Sectional Curvatures and Characteristic Classes

Abstract

We are concerned here with the relationship between the curvature properties of a riemannian manifold X and the global topological and differential invariants of X. An interesting result in this direction is Chern's theorem [6] that if X is compact, orientable, and has constant riemannian sectional curvature, then all Pontrjagin classes of X (with real coefficients) are zero. One of the main results in the present paper is that we can make weaker assumptions on the curvature of the manifold and still conclude that certain Pontrjagin classes must be zero. For each even integer p between 2 and the dimension of the manifold we define, after Allendoerfer [1], a smoothfunction yp on the Grassmann bundle of p-planes tangent to X. This function, called the pth sectional curvature of X, measures the Lipschitz-Killing curvature of geodesic p-dimensional submanifolds of X. The second sectional curvature 72 is the usual riemannian sectional curvature. The higher order sectional curvatures are weaker invariants of the riemannian structure than the riemannian sectional curvature, and it is in terms of these that we state our result mentioned above. The Pontrjagin classes we consider here are cohomology classes with real coefficients.