Extrinsic geodesics and hypersurfaces of type (A) in a complex projective space (Q1004662)

A real hypersurface \(M\) in a complex projective space \(\mathbb C\mathbb{P}^n\) is said to be of type (A) if it is a tube around \(\mathbb C\mathbb{P}^k\), where \(0\leq k\leq n- 1\). If \(k= 0\) or \(k= n- 1\), then \(M\) is called a real hypersurface of type (A\(_1\)). Otherwise, \(M\) is called a hypersurface of type (A\(_2\)). The authors give a characterization of all real hypersurfaces in the class of Hopf hypersurfaces in \(\mathbb C\mathbb{P}^n\). By using this result, they distinguish hypersurfaces of type (A\(_1\)) from hypersurfaces of type (A\(_2\)) in terms of the cardinality of congruence classes of their extrinsic geodesics.