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arXiv · 1404.2577

The index of isolated umbilics on surfaces of non-positive curvature

F. Fontenele·F. Xavier
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Abstract

It is shown that if a $C^2$ surface $M\subset\mathbb R^3$ has negative curvature on the complement of a point $q\in M$, then the $\mathbb Z/2$-valued Poincaré-Hopf index at $q$ of either distribution of principal directions on $M-\{q\}$ is non-positive. Conversely, any non-positive half-integer arises in this fashion. The proof of the index estimate is based on geometric-topological arguments, an index theorem for symmetric tensors on Riemannian surfaces, and some aspects of the classical Poincaré-Bendixson theory.

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F. Fontenele, F. Xavier. 2014-04-09. The index of isolated umbilics on surfaces of non-positive curvature. https://arxiv.org/abs/1404.2577

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