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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Applications of stereographic projections to submanifolds in $E^{m}$ and $S^{m}$


Author: Robert C. Reilly
Journal: Proc. Amer. Math. Soc. 25 (1970), 119-123
MSC: Primary 53.74
DOI: https://doi.org/10.1090/S0002-9939-1970-0254787-2
MathSciNet review: 0254787
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Abstract: In this paper we give a criterion for a compact minimal submanifold of ${S^m}$ to lie in a given great hypersphere in terms of an integral over the stereographic image in ${E^m}$ of the submanifold. We also show that if all the points a certain normal distance $C$ from a compact hypersurface $M$ in ${E^m}$ lie on a sphere of radius $D < C$ then $M$ is a hypersphere. This generalizes a classical result on parallel hypersurfaces. We prove this theorem by showing it to be equivalent, via stereographic projection, to a recent result of Nomizu and Smyth concerning the gauss map for hypersurfaces of ${S^m}$.


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Keywords: Stereographic projection, minimal submanifolds of spheres, submanifolds of Euclidean space, hypersurfaces, integral formulas, gauss map in spheres, parallel hypersurface, toral counterexample
Article copyright: © Copyright 1970 American Mathematical Society