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# Tangentially Positive Isometric Actions and Conjugate Points

Raúl M. Aguilar
Transactions of the American Mathematical Society
Vol. 359, No. 2 (Feb., 2007), pp. 789-825
Stable URL: http://www.jstor.org/stable/20161603
Page Count: 37
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## Abstract

Let (M,g) be a complete Riemannian manifold with no conjugate points and f: $(\text{M},g)\rightarrow (\text{B},g_{\text{B}})$ a principal G-bundle, where G is a Lie group acting by isometries and B the smooth quotient with $g_{\text{B}}$ the Riemannian submersion metric. We obtain a characterization of conjugate point-free quotients $(\text{B},g_{\text{B}})$ 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 $\mu _{G}\colon TM\rightarrow \germ{G}^{\ast}$ , with $\germ{G}$ the Lie Algebra of G, be the moment map of the tangential G-action on TM and let ${\bf G}_{{\bf P}}$ be the canonical pseudo-Riemannian metric on TM defined by the symplectic form dΘ and the map F: TM → M × M, F(z) = (exp(-z),exp(z)). First we prove a theorem, stating that if ${\bf G}_{{\bf P}}$ is not positive definite on the action vector fields for the tangential action along $\mu _{G}{}^{-1}(0)$ then $(\text{B},g_{\text{B}})$ 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 ${\bf G}_{{\bf P}}$ -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.

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