manuscripta mathematica

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

Comparison theorems and hypersurfaces

  • J. -H. Eschenburg


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.


Riemannian Manifold Number Theory Algebraic Geometry Topological Group Fundamental Form 
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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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