A characterization of Einstein real hypersurfaces in quaternionic projective space

Abstract

On a real hypersurface of quatemionic projective space $QP^{m}$ we study the following condition: $\mathfrak{S}(R(X, Y)SZ)=0$ where $\mathfrak{S}$ denotes the cyclic sum, $R$, respectively $S$, the curvature tensor, respectively the Ricci tensor, of the real hypersurface and $X,Y\in \mathscr{D}$, $Z\in \mathscr{D}^{\perp}, \mathscr{D}$ and $\mathscr{D}^{\perp}$ being certain distributions on the real hypersurface. We prove that such a real hypersurface must be Einstein.