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Foliations: Dynamics, Geometry and Topology 2014 ed. [Mīkstie vāki]

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  • Formāts: Paperback / softback, 198 pages, height x width: 240x168 mm, weight: 364 g, 10 Illustrations, color; 10 Illustrations, black and white; IX, 198 p. 20 illus., 10 illus. in color., 1 Paperback / softback
  • Sērija : Advanced Courses in Mathematics - CRM Barcelona
  • Izdošanas datums: 22-Dec-2014
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 3034808704
  • ISBN-13: 9783034808705
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Foliations: Dynamics, Geometry and Topology 2014 ed.Palielināt
  • Formāts: Paperback / softback, 198 pages, height x width: 240x168 mm, weight: 364 g, 10 Illustrations, color; 10 Illustrations, black and white; IX, 198 p. 20 illus., 10 illus. in color., 1 Paperback / softback
  • Sērija : Advanced Courses in Mathematics - CRM Barcelona
  • Izdošanas datums: 22-Dec-2014
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 3034808704
  • ISBN-13: 9783034808705
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783034808705.html
Keywords:
This book is an introduction to several active research topics in Foliation Theory and its connections with other areas. It contains expository lectures showing the diversity of ideas and methods converging in the study of foliations. The lectures by Aziz El Kacimi Alaoui provide an introduction to Foliation Theory with emphasis on examples and transverse structures. Steven Hurder"s lectures apply ideas from smooth dynamical systems to develop useful concepts in the study of foliations: limit sets and cycles for leaves, leafwise geodesic flow, transverse exponents, Pesin Theory and hyperbolic, parabolic and elliptic types of foliations. The lectures by Masayuki Asaoka compute the leafwise cohomology of foliations given by actions of Lie groups, and apply it to describe deformation of those actions. In his lectures, Ken Richardson studies the properties of transverse Dirac operators for Riemannian foliations and compact Lie group actions, and explains a recently proved index formul

a. Besides students and researchers of Foliation Theory, this book will be interesting for mathematicians interested in the applications to foliations of subjects like Topology of Manifolds, Differential Geometry, Dynamics, Cohomology or Global Analysis.

Fundamentals of Foliation Theory.- Foliation Dynamics.- Deformation of Locally Free Actions and Leafwise Cohomology.- Transversal Dirac Operators on Distributions, Foliations, and G-Manifolds.

Recenzijas

This book contains the lecture notes of four courses on several topics with rather different flavor, which are linked by their relation with Foliation Theory. the courses will be very helpful for any reader that wants to get quickly introduced to any of these lines of research. (Jesus A. Įlvarez López, zbMATH 1318.57001, 2015)

1 Deformation of Locally Free Actions and Leafwise Cohomology
1(40)
Masayuki Asaoka
Introduction
1(2)
1.1 Locally free actions and their deformations
3(4)
1.1.1 Locally free actions
3(2)
1.1.2 Rigidity and deformations of actions
5(2)
1.2 Rigidity and deformation of flows
7(4)
1.2.1 Parameter rigidity of locally free R-actions
7(3)
1.2.2 Deformation of flows
10(1)
1.3 Leafwise cohomology
11(8)
1.3.1 Definition and some basic properties
11(3)
1.3.2 Computation by Fourier analysis
14(2)
1.3.3 Computation by a Mayer--Vietoris argument
16(1)
1.3.4 Other examples
17(2)
1.4 Parameter deformation
19(11)
1.4.1 The canonical 1-form
19(3)
1.4.2 Parameter deformation of Rp-actions
22(2)
1.4.3 Parameter rigidity of some non-abelian actions
24(4)
1.4.4 A complete deformation for actions of GA
28(2)
1.5 Deformation of orbits
30(7)
1.5.1 Infinitesimal deformation of foliations
30(1)
1.5.2 Hamilton's criterion for local rigidity
31(1)
1.5.3 Existence of locally transverse deformations
32(2)
1.5.4 Transverse geometric structures
34(3)
Bibliography
37(4)
2 Fundaments of Foliation Theory
41(1)
Aziz El Kacimi Alaoui
Foreword
41(1)
Part I Foliations by Example
41(22)
2.1 Generalities
41(5)
2.1.1 Induced foliations
43(1)
2.1.2 Morphisms of foliations
44(1)
2.1.3 Frobenius Theorem
44(1)
2.1.4 Holonomy of a leaf
45(1)
2.2 Transverse structures
46(7)
2.2.1 Lie foliations
47(1)
2.2.2 Transversely parallelizable foliations
48(1)
2.2.3 Riemannian foliations
48(1)
2.2.4 G/H-foliations
49(3)
2.2.5 Transversely holomorphic foliations
52(1)
2.3 More examples
53(5)
2.3.1 Simple foliations
53(1)
2.3.2 Linear foliation on the torus T2
53(1)
2.3.3 One-dimensional foliations
54(1)
2.3.4 Reeb foliation on the 3-sphere S3
54(1)
2.3.5 Lie group actions
55(3)
2.4 Suspension of diffeomorphism groups
58(3)
2.4.1 General construction
58(1)
2.4.2 Examples
59(2)
2.5 Codimension 1 foliations
61(2)
2.5.1 Existence
61(1)
2.5.2 Topological behavior of leaves
62(1)
Part II A Digression: Basic Global Analysis
63(11)
2.6 Foliated bundles
64(2)
2.6.1 Examples
65(1)
2.7 Transversely elliptic operators
66(4)
2.8 Examples
70(4)
2.8.1 The basic de Rham complex
70(2)
2.8.2 The basic Dolbeault complex
72(2)
Part III Some Open Questions
74
2.9 Transversely elliptic operators
74(1)
2.9.1 Towards a basic index theory
74(1)
2.9.2 Existence of transversely elliptic operators
75(1)
2.9.3 Homotopy invariance of basic cohomology
75(1)
2.10 Complex foliations
75(4)
2.10.1 The ∂F-cohomology
76(3)
2.11 Deformations of Lie foliations
79(8)
2.11.1 Example of a deformation of an Abelian foliation
80(1)
2.11.2 Further questions
80(3)
Bibliography
83(4)
3 Lectures on Foliation Dynamics
87(64)
Steven Hurder
Introduction
87(2)
3.1 Foliation basics
89(2)
3.2 Topological dynamics
91(5)
3.3 Derivatives
96(5)
3.4 Counting
101(5)
3.5 Exponential complexity
106(6)
3.6 Entropy and exponent
112(5)
3.7 Minimal sets
117(3)
3.8 Classification schemes
120(3)
3.9 Matchbox manifolds
123(8)
3.10 Topological shape
131(3)
3.11 Shape dynamics
134(17)
Appendix A Homework
136(1)
Bibliography
137(14)
4 Transversal Dirac Operators on Distributions, Foliations, and G-Manifolds
151
Ken Richardson
Foreword
151(1)
4.1 Introduction to ordinary Dirac operators
152(11)
4.1.1 The Laplacian
152(2)
4.1.2 The ordinary Dirac operator
154(3)
4.1.3 Properties of Dirac operators
157(3)
4.1.4 The Atiyah--Singer Index Theorem
160(3)
4.2 Transversal Dirac operators on distributions
163(4)
4.3 Basic Dirac operators on Riemannian foliations
167(6)
4.3.1 Invariance of the spectrum of basic Dirac operators
167(3)
4.3.2 The basic de Rham operator
170(2)
4.3.3 Poincare duality and consequences
172(1)
4.4 Natural examples of transversal Dirac operators on G-manifolds
173(5)
4.4.1 Equivariant structure of the orthonormal frame bundle
173(3)
4.4.2 Dirac-type operators on the frame bundle
176(2)
4.5 Transverse index theory for G-manifolds and Riemannian foliations
178
4.5.1 Introduction: the equivariant index
178(3)
4.5.2 Stratifications of G-manifolds
181(2)
4.5.3 Equivariant desingularization
183(1)
4.5.4 The fine decomposition of an equivariant bundle
184(2)
4.5.5 Canonical isotropy G-bundles
186(1)
4.5.6 The equivariant index theorem
187(3)
4.5.7 The basic index theorem for Riemannian foliations
190(5)
Bibliography
195