Given a set of pairwise distinct points $A=\{p_{1},p_{2}, \cdots, p_{n}\}$ on some regular surface $S$. How can we prove that there always exists some smooth vector field $\sigma:S \rightarrow TS$, such that $\sigma$ is non-vanishing with respect to $A$ (i.e, $\sigma(p_{i}) \neq 0$ for all $1 \leq i \leq n$)?
2026-04-07 17:51:07.1775584267
Existence of a non-vanishing smooth vector field on a given set of points
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