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Geometric structures on 2-orbifolds : exploration of discrete symmetry / Suhyoung Choi.
Author
Choi, Suhyoung
[Browse]
Format
Book
Language
English
Published/Created
Tokyo, Japan : Mathematical Society of Japan, 2012.
Description
xii, 171 pages : illustrations ; 25 cm.
Availability
Available Online
Project Euclid Prime
Copies in the Library
Location
Call Number
Status
Location Service
Notes
Lewis Library - Stacks
QA641 .C493 2012
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Details
Subject(s)
Geometry, Differential
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Orbifolds
[Browse]
Symmetry (Mathematics)
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Series
MSJ memoirs ; v. 27.
[More in this series]
MSJ memoirs ; v. 27
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Bibliographic references
Includes bibliographical references (p. 163-167) and index.
Contents
Machine generated contents note: 1.Introduction
2.Chapter 2. Manifolds and structures
2.1.The review of topology
2.1.1.Manifolds
2.1.2.Some homotopy theory
2.1.3.Covering spaces and discrete group actions
2.1.4.Simplicial manifolds
2.2.Lie groups
2.2.1.Manifolds and tangent spaces
2.2.2.Lie groups
2.2.3.Lie algebras
2.2.4.Lie groups and Lie algebras
2.2.5.Lie group actions
2.3.Pseudo-groups and G-structures
2.3.1.G-manifolds
2.4.Differential geometry
2.4.1.Riemannian manifolds
2.4.2.Principal bundles and connections: flat connections
2.5.Notes
3.Chapter 3. Geometry and discrete groups
3.1.Geometries
3.1.1.Euclidean geometry
3.1.2.Spherical geometry
3.1.3.Affine geometry
3.1.4.Projective geometry
3.1.5.Conformal geometry
3.1.6.Hyperbolic geometry
3.1.7.Models of hyperbolic geometry
3.2.Discrete groups and discrete group actions
3.2.1.Convex polyhedrons
3.2.2.Convex polytopes
3.2.3.The fundamental domains of discrete group actions
3.2.4.Side pairings and the Poincare fundamental polyhedron theorem
3.2.5.Crystallographic groups
3.3.Notes
4.Chapter 4. Topology of orbifolds
4.1.Compact group actions
4.1.1.Tubes and slices
4.1.2.Locally smooth actions
4.1.3.Manifolds as quotient spaces
4.1.4.Smooth actions are locally smooth
4.1.5.Equivariant triangulation
4.2.The definition of orbifolds
4.2.1.Local groups and the singular set
4.2.2.Examples: good orbifolds
4.2.3.Examples: silvering
4.3.The definition as a groupoid
4.3.1.Groupoids
4.3.2.An abstract definition
4.4.Differentiable structures on orbifolds
4.4.1.Bundles over orbifolds
4.4.2.Tangent bundles and tensor bundles
4.4.3.The existence of a locally finite good covering
4.4.4.Silvering the boundary components
4.4.5.The Gauss-Bonnet theorem
4.5.Triangulation of smooth orbifolds
4.5.1.Triangulation of the stratified spaces
4.5.2.Orbifolds as stratified spaces
4.6.Covering spaces of orbifolds
4.6.1.The fiber product construction by Thurston
4.6.2.Universal covering orbifolds by fiber-products
4.7.The path-approach to the universal covering spaces following Hae-fliger
4.7.1.G-paths
4.7.2.Covering spaces and the fundamental group
4.8.Notes
5.Chapter 5. Topology of 2-orbifolds
5.1.The properties of 2-orbifolds
5.1.1.The triangulation of 2-orbifolds
5.1.2.The classification of 1-dimensional suborbifolds of 2-orbifolds
5.1.3.The orbifold Euler-characteristic for orbifolds due to Satake
5.1.4.The generalized Riemann-Hurwitz formula
5.1.5.A geometrization of 2-orbifolds a partial result
5.1.6.Good, very good, and bad 2-orbifolds
5.2.Topological operations on 2-orbifolds: constructions and decompositions
5.2.1.The definition of splitting and sewing 2-orbifolds
5.2.2.Regular neighborhoods of 1-orbifolds
5.2.3.Splitting and sewing on 2-orbifolds reinterpreted
5.2.4.Identification interpretations of splitting and sewing
5.3.Notes
6.Chapter 6. Geometry of orbifolds: geometric structures on orbifolds
6.1.The definition of geometric structures on orbifolds
6.1.1.An atlas of charts approach
6.1.2.The developing maps and holonomy homomorphisms
6.1.3.The definition as flat bundles with transversal sections
6.1.4.The equivalence of three notions
6.2.The definition of the deformation spaces of (G, X)-structures on orbifolds
6.2.1.The isotopy-equivalence space
6.2.2.The topology of the deformation space
6.2.3.The local homeomorphism theorem
6.3.Notes
7.Chapter 7. Deformation spaces of hyperbolic structures on 2-orbifolds
7.1.The definition of the Teichmuller space of 2-orbifolds
7.2.The geometric cutting and pasting and the deformation spaces
7.2.1.Geometric constructions
7.3.The decomposition of 2-orbifolds into elementary 2-orbifolds
7.3.1.Elementary 2-orbifolds
7.4.The Teichmuller spaces for 2-orbifolds
7.4.1.The strategy of the proof
7.4.2.The generalized hyperbolic triangle theorem
7.4.3.The proof of Proposition 7.4.1
7.4.4.The steps to prove Theorem 7.0.1
7.5.Notes
8.Chapter 8. Deformation spaces of real projective structures on 2-orbifolds
8.1.Introduction to real projective orbifolds
8.1.1.Examples of real projective 2-orbifolds
8.2.A survey of real projective structures on surfaces of negative Euler characteristic
8.2.1.Topological work
8.2.2.The gauge theory and projective structures
8.2.3.Hitchin's conjecture and the generalizations
8.2.4.Group theory and representations
8.3.Real projective structures on 2-orbifolds of negative Euler characteristic
8.3.1.Real projective 2-orbifolds and the Hitchin-Teichmuller components
8.3.2.Understanding the deformation space of real projective structures
8.4.Notes.
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ISBN
9784931469686
493146968X
OCLC
827728535
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Geometric structures on 2-orbifolds : exploration of discrete symmetry / Suhyoung Choi. [electronic resource]
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