Communications of the
Korean Mathematical Society CKMS

We prove that if a real hypersurface with constant mean curvature of a complex space form satisfying $\nabla_{\xi}S=0$ and $S \xi = \sigma \xi$ for a smooth function $\sigma,$ then the structure vector field $\xi$ is principal, where $S$ denotes the Ricci tensor of the hypersurface.

Keywords: $\xi $-parallel Ricci tensor, principal curvature, real hypersurface, mean curvature

MSC numbers: 53C15, 53C45