The volume of a slightly curved submanifold in a convex region
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- by B. V. Dekster PDF
- Proc. Amer. Math. Soc. 68 (1978), 203-208 Request permission
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.References
- B. V. Dekster, Estimates for the volume of a domain in a Riemannian space, Mat. Sb. (N.S.) 88(130) (1971), 61–87 (Russian). MR 0301671
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- B. V. Dekster, Estimates of the length of a curve, J. Differential Geometry 12 (1977), no. 1, 101–117. MR 470906
- D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Mathematics, No. 55, Springer-Verlag, Berlin-New York, 1968 (German). MR 0229177
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 68 (1978), 203-208
- MSC: Primary 53C40
- DOI: https://doi.org/10.1090/S0002-9939-1978-0474147-6
- MathSciNet review: 0474147