Moment Map, a Product Structure, and Riemannian Metrics with no Conjugate Points

Let (M, g) be a complete Riemannian manifold, G a group acting on M freely and properly by isometries with (B = M/G, g_B) its smooth Riemannian quotient. We prove in Theorem 1 the uniqueness of a certain integrable structure on the tangent bundle of M defined in symplectic terms (2.1) and prove in Theorem 2 its naturality with respect to the symplectic reduction corresponding to the tangential action by G. We define the notion of a "tangentially positive" isometric action and show in Theorem 3 how this condition implies that if (M, g) has no conjugate points its quotient (B, g_B) has no conjugate points, and that the strongly stable and unstable distributions in the unit tangent bundle of M are natural under symplectic reduction, by our Theorem 4. In particular, we obtain conditions under which having a geodesic flow of Anosov type is inherited by the Riemannian quotient. This work is followed up by [1] where we prove the converse of Theorem 3 and obtain some curvature restrictions for actions with conjugate point-free quotients.

Full Text (PDF format)

Published 1 January 2005