Flat affine manifolds and their transformations
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
A. Medina · O. Saldarriaga
· A. Villabon
Flat affine manifolds and their transformations Received: 11 December 2019 / Accepted: 30 October 2020 Abstract. We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental form of the bundle of linear frames of the manifold. We show that the group of affine transformations of a real flat affine ndimensional manifold, acts on Rn leaving an open orbit when its dimension is greater than n. Moreover, when the dimension of the group of affine transformations is n, this orbit has discrete isotropy. For any given Lie subgroup H of affine transformations of the manifold, we show the existence of an associative envelope of the Lie algebra of H , relative to the connection. The case when M is a Lie group and H acts on G by left translations is particularly interesting. We also exhibit some results about flat affine manifolds whose group of affine transformations admits a flat affine bi-invariant structure. The paper is illustrated with several examples.
1. Introduction The objects of study of this paper are flat affine paracompact smooth manifolds with no boundary and their affine transformations. A well understanding of the category of Lagrangian manifolds assumes a good knowledge of the category of flat affine manifolds (Theorem 7.8 in [25], see also [8]). Recall that flat affine manifolds with holonomy reduced to G L n (Z) appear naturally in integrable systems and Mirror Symmetry (see [14]). For simplicity, in what follows M is a connected real n-dimensional manifold, P = L(M) its bundle of linear frames, θ its fundamental 1-form, a linear connection on P of connection form ω and ∇ the covariant derivative on M associated to . The pair (M, ∇) is called a flat affine manifold if the curvature and torsion Partially Supported by CODI, Universidad de Antioquia. Project Number 2015-7654. A. Medina: UMR 5149 du CNRS, Institute A. Grothendieck, France and Universidad de Antioquia, Université Montpellier, Montpellier, Colombia. e-mail: [email protected] O. Saldarriaga (B)· A. Villabon: Instituto de Matemáticas, Universidad de Antioquia, Medellín, Colombia. e-mail: [email protected]; [email protected] Mathematics Subject Classification: Primary: 57S20 · 54H15 · Secondary: 53C07 · 17D25
https://doi.org/10.1007/s00229-020-01262-7
A. Medina et al.
tensors of ∇ are both null. In the case where M is a Lie group and the connection is left invariant, we call it a flat affine Lie group. It is well known that a Lie group is flat affine if and only if its Lie algebra is endowed with a left symmetric product compatible with the bracket. For a rather complete overview of flat affine manifolds, the reader can refer to [11]. An affine transforma
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