View source for Poincaré-Hopf theorem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110160/p1101601.png" /> be a smooth compact manifold with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110160/p1101602.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110160/p1101603.png" /> be a [[Vector field|vector field]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110160/p1101604.png" /> with isolated zeros such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110160/p1101605.png" /> points outwards at all points in the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110160/p1101606.png" />.

Then the sum of the indices of the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110160/p1101607.png" /> (see [[Singular point, index of a|Singular point, index of a]]) is equal to the Euler characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110160/p1101608.png" />.

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W. Milnor,   "Topology from the differentiable viewpoint" , Univ. Virginia Press  (1965)  pp. 35</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.G. Lloyd,   "Degree theory" , Cambridge Univ. Press  (1978)  pp. 33</TD></TR></table>