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Variationally complete representations are polar
Author(s):
Antonio
J.
Di Scala;
Carlos
Olmos
Journal:
Proc. Amer. Math. Soc.
129
(2001),
3445-3446.
MSC (1991):
Primary 53C40;
Secondary 53C35
Posted:
May 30, 2001
MathSciNet review:
1845024
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Abstract:
A recent result of C. Gorodski and G. Thorbergsson, involving classification, asserts that a variationally complete representation is polar. The aim of this paper is to give a conceptual and very short proof of this fact, which is the converse of a result of Conlon.
References:
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- [B]
- Bott, R., An application of the Morse theory to the topology of Lie groups, Bull. Soc. Math. France 84 (1956), 251-281. MR 19:291a
- [BS]
- Bott, R. and Samelson, H., Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964-1029. Correction in vol. 83 (1961), 207-208. MR 21:4430; MR 30:589
- [C]
- Conlon, L., Variational completeness and K-transversal domains, J. Differential Geom. 5 (1971), 135-147. MR 45:4320
- [D]
- Dadok, J., Polar coordinates induced by action of compact Lie groups, Trans. Amer. Math. Soc. 288 (1985), 125-137. MR 86k:22019
- [Di]
- Di Scala, A.J., Minimal homogeneous submanifolds in euclidean spaces, to appear in Annals of Global Analysis and Geometry.
- [EH]
- Eschenburg, J.-H. and Heintze, E., Polar representations and symmetric spaces, J. reine angew. Math. 507 (1999), 93-106. MR 2000f:53072
- [GT]
- C. Gorodski and G. Thorbergsson, Representations of compact Lie Groups and the osculating spaces of their orbits, Preprint Mathematisches Institut der Universität zu Köln (September 2000).
- [O]
- Olmos, C., Orbits of rank one and parallel mean curvature, Trans. Amer. Math. Soc. 347, No. 8 (1995), 2927-2939. MR 96a:53069
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- Terng, C. L. and Thorbergsson G., Taut Immersions into Complete Riemannian Manifolds, Tight and Taut Submanifolds, MSRI Publications 32 (1997), 181-228. MR 99m:53122
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Additional Information:
Antonio
J.
Di Scala
Affiliation:
Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina
Email:
discala@mate.uncor.edu
Carlos
Olmos
Affiliation:
Department of Mathematics, Ciudad Universitaria, 5000 Córdoba, Argentina
Email:
olmos@mate.uncor.edu
DOI:
10.1090/S0002-9939-01-06226-8
PII:
S 0002-9939(01)06226-8
Keywords:
Variationally complete,
polar representations,
$s$-representations
Received by editor(s):
November 9, 2000
Received by editor(s) in revised form:
December 6, 2000
Posted:
May 30, 2001
Additional Notes:
This research was supported by Universidad Nacional de Córdoba, CONICET, CONICOR, Secyt-UNC, ANPCyT and CIEM
Communicated by:
Wolfgang Ziller
Copyright of article:
Copyright
2001,
American Mathematical Society
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