# Poincaré-Hopf theorem

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Let $M$ be a smooth compact manifold with boundary $W=\partial M$, and let $X$ be a vector field on $M$ with isolated zeros such that $X$ points outwards at all points in the boundary $W$.

Then the sum of the indices of the zeros of $V$ (see Singular point, index of a) is equal to the Euler characteristic of $M$.

This is the generalization proved by H. Hopf, in 1926, of the two-dimensional version owed to H. Poincaré (1881, 1885) (see Poincaré theorem).

#### References

 [a1] J.W. Milnor, "Topology from the differentiable viewpoint" , Univ. Virginia Press (1965) pp. 35 [a2] N.G. Lloyd, "Degree theory" , Cambridge Univ. Press (1978) pp. 33
How to Cite This Entry:
Poincaré–Hopf theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Poincar%C3%A9%E2%80%93Hopf_theorem&oldid=38716