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In this paper we give a criterion for a compact mini- mal submanif old of Sm to lie in a given great hypersphere in terms of an integral over the stereographic image in Em of the submanifold. We also show that if all the points a certain normal distance C from a compact hypersurface M in Em 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 No- mizu and Smyth concerning the gauss map for hypersurfaces of Sm. Introduction. It is well known that the differential geometry of Euclidean space is closely related, via stereographic projection, to the differential geometry of the standard unit sphere. In this paper we study certain simple aspects of this relation. In ?1 we recall some of the basic formulas relating the geometry of the sphere to that of Euclidean space. In ?2 we use several of these formulas, together with several well known integral formulas, to derive a criterion for telling whether a compact minimal submanifold of the sphere lies within a given hyper- sphere. In ?3 we show that a recent result of Nomizu and Smyth (2 ) which considers the gauss map for hypersurfaces of spheres is equivalent to an extension of a classical theorem concerning parallel hypersurfaces in Euclidean space. NOTATION. Em+l is the Euclidean space of dimension (m+ 1), which we also view in the usual manner as being a vector space with inner product (, ) and norm J J, Sm denotes the unit sphere in Em+1, consisting of all unit vectors in Em+i. We let A be a fixed unit vector in Em+i, we denote the hyperplane passing through the origin in Em+1 which is perpendicular to A by Em and we set Sm-i = EmnSm. We set R = S- A I and denote the stereographic projection from

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