Abstract
The theory of G-structures provides us with a unified framework for a large class of geometric structures, including symplectic, complex and Riemannian structures, as well as foliations and many others. Surprisingly, contact geometry—the “odd-dimensional counterpart” of symplectic geometry—does not fit naturally into this picture. In this paper, we introduce the notion of a homogeneous G-structure, which encompasses contact structures, as well as some other interesting examples that appear in the literature.
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Notes
For more details, we recommend the mathoverflow discussion https://mathoverflow.net/questions/281256/do-contact-and-cr-structures-have-corresponding-g-structures.
References
Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Birkhäuser, Boston (2002)
Cappelletti-Montano, B., De Nicola, A., Yudin, I.: A survey on cosymplectic geometry. Rev. Math. Phys. 25(10), 1343002 (2013)
Eliashberg, Y., Mishachev, N.: Introduction to the \(h\)-principle. Graduate Studies in Math, vol. 48. American Mathematical Society, Providence (2002)
Bruce, A.J., Grabowska, K., Grabowski, J.: Remarks on contact and Jacobi geometry. SIGMA Symmetry Integrability Geom. Methods Appl. 13, 59–80 (2017)
Crainic, M.: Mastermath course differential geometry (Lecture Notes) (2015). http://www.staff.science.uu.nl/~crain101/DG-2015/main10.pdf
Geiges, H.: An Introduction to Contact Topology, Volume 109 of Cambridge Stud. Adv. Math. Cambridge University Press, Cambridge (2008)
Gray, J.W.: Some global properties of contact structures. Ann. Math. 69(2), 421–450 (1959)
Guillemin, V., Miranda, E., Pires, A.R.: Symplectic and Poisson geometry on \(b\)-manifolds. Adv. Math. 264, 864–896 (2014)
Kobayashi, S.: Transformation groups in differential geometry. Springer-Verlag, New York-Heidelberg (1972). Ergeb. Math. Grenzgeb., Band 70
Mackenzie, K.C.H.: General Theory of Lie Groupoids and Lie Algebroids, Volume 213 of London Math. Soc. Lecture Note Ser. Cambridge University Press, Cambridge (2005)
Schnitzer, J., Vitagliano, L.: The local structure of generalized contact bundles. Int. Math. Res. Not. (2019). https://doi.org/10.1093/imrn/rnz009. (in press)
Širola, B.: Normalizers and self-normalizing subgroups II. Cent. Eur. J. Math. 9(6), 1317–1332 (2011)
Sternberg, S.: Lectures on Differential Geometry. Prentice-Hall Inc, Englewood Cliffs (1964)
Vitagliano, L.: Dirac-Jacobi bundles. J. Symplectic Geom. 16(2), 485–561 (2018)
Vitagliano, L., Wade, A.: Holomorphic Jacobi manifolds and holomorphic contact groupoids. Math. Z. (2019). https://doi.org/10.1007/s00209-019-02320-x. (in press)
Yano, K., Kon, M.: Structures on Manifolds, Volume 3 of Ser. Pure Math. World Scientific Publishing Co., Singapore (1984)
Acknowledgements
A. G. Tortorella was supported by an FWO postdoctoral fellowship. L. Vitagliano is a member of the GNSAGA of INdAM. O. Yudilevich was supported by the long-term structural funding— Methusalem grant of the Flemish Government, and by the FWO research project G083118N. The authors would also like to thank the Centre International de Recontre Mathématiques (CIRM) and its staff for their generous hospitality during our stay there as part of the Research in Pairs program, and the anonymous referee for his/her useful comments and suggestions.
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Tortorella, A.G., Vitagliano, L. & Yudilevich, O. Homogeneous G-structures. Annali di Matematica 199, 2357–2380 (2020). https://doi.org/10.1007/s10231-020-00972-9
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DOI: https://doi.org/10.1007/s10231-020-00972-9