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Difference between revisions of "Poincaré-Hopf theorem"

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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" />.
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Let $M$ be a smooth compact manifold with boundary $W=\partial M$, and let $X$ be a [[Vector field|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 <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" />.
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Then the sum of the indices of the zeros of $V$ (see [[Singular point, index of a|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|Poincaré theorem]]).
 
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]]).

Latest revision as of 19:43, 14 August 2014

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://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