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 ${p_{1}, \dots, p_{n}}$ then the sum of their indices equal the Euler characteristic of the surface

\sum_{i} J_{v} (p_{i}) = X (S) = 2 - 2 g

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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
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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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