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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The volume of a slightly curved submanifold in a convex region

Author: B. V. Dekster
Journal: Proc. Amer. Math. Soc. 68 (1978), 203-208
MSC: Primary 53C40
MathSciNet review: 0474147
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Abstract: Let T be a compact convex region in an n-dimensional Riemannian space, $ {k_s}$ be the minimum sectional curvature in T, and $ \kappa > 0$ be the minimum normal curvature of the boundary of T. Denote by $ {P^\nu }(\xi )$ a v-dimensional sphere, plane or hyperbolic plane of curvature $ \xi $. We assume that $ {k_s}$, k are such that on $ {P^2}({k_s})$ there exists a circumference of curvature k. Let $ {R_0} = {R_0}(\kappa ,{k_s})$ be its radius. Now, let Q be a convex (in interior sense) m-dimensional surface in T whose normal curvatures with respect to any normal are not greater than x satisfying $ 0 \leqslant \chi < \kappa $. Denote by $ {L_\chi }$ the length of a circular arc of curvature x in $ {P^2}({k_s})$ with the distance $ 2{R_0}$ between its ends. We prove that the volume of Q does not exceed the volume of a ball in $ {P^m}({k_s} - (n - m){\chi ^2})$ of radius $ \tfrac{1}{2}{L_\chi }$. These volumes are equal when T is a ball in $ {P^n}({k_s})$ and Q is its m-dimensional diameter.

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Article copyright: © Copyright 1978 American Mathematical Society