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

Abstract

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} .