Transitive holonomy group and rigidity in nonnegative curvature

Abstract

In this note, we examine the relationship between the twisting of a vector bundle $\xi$ over a manifold M and the action of the holonomy group of a Riemannian connection on $\xi$ . For example, if there is a holonomy group which does not act transitively on each fiber of the corresponding unit sphere bundle, then for any $f:S^n\to M$ , the pullback $f^*\xi$ of $\xi$ admits a nowhere-zero cross section. These facts are then used to derive a rigidity result for complete metrics of nonnegative sectional curvature on noncompact manifolds.