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Dynamics of the Morse vector field

Yijian Zhang

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.

Dynamics of the Morse vector field

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.
Paper Structure (3 sections, 6 theorems, 15 equations)

This paper contains 3 sections, 6 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications

Key Result

Theorem 2.1

Any two critical points p,q of a Morse function f on M can be connected by multiple orbits of negative gradient flow. Precisely, there exists a sequence of critical points $p_1,\cdots,p_n$ with $p_1=p$, $p_n=q$, such that either $p_i$ flows to $p_{i+1}$, or $p_{i+1}$ flows to $p_i$.

Theorems & Definitions (13)

  • Theorem 2.1
  • proof
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['flow']}
  • Theorem 2.2
  • proof : Preparation for the proof
  • proof
  • Proposition 3.1
  • proof
  • proof
  • ...and 3 more