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# Communications in Analysis and Geometry

## Volume 13 (2005)

### Number 2

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

Pages: 401 – 438

DOI: http://dx.doi.org/10.4310/CAG.2005.v13.n2.a6

#### Author

#### Abstract

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.