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2008-05-18T19:50:37Z

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Vipul: New page: {{index theorem}} ==Name== This result goes by the name of '''Poincaré-Hopf theorem''', '''Hopf index theorem'''. ==Statement== ===For compact manifolds=== Let $M$ be a [[...

2008-02-28T15:12:14Z

New page: {{index theorem}} ==Name== This result goes by the name of '''Poincaré-Hopf theorem''', '''Hopf index theorem'''. ==Statement== ===For compact manifolds=== Let <math>M</math> be a [[...

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{{index theorem}}

==Name==

This result goes by the name of '''Poincaré-Hopf theorem''', '''Hopf index theorem'''.

==Statement==

===For compact manifolds===

Let <math>M</math> be a [[compact manifold|compact]] [[differential manifold]]. Then, the following are true:

# There exists a [[vector field]] on <math>M</math> with isolated zeros. An isolated zero is a point where the vector field vanishes, such that there is an open set containing the point, and not containing any other point where the vector field vanishes. Since <math>M</math> is compact, this is equivalent to demanding that the set of zeros be finite.
# For any vector field with isolated zeros, the sum of the [[index of a zero of a vector field|indices]] of all isolated zeros equals the [[Euler characteristic]] of the manifold.

Poincaré-Hopf index theorem - Revision history

Vipul: 1 revision

Vipul: New page: {{index theorem}} ==Name== This result goes by the name of '''Poincaré-Hopf theorem''', '''Hopf index theorem'''. ==Statement== ===For compact manifolds=== Let M be a [[...

Vipul: New page: {{index theorem}} ==Name== This result goes by the name of '''Poincaré-Hopf theorem''', '''Hopf index theorem'''. ==Statement== ===For compact manifolds=== Let $M$ be a [[...