A characterization of all homogeneous real hypersurfaces in a complex projective space by observing the extrinsic shape of geodesics (Q1961707)

Let \(P_n(\mathbb{C})\) be an \(n\)-dimensional complex projective space with Fubini-Study metric of constant holomorphic sectional curvature 4, and let \(M\) be a real hypersurface of \(P_n(\mathbb{C})\). Then \(M\) has an almost contact metric structure \((\varphi,\xi,\eta,g)\) deduced from the Kähler structure of \(P_n(\mathbb{C})\). In [\textit{R. Takagi}, Osaka J. Math. 10, 495-506 (1973; Zbl 0274.53062)], a classification of homogeneous real hypersurfaces in \(P_n(\mathbb{C})\) as tubes of radius \(r\) over one of six Kähler submanifolds, denoted by \(A_1,A_2,B,C,D\) and \(E\), is given. In this paper, the authors are interested to give a characterization of homogeneous real hypersurfaces in \(P_n(\mathbb{C})\). The main result of this article is the following Theorem. Let \(M\) be a connected real hypersurface of \(P_n(\mathbb{C})\). Then \(M\) is locally congruent to a homogeneous real hypersurface if and only if there exist orthonormal vectors \(v_1,v_2,\dots,v_{2n-2}\) orthogonal to \(\xi\) at each point \(p\) of \(M\) such that all geodesics \(\gamma_i=\gamma_i(s)\) on \(M\) with \(\gamma_i(0)=p\) and \(\dot\gamma_i(0)=v_i\), \(i=1,\dots,2n-2\), are circles in \(P_n(\mathbb{C})\) with positive curvature. The proof of this theorem is given by considering, one by one, the six cases obtained by R. Takagi. For the converse, it is proved that every principal curvature is locally constant on an open dense subset \({\mathcal U}\) of the connected real hypersurface \(M\). Therefore, the continuity of all principal curvatures on \(M\) and Theorem \(B\) mentioned in the paper give the conclusion.