Transitive holonomy group and rigidity in nonnegative curvature (Q5947787)

Let \(p: E \to M\) be an oriented Euclidean (i.e. with a given metric) vector bundle over a manifold \(M\). The authors give sufficient conditions that the holonomy group of any Riemannian connection in \(p\) acts transitively on the unit sphere bundle \(SE \subset E\). Such conditions are consequences of the following simple and useful theorem. Theorem: Let \(p : E \to M\) be an Euclidean oriented rank \(k\) vector bundle. If it admits a Riemannian connection which does not act transitively on the unit sphere bundle \(SE\) then for \(q>1\) there is a short exact sequence \[ 0 \rightarrow \pi_q(S^{k-1}) \rightarrow \pi_q(SE) \rightarrow \pi_q(M)\rightarrow 0. \] Corollary: The action of the holonomy group of any Riemannian connection in \(p\) is transitive on \(SE\) if one of the following conditions holds: i) \(SE\) is a homotopy-sphere, for example, if \(p\) is the canonical line-bundle over complex or quaternionic projective space, ii) \(M\) is a homotopy-sphere and \(p\) doesn't admit a nowhere-zero section, iii) \(p\) is the tangent bundle of the complex or quaternionic projective space. The following application to the geometry of a complete Riemannian manifold \(N\) of nonnegative curvature is given: If the normal bundle \(\nu\) of the soul \(S\) of \(N\) satisfies one of the conditions of the corollary , then the exponential map \(\exp:\nu \to N \) is a diffeomorphism, the metric projection \(N \to S\) is a \(C^{\infty}\) Riemannian submersion and the ideal boundary of \(N\) is a point.