Real hypersurfaces of a complex projective space satisfying a pointwise nullity condition

Abstract

In this paper, we give a classification of real hypersurfaces of a complex projective space $CP^{n}$ satisfying a pointwise nullity condition for the structure vector field $\xi$ i.e., $R(X, Y)\xi=k\{\eta(Y)X-\eta(X)Y\}$, $k$ is a function, and further we prove a local structure theorem of real hypersurfaces of $CP^{n}$ which satisfies $R(X, A\xi)\xi=k\{\eta(A\xi)X-\eta(X)A\xi\}$. The motivation of the present paper is a well-known fact that $CP^{n}$ does not admit a real hypersurface of constant curvature.