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Book DetailsGraduate Studies in MathematicsVolume: 227; 2022; 409 ppMSC: Primary 22; 51; 53; 57
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 3manifolds in this context. The subject of this book is more ambitious in scope. Unlike Thurston's eight 3dimensional 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 interrelationships 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 selfcontained and accessible to students who have taken firstyear graduate courses in topology, smooth manifolds, differential geometry and Lie groups.
ReadershipGraduate students and researchers interested in higher Teichmüller theory, character varieties, and deformations of geometric structures on manifolds.

Table of Contents

Part 1. Affine and projective geometry

Affine geometry

Projective geometry

Duality and nonEuclidean 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

Complexprojective structures

Geometric structures on 3manifolds

Appendices

Appendix A. Transformation groups

Appendix B. Affine connections

Appendix C. Representations of nilpotent groups

Appendix D. 4dimensional 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


Additional Material

Reviews

A constant quality of this book is the abundance of exercises to arouse more curiosity in the reader. ...This book should be of great interest to a wide spectrum of readers, ranging from curious readers to researchers through nongeometric teachers.
Michel Nguiffo B. Boyom, MathSciNet 
'Geometric Structures on Manifolds' is an accessible, sophisticated, modern, and grounded introduction to the theory of (G,X)structures on manifolds, which standardize the notion of spaces locally modeled on manifolds with geometry, aimed at a graduate level audience and above...[and] would make an excellent reference text upon which to base a semesterlong, intermediate level graduate course.
Max Lahn (University of Michigan), MAA Reviews


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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 3manifolds in this context. The subject of this book is more ambitious in scope. Unlike Thurston's eight 3dimensional 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 interrelationships 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 selfcontained and accessible to students who have taken firstyear graduate courses in topology, smooth manifolds, differential geometry and Lie groups.
Graduate students and researchers interested in higher Teichmüller theory, character varieties, and deformations of geometric structures on manifolds.

Part 1. Affine and projective geometry

Affine geometry

Projective geometry

Duality and nonEuclidean 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

Complexprojective structures

Geometric structures on 3manifolds

Appendices

Appendix A. Transformation groups

Appendix B. Affine connections

Appendix C. Representations of nilpotent groups

Appendix D. 4dimensional 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

A constant quality of this book is the abundance of exercises to arouse more curiosity in the reader. ...This book should be of great interest to a wide spectrum of readers, ranging from curious readers to researchers through nongeometric teachers.
Michel Nguiffo B. Boyom, MathSciNet 
'Geometric Structures on Manifolds' is an accessible, sophisticated, modern, and grounded introduction to the theory of (G,X)structures on manifolds, which standardize the notion of spaces locally modeled on manifolds with geometry, aimed at a graduate level audience and above...[and] would make an excellent reference text upon which to base a semesterlong, intermediate level graduate course.
Max Lahn (University of Michigan), MAA Reviews