Dynamics of the Morse vector field

TL;DR

This paper investigates the properties of Morse vector fields on compact manifolds, demonstrating connectedness of critical points via orbits and exponential flow shrinkage, with applications to curvature and map vanishing results.

Contribution

It proves key properties of Morse vector fields, including connectedness of critical points and flow behavior, with new applications in geometry and topology.

Findings

Critical points are connected through orbits.

Flow exhibits exponential shrinkage on stable submanifolds.

Applications include vanishing results for maps and curvature operators.

Abstract

The (negative) gradient vector fields of Morse functions on a compact manifold provide an important example in dynamical system. In this note we prove two important properties of this kind of vector field: Connectedness of critical points through orbits and exponential shrinkage of the flow on stable submanifolds. We also find applications in showing some vanishing results of maps or curvature operators.