Flag structures on real 3-manifolds
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Flag structures on real 3-manifolds E. Falbel1 · J. M. Veloso2 Received: 11 September 2018 / Accepted: 19 March 2020 © Springer Nature B.V. 2020
Abstract We define flag structures on a real three manifold M as the choice of two complex lines on the complexified tangent space at each point of M. We suppose that the plane field defined by the complex lines is a contact plane and construct an adapted connection on an appropriate principal bundle. This includes path geometries and CR structures as special cases. We prove that the null curvature models are given by totally real submanifolds in the flag space SL(3, C)/B, where B is the subgroup of upper triangular matrices. We also define a global invariant which is analogous to the Chern–Simons secondary class invariant for three manifolds with a Riemannian structure and to the Burns–Epstein invariant in the case of CR structures. It turns out to be constant on homotopy classes of totally real immersions in flag space. Keywords CR structure · Path geometry · Flag manifolds · Totally real embedding · Cartan connection
1 Introduction Path geometries and CR structures on real three manifolds were studied by Elie Cartan in a long series of papers (see [10–12] and [5] for a beautiful account of this work). Both geometries have models which are obtained through real forms of a complex group. More precisely, the group SL(3, C) acts by projective transformations on both points in P(C3 ) and ∗ ∗ its lines viewed as P(C3 ). The space of flags F ⊂ P(C3 ) × P(C3 ) of lines containing points is described as the homogeneous space SL(3, C)/B where B is the subgroup of upper triangular matrices. The path geometry of the flag space is defined by the two projections onto points and lines in projective space. Indeed, the kernels of the differentials of each projection
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E. Falbel [email protected] J. M. Veloso [email protected]
1
Sorbonne Université Faculté de Sciences and Institut de Mathématiques de Jussieu-Paris Rive Gauche, CNRS UMR 7586 and INRIA EPI-OURAGAN, Sorbonne Université, 4, Place Jussieu, 75252 Paris Cedex 05, France
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Faculdade de Matemática - ICEN, Universidade Federal do Pará, 66059 Belém, PA, Brazil
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Geometriae Dedicata
define two complex lines in the tangent space of the flag space at each point. It turns out that the planes generated in this way form a contact plane field. The two real models appear as closed orbits of the two non-compact real forms SL(3, R) and SU(2, 1) in the space of flags. In this work we define a structure over a real manifold which interpolates between these two geometries. Namely, the structure is a choice of two complex lines in the complexified tangent space at each point. We call it a flag structure. Path geometry and CR geometry correspond to a particular choice of lines adapted to the real structure of the real forms. Flat structures modeled on SU(2, 1) are known as spherical CR structures and have been studied since Cartan. But it is not known to what extent a 3-manifold may be equipped with such a structure (see [9,15,2
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