Mathematische Zeitschrift

, Volume 237, Issue 2, pp 251–257

Transitive holonomy group and rigidity in nonnegative curvature

  • Luis Guijarro
  • Gerard Walschap
Original article

DOI: 10.1007/PL00004867

Cite this article as:
Guijarro, L. & Walschap, G. Math Z (2001) 237: 251. doi:10.1007/PL00004867

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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Luis Guijarro
    • 1
  • Gerard Walschap
    • 2
  1. 1.Department of Mathematics, Universidad Autonoma de Madrid, 28049 Madrid, Spain (e-mail: lguijarro@mat.ucm.es) ES
  2. 2.Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA (e-mail gerard@aftermath.math.ou.edu) US