Basic transformations of symmetric R-spaces (Q917943)

The irreducible symmetric R-spaces form an important class of compact symmetric spaces. Each of them admits a geometrically natural ``group of basic transformations'', which is greater than the corresponding group of isometries. (This is, e.g., the group of conformal transformations for a sphere, and the group of projective transformations for every classical projective space \(P_ n(F))\). The author gives a characterization of these basic groups in terms of Riemannian geometry. For each compact rank one symmetric space (except the sphere), the basic group is the group of all diffeomorphisms which preserve the system of all so-called Helgason spheres. For any irreducible symmetric R-space of rank greater than one, the basic group is the group of all diffeomorphisms which preserve the arithmetic distance d. (Let us note that the arithmetic distance is a metrical invariant derived from the system of Helgason spheres).