~SL(2,R) geometry

~SL(2,R) geometry
Name	~SL(2,R) geometry
Model	universal cover of SL(2,R)
Curvature	variable (sectional)
Isometry group	Lie group of isometries

~SL(2,R) geometry

~SL(2,R) geometry is a homogeneous three-dimensional geometry arising as the geometry of the universal covering group of SL(2,R) in the context of the Thurston classification of three-manifold geometries. It appears in the study of Seifert fibered spaces, mapping tori of pseudo-Anosov homeomorphisms, and links to the theory of hyperbolic surfaces, Teichmüller theory, and contact topology. Connections to the work of William Thurston, André Weil, Dennis Sullivan, William Goldman, and John Milnor illuminate its role in low-dimensional topology and geometric group theory.

Introduction

The geometry is modeled on the universal cover of SL(2,R), a three-dimensional Lie group closely connected to the geometry of the hyperbolic plane studied by Nikolai Lobachevsky, János Bolyai, and Henri Poincaré. It figures prominently in Thurston's list alongside Euclidean geometry, Spherical geometry, Hyperbolic geometry, S^2 × R geometry, H^2 × R geometry, Nil geometry, and Sol geometry. Important developments have involved interactions with the theories of Riemann surface, Teichmüller space, Fuchsian group, Kleinian group, and Anosov flow.

Model and Metric

A common model uses the universal cover of SL(2,R) endowed with a left-invariant metric induced from the Killing form, paralleling constructions in the work of Élie Cartan and Harish-Chandra. Coordinates relate to the unit tangent bundle of the hyperbolic plane H^2 studied by Henri Poincaré and Ludwig Schläfli; local charts resemble those in descriptions by Charles B. Morrey and Dusa McDuff. The metric is not isotropic like in Spherical geometry or Euclidean geometry and differs from product metrics such as H^2 × R geometry; metric tensors are constructed using structure constants familiar from Sophus Lie and Élie Cartan's theory.

Isometry Group and Symmetries

The full isometry group contains a copy of the left action of the universal cover of SL(2,R) and extends by discrete symmetries comparable to those in the theory of Fuchsian group actions and SL(2,Z). Relations to symmetry groups appear in the works of Élie Cartan, Hermann Weyl, and Felix Klein on transformation groups. Isometry classifications employ methods from Élie Cartan's moving frames and analyses analogous to those used by Wilhelm Killing in Lie algebra classification.

Geodesics and Distance Properties

Geodesic flow on the unit tangent bundle connects with Anosov flows first studied by Dmitri Anosov and explored in the dynamics literature influenced by Stephen Smale and Yakov Sinai. Closed geodesics relate to periodic orbits in the study of Ruelle zeta function and spectral invariants examined by Atle Selberg and Hans Maass. Distance estimates draw on comparisons with Hyperbolic geometry geodesic behavior investigated by Henri Poincaré and rigidity results reminiscent of work by Grigori Margulis and Mostow.

Curvature and Geometric Invariants

Sectional curvatures in ~SL(2,R) geometry vary with direction, echoing curvature analyses from Élie Cartan and modern treatments by Misha Gromov in large-scale geometry. Invariants such as the Ricci tensor and scalar curvature appear in studies by John Milnor on left-invariant metrics and by Richard Hamilton in Ricci flow contexts inspired by Grigori Perelman's work on three-manifolds. Chern--Weil theory contributions from Shiing-Shen Chern and Hermann Weyl inform characteristic class computations for bundles over manifolds modeled on this geometry.

Foliations and Submanifold Geometry

Foliations by fibers mirror the Seifert fibering studied by Herbert Seifert and appear in analyses of contact structures linked to Yakov Eliashberg and William Thurston's work on confoliations. Surface submanifolds correspond to immersed hyperbolic surfaces as in Poincaré's and Ahlfors' investigations of Riemann surfaces; incompressible surfaces relate to Haken manifold theory developed by Wolfgang Haken. Studies of totally geodesic submanifolds reference techniques from Shing-Tung Yau and Jeffrey Weeks in geometric topology.

Relationships to Other Thurston Geometries

~SL(2,R) geometry occupies a niche in Thurston's geometrization framework alongside Nil geometry, Sol geometry, H^2 × R geometry, and S^2 × R geometry, interacting with classification results by William Thurston, John Morgan, and Gang Tian. Transitions between modeled geometries occur in degeneration phenomena explored by Curtis McMullen and Richard Canary in deformation spaces for Kleinian group representations. The geometry underpins examples of Seifert fibered spaces studied in the work of Milnor and Neumann and plays a role in modern studies by Peter Scott and Jaco-Shalen-Johannson decomposition techniques.

Category:Thurston geometries