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manuscripta mathematica

, Volume 59, Issue 3, pp 295–323 | Cite as

Comparison theorems and hypersurfaces

  • J. -H. Eschenburg
Article

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.

Keywords

Riemannian Manifold Number Theory Algebraic Geometry Topological Group Fundamental Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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.

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