Locally homogeneous geometric manifolds

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 S and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of G/H into S. 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 S, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of S. In this talk I will 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.

Invited address for Geometry Section for 2010 International Congress of Mathematicians

References in corpus (11)

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• Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces
• Surface group representations in PU(p,q) and Higgs bundles
• Complex hyperbolic triangle groups
• Convex projective structures on Gromov--Thurston manifolds
• The topology of moduli spaces of free group representations
• Complex Hyperbolic Structures on Disc Bundles over Surfaces
• On dynamics of Out(F_n) on PSL(2,C) characters
• Flag Structures on Seifert Manifolds
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• Affine deformations of a three-holed sphere

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• A Geometric transition from hyperbolic to anti de Sitter geometry
• Families of Geometries, Real Algebras, and Transitions
• Two papers which changed my life: Milnor's seminal work on flat manifolds and flat bundles
• Topology of representation spaces of surface groups in PSL(2,R) with assigned boundary monodromy and nonzero Euler number
• Bi-Lagrangian structures and Teichmüller theory
• Flat vector bundles and open coverings
• Holomorphic geometric structures on Kaehler-Einstein manifolds
• Lifting locally homogeneous geometric structures
• Proof of Chern's conjecture on affine manifolds
• Limit polygons of convex domains in the projective plane
• Castling transformations of projective structures
• The Heisenberg Plane