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Poincare-Hopf index theorem
Let $M$ be a differentiable manifold of dimension $n$ and $V$ a continuous vector field on $M$.
Suppose $z$ is a zero of $V$. Suppose we can choose a closed ball ${\bf B}^n$ centered at $z$ with respect to some local coordinates, so that $z$ is the only zero of $V$ in ${\bf B}^n$. It is possible when all zeros of $V$ are isolated. Now we define the index of vector field $V$ at $z$, ${\rm ind}_V (z)$, to be the degree of the map $q: ∂{\bf B}^n = {\bf S}^{n-1} \to {\bf S}^{n-1}$ given by $q(x)=V(x)/||V(x)||$.
Index Theorem. Suppose $M$ is a compact orientable differentiable manifold and $V$ is a vector field on $M$ with only isolated zeroes. We assume that the direction of $V$ at the boundary $\partial M$ (if it's non-empty) is outward normal. Then $$\Sigma_i {\rm ind}_V(z_i) = \chi(M),$$ where the sum of the indices is over all the isolated zeroes of $V$ and $\chi(M)$ is the Euler characteristic of $M$.
Corollary. A vector field on a manifold with a nonzero Euler characteristic has a zero.
The corollary implies the Hairy Ball Theorem.
