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Transactions of the American Mathematical Society

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Tangentially positive isometric actions and conjugate points

Author: Raúl M. Aguilar
Journal: Trans. Amer. Math. Soc. 359 (2007), 789-825
MSC (2000): Primary 53C20, 53C22, 53D20, 53D25
Published electronically: September 11, 2006
MathSciNet review: 2255197
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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

Keywords: Moment map, isometric action, conjugate points, symplectic reduction.
Received by editor(s): January 8, 2004
Received by editor(s) in revised form: December 16, 2004
Published electronically: September 11, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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