Purity and hybridness of two tensors on a real hypersurface in complex projective space

Abstract

On a real hypersurface $M$ in complex projective space, we can define two tensor fields of type (1, 2), $A_F^{\left(k\right)}$ and $A_T^{\left(k\right)}$, associated with the shape operator $A$ of the real hypersurface, for any nonnull real number $k$. Following Tachibana [Analytic tensor and its generalization, Tohoku Math. J. 12 (1960), 208–221], we study the purity and hybridness of such tensors with respect to $A$.