manuscripta mathematica

, Volume 59, Issue 3, pp 295–323

Comparison theorems and hypersurfaces

  • J. -H. Eschenburg
Article

DOI: 10.1007/BF01174796

Cite this article as:
Eschenburg, J.H. Manuscripta Math (1987) 59: 295. doi:10.1007/BF01174796

Abstract

We compare the second fundamental forms of a family of parallel hypersurfaces in different Riemannian manifolds. This leads to new proofs for the distance and volume comparison theorems in Riemannian geometry. In particular, we get a new result on the volume of the set of points with distance≤r from a totally geodesic submanifold, for any r. The analytic prerequisite is the investigation of the Riccati type ODE which is satisfied by the second fundamental form of a parallel hypersurface family.

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • J. -H. Eschenburg
    • 1
    • 2
  1. 1.Mathematisches Institut der WWUMünster
  2. 2.Mathematisches Institut der Universität FreiburgFreiburg/Br.