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 | References | Similar Articles | Additional Information
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 ${S^m}$.
- Noel J. Hicks, Notes on differential geometry, Van Nostrand Mathematical Studies, No. 3, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR 0179691
- Katsumi Nomizu and Brian Smyth, On the Gauss mapping for hypersurfaces of constant mean curvature in the sphere, Comment. Math. Helv. 44 (1969), 484–490. MR 257939, DOI https://doi.org/10.1007/BF02564548
- Shlomo Sternberg, Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0193578
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Additional Information
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