Difference between revisions of "Poincaré theorem"
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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 |
Poincaré theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_theorem&oldid=23496