| Title:
|
Induced differential forms on manifolds of functions (English) |
| Author:
|
Vizman, Cornelia |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
47 |
| Issue:
|
3 |
| Year:
|
2011 |
| Pages:
|
201-215 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Differential forms on the Fréchet manifold $\mathcal{F}(S,M)$ of smooth functions on a compact $k$-dimensional manifold $S$ can be obtained in a natural way from pairs of differential forms on $M$ and $S$ by the hat pairing. Special cases are the transgression map $\Omega ^p(M)\rightarrow \Omega ^{p-k}(\mathcal{F}(S,M))$ (hat pairing with a constant function) and the bar map $\Omega ^p(M)\rightarrow \Omega ^p(\mathcal{F}(S,M))$ (hat pairing with a volume form). We develop a hat calculus similar to the tilda calculus for non-linear Grassmannians [6]. (English) |
| Keyword:
|
manifold of functions |
| Keyword:
|
fiber integral |
| Keyword:
|
diffeomorphism group |
| MSC:
|
11K11 |
| MSC:
|
22C22 |
| idZBL:
|
Zbl 1249.58005 |
| idMR:
|
MR2852381 |
| . |
| Date available:
|
2011-11-11T08:52:13Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141707 |
| . |
| Reference:
|
[1] Alekseev, A., Strobl, T.: Current algebras and differential geometry.J. High Energy Phys. 03 (2005), 14. MR 2151966 |
| Reference:
|
[2] Bonelli, G., Zabzine, M.: From current algebras for $p$–branes to topological $M$–theory.J. High Energy Phys. 09 (2005), 15. MR 2171351, 10.1088/1126-6708/2005/09/015 |
| Reference:
|
[3] Brylinski, J.–L.: Loop spaces, characteristic classes and geometric quantization.vol. 107, Progr. Math., Birkhäuser, 1993. Zbl 0823.55002, MR 1197353 |
| Reference:
|
[4] Gay–Balmaz, F., Vizman, C.: Dual pairs in fluid dynamics.Ann. Global Anal. Geom. (2011). MR 2860394, 10.1007/s10455-011-9267-z |
| Reference:
|
[5] Greub, W., Halperin, S., Vanstone, R.: Connections, curvature, and cohomology. Vol. I: De Rham cohomology of manifolds and vector bundles.Pure and Applied Mathematics, vol. 47, Academic Press, New York–London, 1972. Zbl 0322.58001, MR 0336650 |
| Reference:
|
[6] Haller, S., Vizman, C.: Non-linear Grassmannians as coadjoint orbits.Math. Ann. 329 (2004), 771–785. MR 2076685, 10.1007/s00208-004-0536-z |
| Reference:
|
[7] Hirsch, M. W.: Differential topology.vol. 33, Springer, 1976, Grad. Texts in Math. Zbl 0356.57001, MR 0448362 |
| Reference:
|
[8] Ismagilov, R. S.: Representations of infinite–dimensional groups.vol. 152, Ams. Math. Soc., 1996, Translations of Mathematical Monographs. Zbl 0856.22001, MR 1393939 |
| Reference:
|
[9] Ismagilov, R. S., Losik, M., Michor, P. W.: A 2–cocycle on a group of symplectomorphisms.Moscow Math. J. 6 (2006), 307–315. MR 2270616 |
| Reference:
|
[10] Kriegl, A., Michor, P. W.: The convenient setting of global analysis.vol. 53, Math. Surveys Monogr., 1997. Zbl 0889.58001, MR 1471480 |
| Reference:
|
[11] Marsden, J. E., Ratiu, T.: Introduction to Mechanics and Symmetry.2nd ed., Springer, 1999. Zbl 0933.70003, MR 1723696 |
| Reference:
|
[12] Marsden, J. E., Weinstein, A.: Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids.Phys. D 7 (1983), 305–323. Zbl 0576.58008, MR 0719058, 10.1016/0167-2789(83)90134-3 |
| Reference:
|
[13] Michor, P. W.: Manifolds of differentiable mappings.vol. 3, Shiva Math. Series, Shiva Publishing Ltd., Nantwich, 1980. Zbl 0433.58001, MR 0583436 |
| Reference:
|
[14] Roger, C.: Extensions centrales d’algèbres et de groupes de Lie de dimension infinie, algèbre de Virasoro et généralisations.Rep. Math. Phys. 35 (1995), 225–266. Zbl 0892.17018, MR 1377323, 10.1016/0034-4877(96)89288-3 |
| Reference:
|
[15] Vizman, C.: Lichnerowicz cocycles and central Lie group extensions.An. Univ. Vest Timis., Ser. Mat.–Inform. 48 (1–2) (2010), 285–297. Zbl 1224.58009, MR 2849342 |
| . |
