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Proceedings of the American Mathematical Society

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

 

 

A characterization of parallel ovaloids


Author: Dimitri Koutroufiotis
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
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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

$\displaystyle [(p - \bar p) - (ck_1^{ - 1} - \bar k_1^{ - 1})][(p - \bar p) - (ck_2^{ - 1} - \bar k_2^{ - 1})] \leq 0$

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.

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DOI: https://doi.org/10.1090/S0002-9939-1974-0353215-X
Keywords: Parallel ovaloids, mapping by equal normals, preservation of principal directions, support function, principal curvatures, sphere
Article copyright: © Copyright 1974 American Mathematical Society