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Full Description
The theory of geometric structures on manifolds which are locally modeled on a homogeneous space of a Lie group traces back to Charles Ehresmann in the 1930s, although many examples had been studied previously. Such locally homogeneous geometric structures are special cases of Cartan connections where the associated curvature vanishes. This theory received a big boost in the 1970s when W. Thurston put his geometrization program for 3-manifolds in this context. The subject of this book is more ambitious in scope. Unlike Thurston's eight 3-dimensional geometries, it covers structures which are not metric structures, such as affine and projective structures.
This book describes the known examples in dimensions one, two and three. Each geometry has its own special features, which provide special tools in its study. Emphasis is given to the inter-relationships between different geometries and how one kind of geometric structure induces structures modeled on a different geometry. Up to now, much of the literature has been somewhat inaccessible and the book collects many of the pieces into one unified work. This book focuses on several successful classification problems. Namely, fix a geometry in the sense of Klein and a topological manifold. Then the different ways of locally putting the geometry on the manifold lead to a ""moduli space"". Often the moduli space carries a rich geometry of its own reflecting the model geometry.
The book is self-contained and accessible to students who have taken first-year graduate courses in topology, smooth manifolds, differential geometry and Lie groups.
Contents
Part 1. Affine and projective geometry: Affine geometry
Projective geometry
Duality and non-Euclidean geometry
Convexity
Part 2. Geometric manifolds: Locally homogeneous geometric structures
Examples of geometric structures
Classification
Completeness
Part 3. Affine and projective structures: Affine structures on surfaces and the Euler characteristic
Affine Lie groups
Parallel volume and completeness
Hyperbolicity
Projective structures on surfaces
Complex-projective structures
Geometric structures on 3-manifolds
Appendices: Appendix A. Transformation groups
Appendix B. Affine connections
Appendix C. Representations of nilpotent groups
Appendix D. 4-dimensional filiform nilpotent Lie algebras
Appendix E. Semicontinuous functions
Appendix F. $\mathsf{SL}(2,\mathbb{C})$ and $O(3,1)$
Appendix G. Lagrangian foliations of symplectic manifolds
Bibliography
Index
