Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Basic differential forms
for actions of Lie groups

Author: Peter W. Michor
Journal: Proc. Amer. Math. Soc. 124 (1996), 1633-1642
MSC (1991): Primary 57S15, 20F55
MathSciNet review: 1307550
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A section of a Riemannian $G$-manifold $M$ is a closed submanifold $\Sigma $ which meets each orbit orthogonally. It is shown that the algebra of $G$-invariant differential forms on $M$ which are horizontal in the sense that they kill every vector which is tangent to some orbit, is isomorphic to the algebra of those differential forms on $\Sigma $ which are invariant with respect to the generalized Weyl group of $\Sigma $, under some condition.

References [Enhancements On Off] (What's this?)

  • 1. Atiyah, M.; Bott, R., The moment map and equivariant cohomology, Topology 23 (1984), 1--28. MR 85e:58041
  • 2. Berline, N.; Getzler, E.; Vergne, M., Heat kernels and differential operators, Grundlehren math. Wiss. 298, Springer-Verlag, Berlin, Heidelberg, New York, 1992. MR 94e:58130
  • 3. Borel, A., Seminar on transformation groups, Annals of Math. Studies, Princeton Univ. Press, Princeton, 1960. MR 22:7129
  • 4. Bott, R.; Samelson, H., Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964--1029. MR 21:4430
  • 5. Cartan, H., Notions d'algèbre differentielle; application aux groupes de Lie et aux variétés où opère un group de Lie, Colloque de Topologie, C.B.R.M., Bruxelles, 1950, pp. 15--27. MR 13:107e
  • 6. Cartan, H., La transgression dans un group de Lie et dans un espace fibré principal, Colloque de Topologie, C.B.R.M., Bruxelles, 1950, pp. 57--71. MR 13:107f
  • 7. Chevalley, C., Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782. MR 17:345d
  • 8. Conlon, L., Variational completeness and K-transversal domains, J. Differential Geom. 5 (1971), 135--147. MR 45:4320
  • 9. Conlon, L., A class of variationally complete representations, J. Differential Geom. 7 (1972), 149--160. MR 51:14123
  • 10. Dadok, J., Polar coordinates induced by actions of compact Lie groups, TAMS 288 (1985), 125--137. MR 86k:22019
  • 11. Greub, Werner; Halperin, Steve; Vanstone, Ray, Connections, Curvature, and Cohomology III, Academic Press, New York and London, 1976. MR 53:4110
  • 12. Humphreys, J. E., Reflection groups and Coxeter groups, Cambridge studies in advanced mathematics 29, Cambridge University Press, Cambridge, 1990, 1992. MR 92h:20002
  • 13. Kunz, Ernst, Kähler Differentials, Viehweg, Braunschweig - Wiesbaden, 1986. MR 88e:14025
  • 14. Luna, D., Fonctions différentiables invariantes sous l'operation d'un groupe réductif, Ann. Inst. Fourier, Grenoble 26 (1976), 33--49. MR 54:11377
  • 15. Nagata, M., On the 14-th problem of Hilbert, Amer. J. Math. 81 (1959), 766--772. MR 21:4151
  • 16. Nagata, M., Lectures on the fourteenth problem of Hilbert, Tata Inst. of Fund. Research, Bombay, 1965. MR 35:6663
  • 17. Onishchik, A. L., On invariants and almost invariants of compact Lie transformation groups, Trudy Mosk. Math. Obshch. 35 (1976), 235--264; Trans. Moscow Math. Soc. N. 1, 1979, pp. 237--267.
  • 18. Palais, R., On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295--323. MR 23:A3802
  • 19. Palais, R. S.; Terng, C. L., A general theory of canonical forms, Trans. AMS 300 (1987), 771-789. MR 88f:57069
  • 20. Palais, R. S.; Terng, C. L., Critical point theory and submanifold geometry, Lecture Notes in Mathematics 1353, Springer-Verlag, Berlin, 1988. MR 90c:53143
  • 21. Popov, V. L., Groups, generators, syzygies, and orbits in invariant theory, Translations of mathematical monographs 100, Amer. Math. Soc., Providence, 1992. MR 93g:14054
  • 22. Ronga, F., Stabilité locale des applications equivariantes, Differential Topology and Geometry, Dijon 1974, Lecture Notes in Math. 484, Springer-Verlag, 1975, pp. 23--35. MR 56:3866
  • 23. Schwarz, G. W., Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63--68. MR 51:6870
  • 24. Schwarz, G. W., Lifting smooth homotopies of orbit spaces, Publ. Math. IHES 51 (1980), 37--136. MR 81h:57024
  • 25. Solomon, L., Invariants of finite reflection groups, Nagoya Math. J. 22 (1963), 57--64. MR 27:4872
  • 26. Szenthe, J., A generalization of the Weyl group, Acta Math. Hungarica 41 (1983), 347--357. MR 85b:57044
  • 27. Szenthe, J., Orthogonally transversal submanifolds and the generalizations of the Weyl group, Period. Math. Hungarica 15 (1984), 281--299. MR 86m:53065
  • 28. Terng, C. L., Isoparametric submanifolds and their Coxeter groups, J. Diff. Geom. 1985 (21), 79--107. MR 87e:53095

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 57S15, 20F55

Retrieve articles in all journals with MSC (1991): 57S15, 20F55

Additional Information

Peter W. Michor
Affiliation: Erwin Schrödinger International Institute of Mathematical Physics, Wien, Austria, Institut für Mathematik, Universität Wien, Austria; Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria

Keywords: Orbits, sections, basic differential forms
Received by editor(s): April 6, 1994
Received by editor(s) in revised form: November 9, 1994
Additional Notes: Supported by Project P 10037–PHY of ‘Fonds zur Förderung der wissenschaftlichen Forschung’.
Communicated by: Roe W. Goodman
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society