, Volume 7, Issue 6, pp 1011-1030

Comparison Geometry with Integral Curvature Bounds

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Abstract.

In this paper we shall generalize a formula of Heintze and Karcher for the volume of normal tubes around geodesics to a situation where one has integral bounds for the sectional curvature. This formula leads to a generalization of Cheeger's lemma for the length of the shortest closed geodesic and to a generalization of the Grove-Petersen finiteness result to a situation where one has integral curvature bounds.

Submitted: August 1996, revised: July 1997