Tangentially positive isometric actions and conjugate points
Author:
Raúl M. Aguilar
Journal:
Trans. Amer. Math. Soc. 359 (2007), 789825
MSC (2000):
Primary 53C20, 53C22, 53D20, 53D25
Published electronically:
September 11, 2006
MathSciNet review:
2255197
Fulltext PDF Free Access
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Abstract: Let be a complete Riemannian manifold with no conjugate points and a principal bundle, where is a Lie group acting by isometries and the smooth quotient with the Riemannian submersion metric. We obtain a characterization of conjugate pointfree quotients in terms of symplectic reduction and a canonical pseudoRiemannian metric on the tangent bundle , from which we then derive necessary conditions, involving and , for the quotient metric to be conjugate pointfree, particularly for a reducible Riemannian manifold. Let , with the Lie Algebra of , be the moment map of the tangential action on and let be the canonical pseudoRiemannian metric on defined by the symplectic form and the map , . First we prove a theorem, stating that if is not positive definite on the action vector fields for the tangential action along then acquires conjugate points. (We proved the converse result in 2005.) Then, we characterize selfparallel vector fields on in terms of the positivity of the length of their tangential lifts along certain canonical subsets of . We use this to derive some necessary conditions, on and , for actions to be tangentially positive on relevant subsets of , which we then apply to isometric actions on complete conjugate pointfree 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
DOI:
http://dx.doi.org/10.1090/S0002994706039201
PII:
S 00029947(06)039201
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.
