Homogeneous manifolds with negative curvature. I

This paper solves the problem of determining which Lie groups act simply transitively on a Riemannian manifold with negative curvature. The results obtained extend those of Heintze for the case of strictly negative curvature. Using results of Wolf and Heintze, it is established that every connected, simply connected, homogeneous manifold M with negative curvature admits a Lie group S acting simply transitively by isometries and every group with this property must be solvable. Formulas for the curvature tensor on M are established and used to show that the Lie algebra of any such group S must satisfy a number of structural conditions. Conversely, given a Lie algebra < satisfying these conditions and any member of an easily constructed family of inner products on i, a metric deforma- tion argument is used to obtain a modified inner product which gives rise to a left invariant Riemannian structure with negative curvature on the associated simply connected Lie group. 1. Introduction. This paper was motivated by the following problem: Which connected Lie groups admit a left invariant Riemannian metric with nega- tive (sectional) curvature? We emphasize that throughout the paper, we under- stand "negative" to mean "less than or equal to zero". Since the property in question is not sensitive to groups linked by a local isomorphism, we deal primar- ily with simply connected groups. Results of J. A. Wolf (13) and E. Heintze (4) show that the above problem is closely linked with the classification of connected, homogeneous Riemannian manifolds with negative curvature. Indeed, if M is such a manifold and if M is simply connected, then M is isometric to a solvable Lie group endowed with a left-invariant metric. In this paper, we give a complete solution to our original problem by show- ing that a necessary and sufficient condition for a group to have the property in question is that its Lie algebra be what we call an "NC algebra". Roughly speak- ing, the crucial properties of an NC algebra $ are that in addition to being solv- able, e must contain an abelian subalgebra a complementary to the derived