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Difference between revisions of "Poincaré theorem"

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m (moved Poincaré theorem to Poincare theorem: ascii title)
m (moved Poincare theorem to Poincaré theorem over redirect: accented title)
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Revision as of 07:55, 26 March 2012

Let a vector field be defined on a smooth closed two-dimensional Riemannian manifold (cf. Vector field on a manifold) and let it have a finite number of isolated singular points . Then

here is the index of the point with respect to (see Singular point, index of a) and is the Euler characteristic of . This was established by H. Poincaré (1881).


Comments

This theorem has since been established for manifolds of all dimensions, [a1].

An immediate consequence is that on a sphere of even dimension there is no continuous vector field without a zero (singularity), the Poincaré–Brouwer theorem, also called the hairy ball theorem. This was established for by Poincaré and for by L.E.J. Brouwer. On the other hand, for the odd-dimensional spheres , , , gives a continuous vector field with no zeros on . More generally one has that there exists a vector field without zero on a manifold if and only if , [a1].

References

[a1] P.S. [P.S. Aleksandrov] Alexandroff, H. Hopf, "Topologie" , Chelsea, reprint (1972) pp. Chapt. XIV, Sect. 4.3
[a2] M.W. Hirsch, "Differential topology" , Springer (1976) pp. Chapt. 6
How to Cite This Entry:
Poincaré theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_theorem&oldid=23496
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article