Characteristic Cohomology Classes in a Riemann Manifold

Abstract

One of the most beautiful applications of combinatorial topology is its solution of the problem of the existence, or non-existence, of a continuous, non-zero function (sometimes called a vector field) on a closed manifold. This question was settled by H. Hopf [5] who proved: THEOREM I. A continuous, non-zero function exists on a closed manifold M if and only if the Euler-Poincare Characteristic, x, of M' is zero. Recent investigations have shown the close relation between this result and the generalized Gauss-Bonnet Theorem [1] which states a relation between the intrinsic differential geometry of a closed Riemann manifold and its EulerPoincar6 Characteristic, namely: THEOREM II. If Mn is a closed, orientable Riemann manifold of class p _ 3, of even dimension, and of total curvature KT, then