a1 Michigan State University, East Lansing, Michigan 48823
Let M be a surface immersed in an m-dimensional space form Rm(c) of curvature c = 1, 0 or −1. Let h be the second fundamental form of this immersion; it is a certain symmetric bilinear mapping for x M, where Tx is the tangent space and the normal space of M at x. Let H be the mean curvature vector of M in Rm(c) and , the scalar product on Rm(c). If there exists a function λ on M such that h(X, Y), H = λX, Y for all tangent vectors X, Y, then M is called a pseudo-umbilical surface of Rm(c). Let D denote the covariant differentiation of Rm(c) and η be a normal vector field. If we denote by D*η the normal component of Dη, then D* defines a connection in the normal bundle. A normal vector field η is said to be parallel in the normal bundle if Dη = 0. The length of mean curvature vector is called the mean curvature.
(Received October 21 1971)
† This work was supported in part by the National Science Foundation under Grant GU-2648.