A characterization of parallel ovaloids
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- by Dimitri Koutroufiotis PDF
- Proc. Amer. Math. Soc. 46 (1974), 86-93 Request permission
Abstract:
Two ovaloids $S,\bar S$ can be mapped diffeomorphically onto each other by equal inner normals. If, under this mapping, principal directions are preserved and \[ [(p - \bar p) - (ck_1^{ - 1} - \bar k_1^{ - 1})][(p - \bar p) - (ck_2^{ - 1} - \bar k_2^{ - 1})] \leq 0\] everywhere on the unit sphere for a certain constant $c$, then $c = 1$ and $p - \bar p$ = constant. Here $p,\bar p$ are the support functions, ${k_1}$, and ${k_2}$ the principal curvatures of $S,{\bar k_1}$ and ${\bar k_2}$ the corresponding principal curvatures of $\bar S$. Various characterizations of the sphere are obtained as corollaries.References
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Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 46 (1974), 86-93
- MSC: Primary 53C40
- DOI: https://doi.org/10.1090/S0002-9939-1974-0353215-X
- MathSciNet review: 0353215