Vector Fields on Manifolds
Michael Francis Atiyah
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Vector Fields on Manifolds

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This paper is a contribution to the topological study of vector fields on manifolds. In particular we shall be concerned with the problems of exist ence of r linearly independent vector fields. For r = 1 the classical result of H. Hopf asserts that the vanishing of the Euler characteristic is the necessary and sufficient condition, and our results will give partial extens ions of Hopf's theorem to the case r 1. Arecent article by E. Thomas [10] gives a good survey of work in this general area. Our approach to these problems is based on the index theory of elliptic differential operators and is...