Applications of stereographic projections to submanifolds in ^{ } and ^{ }
In this paper we give a criterion for a compact minimal submanifold of S m {S^m} to lie in a given great hypersphere in terms of an integral over the stereographic image in E m {E^m} of the submanifold. We also show that if all the points a certain normal distance C C from a compact hypersurface M M...
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| Published in: | Proceedings of the American Mathematical Society Vol. 25; no. 1; pp. 119 - 123 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
01-05-1970
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| Online Access: | Get full text |
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| Summary: | In this paper we give a criterion for a compact minimal submanifold of S m {S^m} to lie in a given great hypersphere in terms of an integral over the stereographic image in E m {E^m} of the submanifold. We also show that if all the points a certain normal distance C C from a compact hypersurface M M in E m {E^m} lie on a sphere of radius D > C D > C then M 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 {S^m} . |
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| ISSN: | 0002-9939 1088-6826 |
| DOI: | 10.1090/S0002-9939-1970-0254787-2 |