Transitive holonomy group and rigidity in nonnegative curvature
- Cite this article as:
- Guijarro, L. & Walschap, G. Math Z (2001) 237: 251. doi:10.1007/PL00004867
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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.