The cut locus and the diastasis of a Hermitian symmetric space of compact type
The main result of this paper is the following theorem: Let M be a Hermitian symmetric space of compact type and D the diastasis of M. Then, for each point p in M, the cut locus \(C_ p(M)\) is equal to the set of points q at which D(p,q) cannot be defined. Here the cut locus \(C_ p(M)\) is determined by that of its maximal totally geodesic flat submanifold [see \textit{T. Sakai,} Hokkaido Math. J. 6, 136-161 (1977; Zbl 0361.53048). The diastasis D of a Kähler manifold M is a real analytic function defined on its domain of real analyticity in \(M\times M\) containing the diagonal set and behaves as the square of the geodesic distance in the small [see \textit{E. Calabi,} Ann. Math., II. Ser. 58, 1-23 (1953; Zbl 0051.131)]. This result gives a relation between the cut locus of a Hermitian symmetric space M of compact type and that of its symmetric Kähler submanifolds \(M_ 1: C_ p(M_ 1)=C_ p(M)\cap M_ 1\) for each point p in \(M_ 1\). In particular, for a complex projective space M the cut locus \(C_ p(M_ 1)\) is a hyperplane section of \(M_ 1\subset M\).
scientific article; zbMATH DE number 3919246