Geodesic Flows on Symmetric Riemann Spaces

Abstract

Let G be a connected non-compact semi-simple Lie group whose center is finite and K a maximal compact subgroup of G. We denote by G/K the homogeneous space of cosets gK, g e G. Then G/K is a real analytic manifold with the natural analytic structure of a homogeneous space. If (M, 9 denote the Lie algebras over the real numbers of U, K respectively, then the tangent space at the point K of the manifold G/K is naturally isomorphic to the linear space (/9. If we denote by e the orthogonal complement of Q in (M with respect to the fundamental bilinear form on (, then (D is the direct sum in the sense of vector spaces of 9 and A, @ = S + @, so that 65/a _. Thus the tangent space at the point K of the manifold G/K is naturally isomorphic to e and we shall simply identify it with A. The restriction of the fundamental bilinear form to e is positive definite and by group translation one obtains a positive definite G-invariant Riemann-metric Q on the manifold G/K. It is a globally symmetric complete Riemann space in the sense (of Elie Cartan [1, 2, 3, 4]) that the geodesic symmetry about each point can be extended to an isometry of the whole Riemann-manifold GIK. Now let r be a discrete subgroup of G. In accordance with a result of C. L. Siegel [14] r acts properly discontinuously on G/K. If no element (o9) of r has' fixed points e G/K, we obtain from G/K by identification modulo r a real analytic manifold r G/K whose points are the double cosets r gK. It is clear that the natural map gK -r gK of G/K onto r G/K is a covering map (in fact G/K is the universal covering space of r G/K) which we can use to define a Riemann-metric on r \U/K. Thus r G/K becomes a locally symmetric Riemann space in the sense that the geodesic symmetry about each point is an isometry of some neighborhood of this point. We wish to study the geodesic flow of the locally symmetric Riemann space r\U/K.