No Access

Geometry of infinite dimensional Cartan developments

Johanna Michor

https://orcid.org/0000-0001-7520-6239

Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria

E-mail Address: Johanna.Michor@univie.ac.at

Search for more papers by this author

and

Peter W. Michor

https://orcid.org/0000-0002-5279-248X

Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria

E-mail Address: Peter.Michor@univie.ac.at

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S2972458924500072Cited by:0 (Source: Crossref)

Abstract

The Cartan development takes a Lie algebra valued 1-form satisfying the Maurer–Cartan equation on a simply connected manifold $M$ to a smooth mapping from $M$ into the Lie group. In this paper, this is generalized to infinite dimensional $M$ for infinite dimensional regular Lie groups. The Cartan development is viewed as a generalization of the evolution map of a regular Lie group. The tangent mapping of a Cartan development is identified as another Cartan development.

Keywords:

AMSC: 58B25

References

1. D. V. Alekseevsky and P. W. Michor, Differential geometry of Cartan connections, Publ. Math. Debrecen 47(3–4) (1995) 349–375. Crossref, Google Scholar
2. E. Cartan, La méthode du repère mobile, la théorie des groupes continus et les espaces généralisées, (Exposés de géométrie V), Actualités Scientifiques et Industrielles, Nr. 194 (Hermann, Paris, 1935), p. 65. Google Scholar
3. E. Cartan, La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile, Leçons professées à la Sorbonne, Rédigées par J. Leray, Cahiers scientifiques, Fasc. 18 (Gauthier-Villars, 1937). Google Scholar
4. T. Diez, A. Futaki and T. Ratiu, Cartan geometry and infinite-dimensional Kempf-Ness theory, preprint (2024), arXiv:2405.20864. Google Scholar
5. A. S. Fokas and I. M. Gelfand, Surfaces on Lie groups, on Lie algebras, and their integrability, Commun. Math. Phys. 177(1) (1996) 203–220, With an appendix by Juan Carlos Alvarez Paiva. Crossref, Google Scholar
6. H. Glöckner, Regularity properties of infinite-dimensional Lie groups, and semiregularity, preprint (2012), arXiv:1208.0715. Google Scholar
7. H. Glöckner, Measurable regularity properties of infinite-dimensional Lie groups, preprint (2015), arXiv:1601.02568. Google Scholar
8. P. Griffiths, On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41 (1974) 775–814. Crossref, Google Scholar
9. M. Hanusch, Differentiability of the evolution map and Mackey continuity, Forum Math. 31(5) (2019) 1139–1177. Crossref, Google Scholar
10. M. Hanusch, The regularity problem for Lie groups with asymptotic estimate Lie algebras, Indag. Math. (N.S.) 31(1) (2020) 152–176. Crossref, Google Scholar
11. M. Hanusch, Regularity of Lie groups, Commun. Anal. Geom. 30(1) (2022) 53–152. Crossref, Google Scholar
12. I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry (Springer-Verlag, Berlin, 1993). Crossref, Google Scholar
13. A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, Vol. 53 (American Mathematical Society, Providence, RI, 1997), http://www.ams.org/online_bks/surv53/. Crossref, Google Scholar
14. A. Kriegl and P. W. Michor, Regular infinite-dimensional Lie groups, J. Lie Theory 7(1) (1997) 61–99. Google Scholar
15. P. W. Michor, Manifolds of Differentiable Mappings, Shiva Mathematics Series, Vol. 3 (Shiva Publishing, Nantwich, 1980). Google Scholar
16. P. W. Michor, A convenient setting for differential geometry and global analysis. I, Cah. Topol. Géom. Différ. 25(1) (1984) 63–109. Google Scholar
17. P. W. Michor, A convenient setting for differential geometry and global analysis. II, Cah. Topol. Géom. Différ. 25(2) (1984) 113–178. Google Scholar
18. P. W. Michor, Manifolds of mappings for continuum mechanics, in Geometric Continuum Mechanics, Advances in Continuum Mechanics, Vol. 42 (Birkhäuser Basel, 2020), pp. 3–75. Crossref, Google Scholar
19. J. Milnor, Remarks on infinite-dimensional Lie groups, in Relativity, Groups and Topology, II (Les Houches, 1983) (North-Holland, Amsterdam, 1984), pp. 1007–1057. Google Scholar
20. H. Omori, Y. Maeda and A. Yoshioka, On regular Fréchet Lie groups I. Some differential geometric expressions of Fourier integral operators on a Riemannian manifold, Tokyo J. Math. 3 (1980) 353–390. Google Scholar
21. H. Omori, Y. Maeda and A. Yoshioka, On regular Fréchet Lie groups II. Composition rules of Fourier Integral operators on a Riemannian manifold, Tokyo J. Math. 4 (1981) 221–253. Google Scholar
22. H. Omori, Y. Maeda and A. Yoshioka, On regular Fréchet Lie groups III, Tokyo J. Math. 4 (1981) 255–277. Google Scholar
23. H. Omori, Y. Maeda and A. Yoshioka, On regular Fréchet Lie groups IV. Definitions and fundamental theorems, Tokyo J. Math. 5 (1982) 365–398. Google Scholar
24. H. Omori, Y. Maeda and A. Yoshioka, On regular Fréchet Lie groups V. Several basic properties, Tokyo J. Math. 6 (1983) 39–64. Google Scholar
25. H. Omori, Y. Maeda, A. Yoshioka and O. Kobayashi, On regular Fréchet Lie groups VI. Infinite dimensional Lie groups which appear in general relativity, Tokyo J. Math. 6 (1983) 217–246. Google Scholar
26. A. L. Onishchik, On the classification of fiber spaces, Sov. Math. Doklady 2 (1961) 1561–1564. Google Scholar
27. A. L. Onishchik, Connections with zero curvature and the de Rham theorem, Sov. Math. Doklady 5 (1964) 1654–1657. Google Scholar
28. A. L. Onishchik, Some concepts and applications of non-abelian cohomology theory, Trans. Moscow Math. Soc. 17 (1967) 49–98. Google Scholar
29. Wikipedia, Convenient vector space — Wikipedia, the free encyclopedia, 2015, accessed on 25 February 2024. Google Scholar