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Proceedings of the American Mathematical Society

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

 

 

Applications of stereographic projections to submanifolds in $ E\sp{m}$ and $ S\sp{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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DOI: https://doi.org/10.1090/S0002-9939-1970-0254787-2
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