proceedings of the american mathematical society Volume 29, No. 2, July 1971 MINIMAL HYPERSURFACES IN AN w-SPHERE BANG-YEN CHEN Abstract. (1) A submanifold Mn of a euclidean space En+i of codimension 2 is a pseudo-umbilical submanifold with constant mean curvature if and only if it is a minimal hypersurface of a hypersphere of £n+2. (2) A complete oriented minimal surface Mï of a 3-sphere S3 on which the Gauss curvature does not change its sign is either an equatorial sphere or a Clifford flat torus. 1. Introduction. Let x'. Mn—>Pm be an isometric immersion of a Riemannian manifold Mn of dimension n into an oriented Rieman- nian manifold Rm of dimension m (m>n). For a unit normal vector e at x(p), pEMn, there corresponds a selfadjoint transformation A (e) of the tangent space TP(M") at p into itself, called the second fundamental form at e. If e„+i, • • • , em is an orthonormal basis of the normal space of M" in Rm at x(p), then the mean curvature vector H is given by m (1) H=(l/n) £ (trace ¿(er))er. r=n+l It is easy to verify that H is independent of the choice of the ortho- normal basis e„+i, • • • , em. The length of the mean curvature vector H is called the mean curvature. If the mean curvature vector H=0 identically, then the immersion x:M"—*Rm is called a minimal im- mersion and Mn is called a minimal submanifold of Rm. If the mean curvature vector H is nowhere zero and the second fundamental form at the direction of the mean curvature vector is proportional to the identity transformation of the tangent space of Mn everywhere, then the immersion x:Mn-+Rm is called a pseudo-umbilical immersion and Mn is called a pseudo-umbilical submanifold of Rm. In this paper we prove the following theorems: Theorem 1. Let x : M"—>En+2 be an isometric immersion of a Rie- mannian manifold MH of dimension n into a euclidean space En+2 of dimension n-\-2. Then Mn is a pseudo-umbilical submanifold of En+2 with constant mean curvature if and only if Mn is a minimal hyper- surface of a hypersphere of En+2. Received by the editors September 25, 1970. AMS 1970 subject classifications. Primary 53A10, 53C40; Secondary53B25,53A05. Key words and phrases. Second fundamental form, mean curvature vector, mean curvature, pseudo-umbilical submanifold, minimal hypersurface, equatorial sphere, Clifford flat torus, Gauss curvature. Copyright ® 1971, American Mathematical Society 375 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use