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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Note on vector fields in manifolds


Author: Hans Samelson
Journal: Proc. Amer. Math. Soc. 36 (1972), 272-274
MSC: Primary 57D25
MathSciNet review: 0314068
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Abstract: We give a direct geometric proof of Hopf's theorem on the sum of indices at the zeros of a vector field in a manifold M, or rather of that part of the theorem that says that the sum is the same for any two vector fields. The main idea is to connect the two fields by a one-parameter family of fields and to make everything transversal (to $ M \times I$). The resulting system of curves permits one to read off the theorem.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1972-0314068-7
PII: S 0002-9939(1972)0314068-7
Keywords: Manifold, vector field, index, transversality
Article copyright: © Copyright 1972 American Mathematical Society