Invariant homogeneous structures on homogeneous real hypersurfaces in a complex projective space and an odd-dimensional sphere (Q5939529)

The Hopf fibration is a useful tool for studying the geometry of submanifolds in the complex projective space \(\mathbb{C} P_n\). The total space of such fibration is the odd-dimensional unit sphere \(S^{2n+1}\), so informations of submanifolds in \(\mathbb{C} P_n\) can be translated into informations in \(S^{2n+1}\) and vice versa. By using such method, R. Takagi classified homogeneous real hypersurfaces in \(\mathbb{C} P_n\) into five types (A)--(E). NEWLINENEWLINENEWLINEHomogeneous Riemannian manifolds can be characterized by means of the existence of homogeneous structure tensor; by using the theory of the homogeneous structures the author, in his previous papers, studied the naturally reductivity of homogeneous real hypersurfaces of type (A) and proved that the only natural reductive homogeneous real hypersurfaces in \(\mathbb{C} P_n\) are of type (A). In this paper the author determines a homogeneous structure on a real hypersurface of type (B) by using the almost contact metric structure and the shape operator. After, he investigated relations between homogeneous structures on submanifolds in \(\mathbb{C} P_n\) and \(S^{2n+1}\).