, Volume 237, Issue 2, pp 251-257

Transitive holonomy group and rigidity in nonnegative curvature

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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.

Received July 27, 1999; in final form November 28, 1999 / Published online February 5, 2001