Some Conditions Concerning the Shape Operator of a Real Hypersurface in Complex Projective Space

Abstract

Let M be a real hypersurface of a complex projective space. For any operator B on M and any nonnull real number k, we can define two tensor fields of type (1,2) on M, B_F^{(k)} and B_T^{(k)}. We will classify real hypersurfaces in complex projective space for which B_F^{(k)} and B_T^{(k)} either take values in the maximal holomorphic distribution mathbb {D} or are parallel to the structure vector field xi , in the particular case of B=A, where A denotes the shape operator of M. We also introduce the concept of A_F^{(k)} and A_T^{(k)} being mathbb {D}-recurrent and classify real hypersurfaces such that either A_F^{(k)} or A_T^{(k)} are mathbb {D}-recurrent.