduces a geometric heat flow method to identify Killing vector fields on closed Riemannian manifolds with positive curvature, exploring its properties, convergence, and applications in geometry and physics.

Contribution

It presents a novel heat flow approach for finding Killing vector fields, including proofs of global existence, convergence analysis, and new criteria for their existence.

Findings

Proved global existence of the flow solution.

Established convergence properties of the flow.

Derived new criteria for the existence of Killing vector fields.

Abstract

In this paper we introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution of this flow, discuss its convergence and possible applications, and its relation to the Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. The similar flow to finding holomorphic vector fields on K\"ahler manifolds will be studied in \cite{LL2}.