Vector fields on non-compact manifolds

Tsuyoshi Kato; Daisuke Kishimoto; Mitsunobu Tsutaya

doi:10.2140/agt.2024.24.3985

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arXiv · 2211.00512

Vector fields on non-compact manifolds

Tsuyoshi Kato·Daisuke Kishimoto·Mitsunobu Tsutaya

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Abstract

Let $M$ be a non-compact connected manifold with a cocompact and properly discontinuous action of a discrete group $G$. We establish a Poincar\'{e}-Hopf theorem for a bounded vector field on $M$ satisfying a mild condition on zeros. As an application, we show that such a vector field must have infinitely many zeros whenever $G$ is amenable and the Euler characteristic of $M/G$ is non-zero.

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Tsuyoshi Kato, Daisuke Kishimoto, Mitsunobu Tsutaya. 2022-11-01. Vector fields on non-compact manifolds. https://doi.org/10.2140/agt.2024.24.3985

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