Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 099, 46 pages      arXiv:0911.0372      http://dx.doi.org/10.3842/SIGMA.2009.099

Geometric Structures on Spaces of Weighted Submanifolds

Brian Lee
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4 Canada

Received May 31, 2009, in final form October 25, 2009; Published online November 02, 2009

Abstract
In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on ''convenient'' vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold (M,ω), we construct a weak symplectic structure on each leaf Iw of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weighted Lagrangian submanifolds is equivalent to a heuristic weak symplectic structure of Weinstein [Adv. Math. 82 (1990), 133-159]. When the weightings are positive, these symplectic spaces are symplectomorphic to reductions of a weak symplectic structure of Donaldson [Asian J. Math. 3 (1999), 1-15] on the space of embeddings of a fixed compact oriented manifold into M. When M is compact, by generalizing a moment map of Weinstein we construct a symplectomorphism of each leaf Iw consisting of positive weighted isotropic submanifolds onto a coadjoint orbit of the group of Hamiltonian symplectomorphisms of M equipped with the Kirillov-Kostant-Souriau symplectic structure. After defining notions of Poisson algebras and Poisson manifolds, we prove that each space Iw can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds.

Key words: infinite dimensional manifolds; weakly symplectic structures; convenient vector spaces; Lagrangian submanifolds; isodrastic foliation.

pdf (520 kb)   ps (336 kb)   tex (51 kb)

References

  1. Binz E., Fischer H.R., The manifold of embeddings of a closed manifold, in Differential Geometric Methods in Mathematical Physics (Proc. Internat. Conf., Tech. Univ. Clausthal, Clausthal-Zellerfeld, 1978), Editor H.D. Doebner, Lecture Notes in Phys., Vol. 139, Springer, Berlin - New York, 1981, 310-329.
  2. Cushman R., Sniatycki J., Differential structure of orbit spaces, Canad. J. Math. 53 (2001), 715-755.
  3. Donaldson S., Moment maps and diffeomorphisms, Asian J. Math. 3 (1999), 1-15.
  4. Donaldson S., Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), 479-522.
  5. Ebin D., The manifold of Riemannian metrics, in Proceedings of Symposia in Pure Mathematics (1968, Berkeley), Global Analysis, American Mathematical Society, Providence, RI, 1970, 11-40.
  6. Frölicher A., Kriegl A., Linear spaces and differentiation theory, Pure and Applied Mathematics, John Wiley & Sons, Chichester, 1988.
  7. Hamilton R.S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 65-222.
  8. Khesin B., Lee P., Poisson geometry and first integrals of geostrophic equations, Phys. D 237 (2008), 2072-2077, arXiv:0802.4439.
  9. Kriegl A., Michor P.W., The convenient setting of global analysis, Mathematical Surveys and Monographs, Vol. 53, American Mathematical Society, Providence, RI, 1997.
  10. Moser J., On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 280-296.
  11. Ortega J.P., Ratiu T.S., The optimal momentum map, in Geometry, Mechanics, and Dynamics, Springer, New York, 2002, 329-362.
  12. Polterovich L., The geometry of the group of symplectic diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001.
  13. Sikorski R., Introduction to differential geometry, Biblioteka Matematyczna, Tom 42, PWN, Warsaw, 1972 (in Polish).
  14. Smolentsev N.K., Natural weak Riemannian structures in the space of Riemannian metrics, Sibirsk. Mat. Zh. 35 (1994), 439-445 (English transl.: Siberian Math. J. 35 (1994), 396-402).
  15. Souriau J.M., Groupes différentiels, in Differential Geometrical Methods in Mathematical Physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math., Vol. 836, Springer, Berlin - New York, 1980, 91-128.
  16. Stefan P., Accessibility and foliations with singularities, Bull. Amer. Math. Soc. 80 (1974), 1142-1145.
  17. Stefan P., Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc. (3) 29 (1974), 699-713.
  18. Sussmann H.J., Orbits of families of vector fields and integrability of systems with singularities, Bull. Amer. Math. Soc. 79 (1973), 197-199.
  19. Sussmann H.J., Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171-188.
  20. Warner F.W., Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, Vol. 94, Springer-Verlag, New York - Berlin, 1983.
  21. Weinstein A., Symplectic manifolds and their Lagrangian submanifolds, Adv. Math. 6 (1971), 329-346.
  22. Weinstein A., Neighborhood classification of isotropic embeddings, J. Differential Geom. 16 (1981), 125-128.
  23. Weinstein A., Connections of Berry and Hannay type for moving Lagrangian submanifolds, Adv. Math. 82 (1990), 133-159.

Previous article   Next article   Contents of Volume 5 (2009)