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Geometric Structures on Manifolds
 
William M. Goldman University of Maryland, College Park, MD
Geometric Structures on Manifolds
Hardcover ISBN:  978-1-4704-7103-3
Product Code:  GSM/227
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Softcover ISBN:  978-1-4704-7198-9
Product Code:  GSM/227.S
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-7197-2
EPUB ISBN:  978-1-4704-7704-2
Product Code:  GSM/227.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-1-4704-7103-3
eBook: ISBN:  978-1-4704-7197-2
Product Code:  GSM/227.B
List Price: $210.00 $167.50
MAA Member Price: $189.00 $150.75
AMS Member Price: $168.00 $134.00
Softcover ISBN:  978-1-4704-7198-9
eBook: ISBN:  978-1-4704-7197-2
Product Code:  GSM/227.S.B
List Price: $170.00 $127.50
MAA Member Price: $153.00 $114.75
AMS Member Price: $136.00 $102.00
Please Note: Purchasing the eBook version includes access to both a PDF and EPUB version
Geometric Structures on Manifolds
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Geometric Structures on Manifolds
William M. Goldman University of Maryland, College Park, MD
Hardcover ISBN:  978-1-4704-7103-3
Product Code:  GSM/227
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Softcover ISBN:  978-1-4704-7198-9
Product Code:  GSM/227.S
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-7197-2
EPUB ISBN:  978-1-4704-7704-2
Product Code:  GSM/227.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-1-4704-7103-3
eBook ISBN:  978-1-4704-7197-2
Product Code:  GSM/227.B
List Price: $210.00 $167.50
MAA Member Price: $189.00 $150.75
AMS Member Price: $168.00 $134.00
Softcover ISBN:  978-1-4704-7198-9
eBook ISBN:  978-1-4704-7197-2
Product Code:  GSM/227.S.B
List Price: $170.00 $127.50
MAA Member Price: $153.00 $114.75
AMS Member Price: $136.00 $102.00
Please Note: Purchasing the eBook version includes access to both a PDF and EPUB version
  • Book Details
     
     
    Graduate Studies in Mathematics
    Volume: 2272022; 409 pp
    MSC: 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 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 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 self-contained and accessible to students who have taken first-year graduate courses in topology, smooth manifolds, differential geometry and Lie groups.

    Readership

    Graduate 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 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
  • 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 non-geometric 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 semester-long, intermediate level graduate course.

      Max Lahn (University of Michigan), MAA Reviews
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 2272022; 409 pp
MSC: 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 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 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 self-contained and accessible to students who have taken first-year graduate courses in topology, smooth manifolds, differential geometry and Lie groups.

Readership

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 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
  • 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 non-geometric 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 semester-long, intermediate level graduate course.

    Max Lahn (University of Michigan), MAA Reviews
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
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