Poincaré-Hopf theorem

See @needham2021visual page 206.

Theorem

If a vector field $v$ on a smooth surface $S$ of genus $g$ has only a finite number of singular points $\{p_1,\cdots ,p_n\}$ then the sum of their indices equal the Euler characteristic of the surface

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Proof

A wonderful proof is in @needham2021visual page 207

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As a immediate consequence, a vector field with no singular points can only exist in a surface of genus 0, i.e., in a topological torus

Another conclusion is the Hairy Ball Theorem.

And another conclusion is a proof for the generalized Euler formula (see @needham2021visual page 209):

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es