Note on vector fields in manifolds

Abstract

We give a direct geometric proof of Hopf's theorem on the sum of indices at the zeros of a vector 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 A'M x I). The resulting system of curves permits one to read off the theorem. The classical theorem of H. Hopf on vector fields in a manifold [2] (see also Lefschetz [3]) can be stated as follows: Let M be a differentiable (say C) compact connected closed manifold, and let V be a (differentiable) vector in M with a finite number of zeros; then the sum of the indices of the zeros of V equals the Euler characteristic of M. We propose to give a short proof of that part of the theorem that says that the sum of the indices of the zeros is the same for any two vector fields. (That the common value is the Euler characteristic of M can then be seen by taking for V the gradient field of a Morse function.) The main tool will be transversality (general position) ([1], [4], [5]). We take as known the definition and basic properties of the index of a zero: in any coordinate system, with the zero as origin, V defines a map of the unit sphere into itself; the index is the degree of this map. If we deform V slightly to V', then any zero of V' must appear near a zero of V, and the index of any zero of V equals the sum of the indices of the nearby zeros of V'. We regard a vector as a section of the tangent bundle TM of M. The transversality theorem ([1], [4], [5]) and the last remark above allow us to restrict ourselves to transversal vector fields, that is, vector fields that are transversal to the 0-section of TM (which we identify with M itself). For such a vector the zeros are automatically isolated; Received by the editors January 10, 1972 and, in revised form, March 24, 1972. AMS 1969 subject classJifcations. Primary 5734, 5732.