New conditions on some derivatives of the structure Lie operator on a real hypersurface in complex projective space

Abstract Let M be a real hypersurface in complex projective space equipped with both the Levi-Civita and k th generalized Tanaka-Webster connections, for any nonnull real number k . For any operator B on M we can define two tensor fields of type (1,2) on M , $$\displaystyle B_F^{(k)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mi>F</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:mstyle> </mml:math> and $$\displaystyle B_T^{(k)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mi>T</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:mstyle> </mml:math> , related to both connections. In the particular case of $$\displaystyle B=L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> </mml:mstyle> </mml:math> , the structure Lie operator on M , we study purity and hybridness of $$\displaystyle L_F^{(k)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>F</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:mstyle> </mml:math> and $$\displaystyle L_T^{(k)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>T</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:mstyle> </mml:math> with respect to the structure operator $$\displaystyle \phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle> <mml:mi>ϕ</mml:mi> </mml:mstyle> </mml:math> .