Tangentially positive isometric actions and conjugate points

Abstract

Let (M, g) be a complete Riemannian manifold with no conjugate points and f: (M,g) → (B,g B ) a principal G-bundle, where G is a Lie group acting by isometries and B the smooth quotient with g B the Riemannian submersion metric. We obtain a characterization of conjugate point-free quotients (B, g B ) in terms of symplectic reduction and a canonical pseudo-Riemannian metric on the tangent bundle TM, from which we then derive necessary conditions, involving G and M, for the quotient metric to be conjugate point-free, particularly for M a reducible Riemannian manifold. Let μ G : TM → ®*, with ® the Lie Algebra of G, be the moment map of the tangential G-action on TM and let Gp be the canonical pseudo-Riemannian metric on TM defined by the symplectic form dΘ and the map F: TM → M x M, F(z) = (exp(-z),exp(z)). First we prove a theorem, stating that if Gp is not positive definite on the action vector fields for the tangential action along μG -1 (0) then (B, g B ) acquires conjugate points. (We proved the converse result in 2005.) Then, we characterize self-parallel vector fields on M in terms of the positivity of the Gp-length of their tangential lifts along certain canonical subsets of TM. We use this to derive some necessary conditions, on G and M, for actions to be tangentially positive on relevant subsets of TM, which we then apply to isometric actions on complete conjugate point-free reducible Riemannian manifolds when one of the irreducible factors satisfies certain curvature conditions.