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 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 point-free quotients in terms of symplectic reduction and a canonical pseudo-Riemannian metric on the tangent bundle , from which we then derive necessary conditions, involving and , for the quotient metric to be conjugate point-free, 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 pseudo-Riemannian 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 self-parallel 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 point-free reducible Riemannian manifolds when one of the irreducible factors satisfies certain curvature conditions.

**1.**Raúl M. Aguilar,*Moment map, a product structure, and Riemannian metrics with no conjugate points*, Comm. Anal. Geom.**13**(2005), no. 2, 401–438. MR**2154825****2.**Raúl M. Aguilar,*Symplectic reduction and the homogeneous complex Monge-Ampère equation*, Ann. Global Anal. Geom.**19**(2001), no. 4, 327–353. MR**1842574**, 10.1023/A:1010715415333**3.**S. Bochner,*Vector fields and Ricci curvature*, Bull. Amer. Math. Soc.**52**(1946), 776–797. MR**0018022**, 10.1090/S0002-9904-1946-08647-4**4.**Nicolas Bourbaki,*Lie groups and Lie algebras. Chapters 1–3*, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1975 edition. MR**979493****5.**Peter Dombrowski,*On the geometry of the tangent bundle*, J. Reine Angew. Math.**210**(1962), 73–88. MR**0141050****6.**Victor Guillemin and Matthew Stenzel,*Grauert tubes and the homogeneous Monge-Ampère equation*, J. Differential Geom.**34**(1991), no. 2, 561–570. MR**1131444****7.**Robert Hermann,*A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle*, Proc. Amer. Math. Soc.**11**(1960), 236–242. MR**0112151**, 10.1090/S0002-9939-1960-0112151-4**8.**N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček,*Hyper-Kähler metrics and supersymmetry*, Comm. Math. Phys.**108**(1987), no. 4, 535–589. MR**877637****9.**Wilhelm Klingenberg,*Riemannian geometry*, de Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter & Co., Berlin-New York, 1982. MR**666697****10.**Shoshichi Kobayashi,*Transformation groups in differential geometry*, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1972 edition. MR**1336823****11.**Shoshichi Kobayashi and Katsumi Nomizu,*Foundations of differential geometry. Vol I*, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR**0152974****12.**Shoshichi Kobayashi and Katsumi Nomizu,*Foundations of differential geometry. Vol. II*, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR**0238225****13.**László Lempert and Róbert Szőke,*Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundle of Riemannian manifolds*, Math. Ann.**290**(1991), no. 4, 689–712. MR**1119947**, 10.1007/BF01459268**14.**Jerrold Marsden and Alan Weinstein,*Reduction of symplectic manifolds with symmetry*, Rep. Mathematical Phys.**5**(1974), no. 1, 121–130. MR**0402819****15.**Barrett O’Neill,*The fundamental equations of a submersion*, Michigan Math. J.**13**(1966), 459–469. MR**0200865**

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Additional Information

**Raúl M. Aguilar**

Affiliation:
Massachusetts Maritime Academy, Buzzards Bay, Massachusetts 02562

Email:
raguilar@maritime.edu

DOI:
https://doi.org/10.1090/S0002-9947-06-03920-1

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.