Difference between revisions of "Poincaré-Hopf theorem"
From Encyclopedia of Mathematics
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| − | Let | + | {{TEX|done}} |
| + | 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 | + | 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=22929
Poincaré-Hopf theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9-Hopf_theorem&oldid=22929
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article