Homogeneous G-structures
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Homogeneous G‑structures Alfonso Giuseppe Tortorella1 · Luca Vitagliano2 · Ori Yudilevich1 Received: 20 July 2019 / Accepted: 29 February 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords G-structures · Contact structures · Atiyah algebroid Mathematics Subject Classification 53C10 (Primary) · 53D10
1 Introduction The theory of G-structures places a variety of geometric structures on equal footing, the idea being to encode a structure on a manifold M by its set of compatible frames, which (in many interesting examples) forms a reduction of the frame bundle of the manifold to a structure group G ⊂ GLn (ℝ) (with n = dim M ). The group G plays a key role in the theory, namely that of the linear model for the geometric structure. For example, a symplectic manifold induces a reduction of its frame bundle to the symplectic group, complex manifolds are modeled by the complex general linear group, Riemannian manifolds by the orthogonal group, volume forms by the special linear group, and so forth (see [5, 9, 13] for introductions to the theory of G-structures). The pattern that repeats itself in each example is as follows: Every structure, say one modeled by the group G, has a corresponding almost structure where the integrability * Luca Vitagliano [email protected] Alfonso Giuseppe Tortorella [email protected] Ori Yudilevich [email protected] 1
Department of Mathematics, KU Leuven, Celestijnenlaan 200B ‑ 3001, Leuven, Belgium
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DipMat, Università degli Studi di Salerno, via Giovanni Paolo II no 123, 84084 Fisciano, SA, Italy
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axiom is removed. The instances of the almost structure that a given manifold admits are in one-to-one correspondence with reductions of the frame bundle of the manifold to G, and, of those, the instances of the structure correspond to so-called integrable reductions, which means that the manifold admits an atlas of coordinate charts that are compatible with the reduction (see Sect. 4 for more details). For example, almost symplectic structures (i.e., non-degenerate 2-forms) on a given manifold are in one-to-one correspondence with reductions of the frame bundle of the manifold to the symplectic group, and the symplectic structures (i.e., closed non-degenerate 2-forms) correspond to the integrable reductions. In this case, integrability is equivalent to the existence of an atlas consisting of Darboux charts. Contact s
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