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

Abstract

AbstractLet 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, $$B_F^{(k)}$$ B F ( k ) and $$B_T^{(k)}$$ B T ( k ) . We will classify real hypersurfaces in complex projective space for which $$B_F^{(k)}$$ B F ( k ) and $$B_T^{(k)}$$ B T ( k ) either take values in the maximal holomorphic distribution $$\mathbb {D}$$ D or are parallel to the structure vector field $$\xi $$ ξ , in the particular case of $$B=A$$ B = A , where A denotes the shape operator of M. We also introduce the concept of $$A_F^{(k)}$$ A F ( k ) and $$A_T^{(k)}$$ A T ( k ) being $$\mathbb {D}$$ D -recurrent and classify real hypersurfaces such that either $$A_F^{(k)}$$ A F ( k ) or $$A_T^{(k)}$$ A T ( k ) are $$\mathbb {D}$$ D -recurrent.