H-Recurrent Surfaces in Euclidean Space E^{m}

The object of the paper is to study some smooth surfaces M whose mean curvature vector H satisfies the H-recurrent condition D_{X}H=λ(X)H in m-dimensional Euclidean space E^{m}, where X is a tangent vector field of M and λ is a 1-form. First of all,we prove that the surfaces which satisfy the H-recurrent condition in E^{m} are R^{}-parallel (i.e., R^{}H=0). Then, we show that H-parallel surfaces in E are either totally umbilical or normally flat.