On conjugate loci and cut loci of compact symmetric spaces I

Abstract

In the present paper, we shall first study the topological structures of the cut locus and the conjugate locus of a compact symmetric space. We shall show that the cut locus C is the disjoint union of finite cell bundles Ex over compact manifolds B*. Here each Ex is the interior of a disk bundle Ex over B\ and C is obtained by the successive attachments of Ex at boundaries dEx of Ex. The conjugate locus can be also stratified in a similar way. We shall next give another stratification of the cut locus of an irreducible symmetric i?-space M=G/U by means of orbits of a certain subgroup U of G. We shall prove the following facts : M has finite[/-orbits, say Vo, Vu ・・・, Vr; There exists a unique open 17-orbit, say Vo, among them; Then the cut locus is the union of Vi, ・■・,Vr; Each Vi is described by means of generalized Schubert cells of M, and it has the structure of a vector bundle over a symmetric i?-space Bt. These results include those on cut loci of Grassmann manifolds, U(ri)}O(ri),U(n), SO(n) and U(2n)ISp(n) by Wong [12], [13], Sakai [5], [6]. We retain the. definitions and notations in Part I.