Transformation groups on compact symmetric spaces

Abstract

The symmetric spaces constitute the most important class of Riemannian manifolds; some of them have been the standard spaces in various branches of geometry, and many authors threw light upon their deep properties. Still there seems to be no thorough study of general transformation groups L (other than the isometry groups H, but containing G) of compact(2) symmetric spaces M, which we call geometric transformation groups of M in this introduction. Its need will be patent if it will reveal interrelations of symmetric spaces, and if L will be the automorphism group of some geometric structure of M, more or less closely related with the Riemannian structure of M, by which M is geometrically distinguished from the other symmetric spaces. Let us observe a few examples. Let M be the sphere as a symmetric space. M has the projective [respectively, conformal] structure; it can be thought of as the set of all geodesics [respectively, the function which gives the angles between two tangent vectors at the same points] of M. This can be defined, of course, for any Riemannian manifold, but the automorphism group L, or the projective [respectively, conformal] transformation group, differs from the isometry group G for the sphere M, and by this fact, M is distinguished from all other symmetric spaces (except the real projective space which is locally isometric with M), as asserted by E. Cartan [Oeuvres completes, Partie I, Vol. II, Gauthier-Villars, Paris, 1952, p. 659]. (See [7], [8] for the proof.) M is the standard space in the projective [respectively, conformal] differential geometry. And, 'a la F. Klein, this group L on M gives rise to the (real) projective [respectively, conformal or Moebius] geometry. Next, to observe another example, we select a compact hermitian symmetric space for M. The structure is the complex structure connected with the Riemannian metric in a certain way. The automorphism group L is the holomorphic transformation group. L is a complex Lie group whose complex structure essentially determines that of M.