Vol. 23, No. 2, 1967

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The volume of a totally-geodesic hypersurface in a pinched manifold

Nathaniel Grossman

Vol. 23 (1967), No. 2, 257–262

We will give necessary conditions for a compact hypersurface to be totally-geodesic in a manifold of strictly positive curvatures. These conditions relate the volume of the hypersurface to the pinching of the manifold. The method consists of obtaining estimates from above and below for the volume of the manifold. Comparison of these estimates gives inequalities for the volume of the hypersurface. The method appears on the surface to apply to totally-geodesic submanifolds of arbitrary codimension with but a little modification. We are grateful to R. L. Bishop for pointing out that below the surface are snags that we have not yet been able to avoid.

Mathematical Subject Classification
Primary: 53.75
Received: 16 November 1966
Published: 1 November 1967
Nathaniel Grossman