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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Tangentially positive isometric actions and conjugate points
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by Raúl M. Aguilar PDF
Trans. Amer. Math. Soc. 359 (2007), 789-825 Request permission

Abstract:

Let $\left ({\mathrm {M}}, g\right )$ be a complete Riemannian manifold with no conjugate points and $f\colon \left ({\mathrm {M}}, g\right ) \to \left ({\mathrm {B}}, g_{\mathrm {B}}\right )$ a principal $G$-bundle, where $G$ is a Lie group acting by isometries and ${\mathrm {B}}$ the smooth quotient with $g_{\mathrm {B}}$ the Riemannian submersion metric. We obtain a characterization of conjugate point-free quotients $\left ({\mathrm {B}}, g_{\mathrm {B}}\right )$ in terms of symplectic reduction and a canonical pseudo-Riemannian metric on the tangent bundle $T{\mathrm {M}}$, from which we then derive necessary conditions, involving $G$ and ${\mathrm {M}}$, for the quotient metric to be conjugate point-free, particularly for ${\mathrm {M}}$ a reducible Riemannian manifold. Let ${\mu _G}\colon T{\mathrm {M}}\to {\mathfrak {G}}^*$, with ${\mathfrak {G}}$ the Lie Algebra of $G$, be the moment map of the tangential $G$-action on $T{\mathrm {M}}$ and let ${\mathbf {G}}_{\mathbf {P}}$ be the canonical pseudo-Riemannian metric on $T{\mathrm {M}}$ defined by the symplectic form $d\Theta$ and the map $F\colon T{\mathrm {M}}\to {\mathrm {M}}\times {\mathrm {M}}$, $F(z)=\left ( \exp (-z), \exp (z)\right )$. First we prove a theorem, stating that if ${\mathbf {G}}_{\mathbf {P}}$ is not positive definite on the action vector fields for the tangential action along ${\mu _G}^{-1}(0)$ then $\left ({\mathrm {B}},g_{\mathrm {B}}\right )$ acquires conjugate points. (We proved the converse result in 2005.) Then, we characterize self-parallel vector fields on ${\mathrm {M}}$ in terms of the positivity of the ${\mathbf {G}}_{\mathbf {P}}$-length of their tangential lifts along certain canonical subsets of $T{\mathrm {M}}$. We use this to derive some necessary conditions, on $G$ and ${\mathrm {M}}$, for actions to be tangentially positive on relevant subsets of $T{\mathrm {M}}$, 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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Additional Information
  • Raúl M. Aguilar
  • Affiliation: Massachusetts Maritime Academy, Buzzards Bay, Massachusetts 02562
  • Email: raguilar@maritime.edu
  • Received by editor(s): January 8, 2004
  • Received by editor(s) in revised form: December 16, 2004
  • Published electronically: September 11, 2006
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 789-825
  • MSC (2000): Primary 53C20, 53C22, 53D20, 53D25
  • DOI: https://doi.org/10.1090/S0002-9947-06-03920-1
  • MathSciNet review: 2255197