On two tensors associated to the structure Jacobi operator of a real hypersurface in complex projective space (Q6057470)

A generic real hypersurface \(M\) in complex projective space \(\mathbb{C}P^n\) inherits a contact metric structure from the complex structure of \(\mathbb{C}P^n\) and the Fubini-Study metric. Homogeneous hypersurfaces of this type have been classified by R.\ Takagi into six classes \(A_1\), \(A_2\), \(B\), \(C\), \(D\) and \(E\). The hypersurface \(M\) inherits a Webster-Tanaka connection, which can be deformed to a one-parameter family parametrized by a non-zero real number \(k\). It also inherits a family of so-called Cho-operators parametrized in the same way. Now the Riemann curvature of \(M\) gives rise to a Jacobi operator, which can be modified using either the Cho-operators or the torsion of the deformed Webster-Tanaka connections. The main results of the article are that symmetry properties of these tensors (for appropriate values of \(k\)) can be used to locally characterize hypersurfaces of type \(A\) and \(B\), respectively.