Namespaces
Variants
Actions

Poincaré-Hopf theorem

From Encyclopedia of Mathematics
Jump to: navigation, search

Let be a smooth compact manifold with boundary , and let be a vector field on with isolated zeros such that points outwards at all points in the boundary .

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

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://encyclopediaofmath.org/index.php?title=Poincar%C3%A9-Hopf_theorem&oldid=23478
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article