AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
The Global Solutions to a Cartan’s Realization Problem
About this Title
Rui Loja Fernandes and Ivan Struchiner
Publication: Memoirs of the American Mathematical Society
Publication Year:
2025; Volume 306, Number 1548
ISBNs: 978-1-4704-7341-9 (print); 978-1-4704-8056-1 (online)
DOI: https://doi.org/10.1090/memo/1548
Published electronically: January 27, 2025
Keywords: Cartan realization,
Lie groupoid,
Lie algebroid,
G-structure
Table of Contents
Chapters
- 1. Introduction
- 2. Prelude: An example
- 3. $G$-structure groupoids and connections
- 4. $G$-structure algebroids and connections
- 5. Construction of solutions
- 6. $G$-integrability
- 7. Solutions to Cartan’s Realization Problem
- 8. Symmetries and moduli space of solutions
- 9. $G$-structure algebroids and geometric structures
- 10. Extremal Kähler metrics on surfaces
Abstract
We introduce a systematic method to solve a type of Cartan’s realization problem. Our method builds upon a new theory of Lie algebroids and Lie groupoids with structure group and connection. This approach allows to find local as well as complete solutions, their symmetries, and to determine the moduli spaces of local and complete solutions. We illustrate our method with the problem of classification of extremal Kähler metrics on surfaces.- Henrique Bursztyn, David Iglesias-Ponte, and Jiang-Hua Lu, Dirac geometry and integration of Poisson homogeneous spaces, J. Differential Geom. 126 (2024), no. 3, 939–1000. MR 4753491, DOI 10.4310/jdg/1717348869
- Henrique Bursztyn, Francesco Noseda, and Chenchang Zhu, Principal actions of stacky Lie groupoids, Int. Math. Res. Not. IMRN 16 (2020), 5055–5125. MR 4139032, DOI 10.1093/imrn/rny142
- Robert L. Bryant, Bochner-Kähler metrics, J. Amer. Math. Soc. 14 (2001), no. 3, 623–715. MR 1824987, DOI 10.1090/S0894-0347-01-00366-6
- R. L. Bryant, Notes on exterior differential systems, Preprint, arXiv:1405.3116, (2014).
- Henrique Bursztyn, A brief introduction to Dirac manifolds, Geometric and topological methods for quantum field theory, Cambridge Univ. Press, Cambridge, 2013, pp. 4–38. MR 3098084
- Marius Crainic and Rui Loja Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), no. 2, 575–620. MR 1973056, DOI 10.4007/annals.2003.157.575
- Marius Crainic and Rui Loja Fernandes, Integrability of Poisson brackets, J. Differential Geom. 66 (2004), no. 1, 71–137. MR 2128714
- Marius Crainic and Rui Loja Fernandes, Lectures on integrability of Lie brackets, Lectures on Poisson geometry, Geom. Topol. Monogr., vol. 17, Geom. Topol. Publ., Coventry, 2011, pp. 1–107. MR 2795150, DOI 10.2140/gt
- Quo-Shin Chi, Sergey A. Merkulov, and Lorenz J. Schwachhöfer, On the existence of infinite series of exotic holonomies, Invent. Math. 126 (1996), no. 2, 391–411. MR 1411138, DOI 10.1007/s002220050104
- Michel Cahen and Lorenz J. Schwachhöfer, Special symplectic connections and Poisson geometry, Lett. Math. Phys. 69 (2004), 115–137. MR 2104439, DOI 10.1007/s11005-004-0474-4
- Michel Cahen and Lorenz J. Schwachhöfer, Special symplectic connections, J. Differential Geom. 83 (2009), no. 2, 229–271. MR 2577468
- Marius Crainic and Ori Yudilevich, Lie pseudogroups à la Cartan, Mem. Amer. Math. Soc. 303 (2024), no. 1527, v+99. MR 4837619, DOI 10.1090/memo/1527
- J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer-Verlag, Berlin, 2000. MR 1738431, DOI 10.1007/978-3-642-56936-4
- Rui Loja Fernandes and Ivan Struchiner, The classifying Lie algebroid of a geometric structure I: Classes of coframes, Trans. Amer. Math. Soc. 366 (2014), no. 5, 2419–2462. MR 3165644, DOI 10.1090/S0002-9947-2014-05973-4
- Rui Loja Fernandes and Ivan Struchiner, The classifying Lie algebroid of a geometric structure II: $G$-structures with connection, São Paulo J. Math. Sci. 15 (2021), no. 2, 524–570. MR 4341118, DOI 10.1007/s40863-021-00272-x
- Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1393940
- I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, vol. 91, Cambridge University Press, Cambridge, 2003. MR 2012261, DOI 10.1017/CBO9780511615450
- Sergei Merkulov and Lorenz Schwachhöfer, Classification of irreducible holonomies of torsion-free affine connections, Ann. of Math. (2) 150 (1999), no. 1, 77–149. MR 1715321, DOI 10.2307/121098
- Behrang Noohi, Fibrations of topological stacks, Adv. Math. 252 (2014), 612–640. MR 3144243, DOI 10.1016/j.aim.2013.11.008
- Lorenz J. Schwachhöfer, Connections with irreducible holonomy representations, Adv. Math. 160 (2001), no. 1, 1–80. MR 1831947, DOI 10.1006/aima.2000.1973
- Gábor Székelyhidi, An introduction to extremal Kähler metrics, Graduate Studies in Mathematics, vol. 152, American Mathematical Society, Providence, RI, 2014. MR 3186384, DOI 10.1090/gsm/152