New characterizations of ruled real hypersurfaces in complex projective space (Q6061196)

Let \(M\) be a real hypersurface in the complex projective space \(\mathbb{C}P^n\). Then there are several canonical connections on \(M\). On the one hand, there is the Levi-Civita connection of the restriction of the Fubini-Study metric. On the other hand, \(M\) inherits a contact metric structure, which in turn defines a family of generalized Webster-Tanaka connections parametrized by a non-zero real number. These connections can be used to associate to any symmetric \(\binom11\)-tensor field on \(M\) two families of \(\binom12\)-tensor fields. This is applied to the shape operator to obtain two families of such tensor fields. The main results of the article are characterizations of hypersurfaces for which the map on the contact distribution induced by multiplication by \(i\) is symmetric respectively skew-symmetric with respect to these tensor fields. In particular, this leads to new characterizations of ruled hypersurfaces.