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

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 ${S^m}$.