Mh. Math. 90, 185--194 (1980) Moaatah~e liiz Maihemalik 9 by Springer-Verlag 1980 Surfaces with Parallel Normalized Mean Curvature Vector By Bang-Yen Chen, East Lansing, Michigan (Received 5 February 1980) Abstract. A surface M in a Riemannian manifold is said to have parallel normalized mean curvature vector if the mean curvature vector is nonzero and the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. In this paper, it is proved that every analytic surface in a euclidean m-space E m with parallel normalized mean curvature vector must either lies in a E 4or lies in a hypersphere of E m as a minimal surface. Moreover, it is proved that if a l~iemann sphere in E ~ has parallel normalized mean curvature vector, then it lies either in a E a or in a hypersphere ofE m as a minimal surfaces. Applications to the classification of surfaces with constant Gauss curvature and with parallel normalized mean curvature vector are also given. I. Introduction ~ Let M be a connected analytic surface 2 in an analytic Riemann- ian manifold R m of m dimensions. Let V and V' be the covariant differentiations of M and R ~, respectively. Let X and Y be two tangent vector fields on M. Then the second fundamental form h is given by V'x Y = Vx Y + h(X, Y). (1.1) It is well-known that h (X, Y) is a normal vector field on M and it is symmetric on X and Y. Let } be a normal vector field on M, we write V'x} = - As(X) -I- V}}, (1.2) where -As(X ) and V~:$ denote the tangential and normal com- ponents of V} }, respectively. Then we have <As(X), Y> = <h(X, Y),}>, (1.3) i We shall follow the notations and definitions of [1]. 2 All manifolds and related differentiable geometric structures are assumed to be real analytic. Moreover, manifolds are assumed to be connected unless mentioned otherwise. 0026--9255/80/0090/0185/$02.00