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.
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Eschenburg, J.H. Comparison theorems and hypersurfaces. Manuscripta Math 59, 295–323 (1987). https://doi.org/10.1007/BF01174796
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DOI: https://doi.org/10.1007/BF01174796