Certain types of real hypersurfaces in complex space forms

Abstract

A nice characterization of real hypersurfaces of a nonflat complex space form is very much familiar when its structure vector field $$\xi $$ is Killing. This motivates us to classify a real hypersurface of a nonflat complex space form when the structure vector field $$\xi $$ is 2-Killing (i.e., $$\pounds _{\xi }\pounds _{\xi }g = 0)$$ , where $$\pounds _{\xi }$$ denotes the Lie derivative along the vector field $$\xi $$ . Further, we classify real hypersurfaces of a nonflat complex space form when the tensor field $$T (= \pounds _{\xi }g)$$ is (i) weakly Lie $$\xi $$ -parallel, and (ii) weakly parallel.