Flag structures on Seifert manifolds

Barbot, Thierry

doi:10.2140/gt.2001.5.227

Cited by 6 publications

(28 citation statements)

References 28 publications

(45 reference statements)

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“…More generally, we prove that all the deformations W ! PGL.3; R/ studied in our paper [2] are .G; Y /-Anosov. As a corollary, we obtain all the main results of [2] and extend them to any small deformation of 0 , not necessarily preserving a point or a projective line in the projective space: in particular, there is a ./-invariant solid torus in the flag variety.…”

mentioning

confidence: 79%

“…PGL.3; R/ studied in our paper [2] are .G; Y /-Anosov. As a corollary, we obtain all the main results of [2] and extend them to any small deformation of 0 , not necessarily preserving a point or a projective line in the projective space: in particular, there is a ./-invariant solid torus in the flag variety. The quotient space ./n is a flag manifold, naturally equipped with two 1-dimensional transversely projective foliations arising from the projections of the flag variety on the projective plane and its dual; if is strongly irreducible, these foliations are not minimal.…”

mentioning

confidence: 79%

“…In [2], we considered representations W ! G near horocyclic representations, ie, obtained from a faithful discrete representation !…”

Section: Introductionmentioning

confidence: 99%

“…H D SL.2; R/ composed with the natural morphism identifying H with the commutator subgroup of the stabilizer in G of a point p 0 and a projective line d 0 in the projective plane, with p 0 … d 0 . Actually, in [2], we only considered some deformations of horocyclic representations that we called hyperbolic representations. They are characterized by the following properties:…”

Section: Introductionmentioning

confidence: 99%

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Three-dimensional Anosov flag manifolds

Barbot

2010

Geom. Topol.

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Let be a surface group of higher genus. Let 0 W ! PGL.V / be a discrete faithful representation with image contained in the natural embedding of SL.2; R/ in PGL.3; R/ as a group preserving a point and a disjoint projective line in the projective plane. We prove that 0 is .G; Y /-Anosov (following the terminology of Labourie [15]), where Y is the frame bundle. More generally, we prove that all the deformations W ! PGL.3; R/ studied in our paper [2] are .G; Y /-Anosov. As a corollary, we obtain all the main results of [2] and extend them to any small deformation of 0 , not necessarily preserving a point or a projective line in the projective space: in particular, there is a ./-invariant solid torus in the flag variety. The quotient space ./n is a flag manifold, naturally equipped with two 1-dimensional transversely projective foliations arising from the projections of the flag variety on the projective plane and its dual; if is strongly irreducible, these foliations are not minimal. More precisely, if one of these foliations is minimal, then it is topologically conjugate to the strong stable foliation of a double covering of a geodesic flow, and preserves a point or a projective line in the projective plane. All these results hold for any .G; Y /-Anosov representation which is not quasi-Fuchsian, ie, does not preserve a strictly convex domain in the projective plane. 57M50

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confidence: 79%

mentioning

confidence: 79%

“…In [2], we considered representations W ! G near horocyclic representations, ie, obtained from a faithful discrete representation !…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Three-dimensional Anosov flag manifolds

Barbot

2010

Geom. Topol.

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“…Recently [102] they have also shown that a very wide class of Anosov representations as defined by Labourie [120], correspond to geometric structures on closed manifolds. (A much different class of Anosov representations of surface groups has recently been studied by Barbot [6,7].…”

Section: Complex Projective 1-manifolds Flat Conformal Structures Anmentioning

confidence: 99%

Locally Homogeneous Geometric Manifolds

Goldman

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Abstract.Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology Σ and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of X into Σ. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on Σ, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of Σ.We survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension, and how it leads to a general study of surface group representations. In particular geometric structures are a useful tool in understanding local and global properties of deformation spaces of representations of fundamental groups.Mathematics Subject Classification (2000). Primary 57M50; Secondary 57N16.Keywords. connection, curvature, fiber bundle, homogeneous space, Thurston geometrization of 3-manifolds, uniformization, crystallographic group, discrete group, proper action, Lie group, fundamental group, holonomy, completeness, development, geodesic, symplectic structure, Teichmüller space, Fricke space, hypebolic structure, Riemannian metric, Riemann surface, affine structure, projective structure, conformal structure, spherical CR structure, complex hyperbolic structure, deformation space, mapping class group, ergodic action. Historical backgroundWhile geometry involves quantitative measurements and rigid metric relations, topology deals with the loose quantitative organization of points. Felix Klein proposed in his 1872 Erlangen Program that the classical geometries be considered as the properties of a space invariant under a transitive Lie group action. Therefore one may ask which topologies support a system of local coordinates modeled on a fixed homogeneous space X = G/H such that on overlapping coordinate patches, the coordinate changes are locally restrictions of transformations from G. W. GoldmanIn this generality this question was first asked by Charles Ehresmann [55] at the conference "Quelques questions de Geométrie et de Topologie," in Geneva in 1935. Forty years later, the subject of such locally homogeneous geometric structures experienced a resurgence when W. Thurston placed his 3-dimensional geometrization program [158] in the context of locally homogeneous (Riemannian) structures. The rich diversity of geometries on homogeneous spaces brings in a wide range of techniques, and the field has thrived through their interaction.Before Ehresmann, the subject may be traced to several independent threads in the 19th century:• The theory of monodromy of Schwarzian differential equ...

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Flag structures on real 3-manifolds

Falbel

Veloso

2020

Geom Dedicata

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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.

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Flag structures on Seifert manifolds

Cited by 6 publications

References 28 publications

Three-dimensional Anosov flag manifolds

Three-dimensional Anosov flag manifolds

Locally Homogeneous Geometric Manifolds

Flag structures on real 3-manifolds

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