Abstract
This paper presents the theory of non-smooth Lie group actions on chains of Banach manifolds. The rigorous functional analytic spaces are given to deal with quotients of such actions. A hydrodynamical example is studied in detail.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abraham, R., Marsden, J.E.: Foundations of Mechanics. Benjamin-Cummings Publ. Co, Updated 1985 version, reprinted by Perseus Publishing, 2nd edn (1978)
Bourbaki N. (1972) Groupes et algèbres de Lie. Chapitres 2 et 3. Hermann, Paris
Chernoff P.R., Marsden J.E. (1974) Properties of Infinite Dimensional Hamiltonian Systems, vol. 425 of Lecture Notes in Math. Springer, New York
Cushman R.H., Bates L.M. (1997) Global Aspects of Classical Integrable Systems. Birkäuser, Boston
Ebin D.G. (1968) The manifold of Riemannian metrics. Proc. Sympos. Pure Math. 15, 11–40
Ebin D.G., Marsden J.E. (1970) Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92: 102–163
Gay-Balmaz F., Ratiu T.S. (2005) The Lie–Poisson structure of the LAE-α equation. Dyn. PDE 2(1): 25–57
Isenberg J., Marsden J.E. (1982) A slice theorem for the space of solutions of Einstein equations. Phys. Rep. 89, 179–222
Karcher H. (1970) On the Hilbert manifold H 1(S 1, M) of closed curves. Comm. Pure Appl. Math. 23, 201–219
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I. Interscience Publishers, a division of John Wiley & Sons, New York (1963)
Loja Fernandes, R., Ortega, J.-P., Ratiu, T.S.: Momentum maps in Poisson geometry, preprint (2006)
Marsden, J.E., Misiolek, G., Ortega, J.-P., Perlmutter, M., Ratiu, T.S.: Hamiltonian Reduction by Stages, Lecture Notes in Mathematics (2006) (to appear)
Meyers S.B., Steenrod N. (1939) The group of isometries of a Riemannian manifold. Ann. Math. 40, 400–456
Ortega J-P., Ratiu T.S. (2004) Momentum Maps and Hamiltonian Reduction. Progress in Mathematics, vol. 222. Birkäuser, Boston
Palais R.S. (1968) On the existence of slices for actions of noncompact Lie groups. Ann. Math. 73, 295–323
Vasylkevych S., Marsden J.E. (2005) The Lie–Poisson structure of the Euler equations of an ideal fluid. Dyn PDE 2(4): 281–300
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gay-Balmaz, F., Ratiu, T.S. Group actions on chains of Banach manifolds and applications to fluid dynamics. Ann Glob Anal Geom 31, 287–328 (2007). https://doi.org/10.1007/s10455-007-9061-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-007-9061-0