Isometry Groups of Manifolds of Negative Curvature

W. Byers
1976 Proceedings of the American Mathematical Society  
Solvable subgroups of the isometry groups of a simply-connected manifold of negative curvature are characterized and this characterization is used to show that the isometry group of the universal Riemannian covering of a compact manifold of negative curvature is either discrete or semisimple. Introduction. A number of recent papers have related the geometry of manifolds of negative curvature to the algebra of various groups of isometries (for example [4], [8]). In this paper we study various
more » ... we study various groups of isometries of a simply-connected manifold M of negative curvature. In Theorem 5 we use results of Bishop and O'Neill [2] to show that a solvable group of isometries either leave a single geodesic invariant, permute a class of asymptotic geodesies, or else have a nonempty fixed point set. If the total isometry group I(M) does not satisfy either of two former conditions, we show in Theorem 7 that there is a compact normal subgroup K such that I(M)/K is semisimple and acts effectively on a closed, connected, totally convex submanifold of M. Using these results we show in Theorem 9 that if M is the universal Riemannian covering of a compact manifold of negative curvature, then the isometry group 7(Af) is either discrete or semisimple. This may be viewed as an extension of the classical situation where the compact manifold may be considered as a double coset space T\G/7<" of a connected semisimple Lie group G and where the symmetric space G/K can be given an invariant metric of nonpositive curvature so that G is isomorphic to the identity component of the isometry group [5] .
doi:10.2307/2040801 fatcat:musfi7qhmnahnn4xreu6uzxzb4