Dynamics of the Morse vector field

Yijian Zhang

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arXiv · 2512.23268

Dynamics of the Morse vector field

Yijian Zhang

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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.

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Yijian Zhang. 2025-12-29. Dynamics of the Morse vector field. https://arxiv.org/abs/2512.23268

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