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This paper gives a sufficient condition for a submanifold of a Riemannian manifold of nonnegative constant curvature to be totally umbilical. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature. Introduction. In a previous paper [4] the present author proved the following theorem. Theorem A. Let M be an n-dimensional, connected, compact hypersurface with constant mean curvature immersed in an (n + l)-dimensional Riemannian manifold of nonnegative constant curvature. If the second fundamental tensor H satisfies (n 1) trace H2 < (trace H)2, then M is a totally umbilical hypersurface and consequently a sphere. In this theorem, if the length of the second fundamental tensor is constant, then we can replace the condition that M is compact by requiring that M is complete. In this case, the estimation in Theorem A is the best possible since Sn~ x E1 in F" + I satisfies (22 l) trace H2 = (trace H)2. In this paper, the author tries to generalize this theorem to a submanifold of any codimension which is immersed in a Riemannian manifold of nonnegative constant curvature. 1. Preliminaries. Let M be an 72-dimensional submanifold of an (22 +p)-dimensional Riemannian manifold M of constant curvature c. The Riemannian connections of M and M ate denoted by V and V respectively, whereas the connection in the normal bundle of M in M is denoted by D. Let N., • ■ • , N be mutually orthogonal unit normal vectors at a point P £ M and extend them to vector fields in a neighborhood of P. We define H . X (A = 1, 2, • • • , p) to be the tangential component of ^x^ A ^or ^ £ "^f/^ an<^ ca^ ^A tne second fundamental Received by the editors April 13, 1972. AMS (A/OS) subject classifications (1970). Primary 53C40, 53C20.

Submanifolds and a pinching problem on the second fundamental tensors