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Let (M, g) be a complete Riemannian manifold, G a group acting on M freely and properly by isometries with (B = M/G, gB) 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, gB) 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.

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