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Geometry and topology of complete Lorentz spacetimes of constant curvature
, 2013
"... Abstract. We study proper, isometric actions of nonsolvable discrete groups Γ on the 3dimensional Minkowski space R2,1 as limits of actions on the 3dimensional antide Sitter space AdS3. To each such action is associated a deformation of a hyperbolic surface group Γ0 inside O(2, 1). When Γ0 is con ..."
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Abstract. We study proper, isometric actions of nonsolvable discrete groups Γ on the 3dimensional Minkowski space R2,1 as limits of actions on the 3dimensional antide Sitter space AdS3. To each such action is associated a deformation of a hyperbolic surface group Γ0 inside O(2, 1). When Γ0 is convex cocompact, we prove that Γ acts properly on R2,1 if and only if this grouplevel deformation is realized by a deformation of the quotient surface that everywhere contracts distances at a uniform rate. We give two applications in this case. (1) Tameness: A complete flat spacetime is homeomorphic to the interior of a compact manifold. (2) Geometric transition: A complete flat spacetime is the rescaled limit of collapsing AdS spacetimes. 1.
The infinitesimalization and reconstruction of locally homogeneous manifolds
 SIGMA
"... Abstract. A linear connection on a Lie algebroid is called a Cartan connection if it is suitably compatible with the Lie algebroid structure. Here we show that a smooth connected manifold M is locally homogeneous – i.e., admits an atlas of charts modeled on some homogeneous space G/H – if and only ..."
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Abstract. A linear connection on a Lie algebroid is called a Cartan connection if it is suitably compatible with the Lie algebroid structure. Here we show that a smooth connected manifold M is locally homogeneous – i.e., admits an atlas of charts modeled on some homogeneous space G/H – if and only if there exists a transitive Lie algebroid over M admitting a flat Cartan connection that is ‘geometrically closed’. It is shown how the torsion and monodromy of the connection determine the particular form of G/H. Under an additional completeness hypothesis, local homogeneity becomes global homogeneity, up to cover. Key words: locally homogeneous; Lie algebroid; Cartan connection; completeness 2010 Mathematics Subject Classification: 53C30; 53C15; 53C07
Symmetry, Integrability and Geometry: Methods and Applications The Infinitesimalization and Reconstruction of Locally Homogeneous Manifolds
"... Abstract. A linear connection on a Lie algebroid is called a Cartan connection if it is suitably compatible with the Lie algebroid structure. Here we show that a smooth connected manifold M is locally homogeneous – i.e., admits an atlas of charts modeled on some homogeneous space G/H – if and only i ..."
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Abstract. A linear connection on a Lie algebroid is called a Cartan connection if it is suitably compatible with the Lie algebroid structure. Here we show that a smooth connected manifold M is locally homogeneous – i.e., admits an atlas of charts modeled on some homogeneous space G/H – if and only if there exists a transitive Lie algebroid over M admitting a flat Cartan connection that is ‘geometrically closed’. It is shown how the torsion and monodromy of the connection determine the particular form of G/H. Under an additional completeness hypothesis, local homogeneity becomes global homogeneity, up to cover. Key words: locally homogeneous; Lie algebroid; Cartan connection; completeness 2010 Mathematics Subject Classification: 53C30; 53C15; 53C07
Surface group representations, Higgs bundles, and
, 2002
"... holomorphic triples ..."
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Rational curves and parabolic geometries
, 2008
"... The twistor transform of a parabolic geometry has two steps: lift up to a geometry of higher dimension, and then drop to a geometry of lower dimension. The first step is a functor, but the second requires some compatibility conditions. Local necessary conditions were uncovered by Andreas Čap [9]. I ..."
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The twistor transform of a parabolic geometry has two steps: lift up to a geometry of higher dimension, and then drop to a geometry of lower dimension. The first step is a functor, but the second requires some compatibility conditions. Local necessary conditions were uncovered by Andreas Čap [9]. I prove necessary and sufficient global conditions for complex parabolic geometries: rationality of curves defined by certain ordinary differential equations. I harness Mori’s bend–andbreak to show that any parabolic geometry on any closed Kähler manifold containing a rational curve is inherited from a parabolic geometry on a lower dimensional closed Kähler manifold. These results yield global theorems on complex ordinary differential equations, holomorphic 2plane fields on 5folds, and other complex geometric structures on
2.3. Developing maps and holonomy morphisms 2
"... Abstract. On all compact complex surfaces (modulo finite unramified coverings), we classify all of the locally homogeneous geometric structures which are locally isomorphic to the “exotic ” homogeneous surfaces of Lie. ..."
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Abstract. On all compact complex surfaces (modulo finite unramified coverings), we classify all of the locally homogeneous geometric structures which are locally isomorphic to the “exotic ” homogeneous surfaces of Lie.