Vector Fields on Manifolds

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.