Geometric structures on manifolds / William M. Goldman.

Format:

Book

Author/Creator:

Goldman, William Mark, author.

Series:

Graduate studies in mathematics ; v. 227.

Graduate studies in mathematics, 1065-7339 ; 227

Language:

English

Subjects (All):

Manifolds (Mathematics).

Physical Description:

li, 409 pages : illustrations ; 26 cm.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2022].

Summary:

"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"-- Page 4 of cover.

Contents:

Affine and projective geometry. Affine geometry

Projective geometry

Duality and Non-Euclidean geometry

Convexity

Geometric manifolds. Locally homogeneous geometric structures

Examples of geometric structures

Classification

Completeness

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. Transformation groups

Affine connections

Representations of nilpotent groups

4-dimensional filiform nilpotent Lie algebras

Semicontinuous functions

SL(2,C) and O(3.1)

Lagrangian foliations of symplectic manifolds.

Notes:

Includes bibliographical references and index.

ISBN:

9781470471033

1470471035

9781470471989

1470471981

OCLC:

1345242805