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Proceedings of the London Mathematical Society Advance Access originally published online on April 24, 2009
Proceedings of the London Mathematical Society 2009 99(3):609-632; doi:10.1112/plms/pdp012
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© 2009 London Mathematical Society

Cohomogeneity one disk bundles with normal homogeneous collars

Lorenz J. Schwachhöfer

Department of Mathematics
TU Dortmund University
Vogelpothsweg 87
44221 Dortmund
Germany

Kristopher Tapp

Department of Mathematics
Saint Joseph University
5600 City Avenue
Philadelphia, PA 19131
USA
ktapp@sju.edu

Received 1 October 2008. Revision received 24 February 2009.

We consider cohomogeneity one homogeneous disk bundles and address the question when these admit a nonnegatively curved{dagger} invariant metric with normal collar, that is, such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundle, then we show that its rank has to be in {2, 3, 4, 6, 8}. Moreover, we give a complete classification of such bundles of rank 6 and 8, and a partial classification for rank 3.

{dagger} Throughout this article, the term ‘curvature’ refers to the sectional curvature.

The first author was supported by the Schwerpunktprogramm Differentialgeometrie of the Deutsche Forschungsgesellschaft.

2000 Mathematics Subject Classification 53C20 (primary), 53C30 (secondary).


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