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Overview"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." Full Product DetailsAuthor: William M. GoldmanPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.272kg ISBN: 9781470471989ISBN 10: 1470471981 Pages: 437 Publication Date: 30 January 2023 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsPart 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 IndexReviewsAuthor InformationWilliam M. Goldman, University of Maryland, College Park, MD. Tab Content 6Author Website:Countries AvailableAll regions |
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