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Lectures On The Geometry Of Manifolds (Third Edition) [Minkštas viršelis]

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(Univ Of Notre Dame, Usa)
  • Formatas: Paperback / softback, 700 pages
  • Išleidimo metai: 22-Oct-2020
  • Leidėjas: World Scientific Publishing Co Pte Ltd
  • ISBN-10: 9811215952
  • ISBN-13: 9789811215957
Kitos knygos pagal šią temą:
Lectures On The Geometry Of Manifolds (Third Edition)Didesnis paveikslėlis
  • Formatas: Paperback / softback, 700 pages
  • Išleidimo metai: 22-Oct-2020
  • Leidėjas: World Scientific Publishing Co Pte Ltd
  • ISBN-10: 9811215952
  • ISBN-13: 9789811215957
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9789811215957.html
Raktažodžiai:

The goal of this book is to introduce the reader to some of the main techniques, ideas and concepts frequently used in modern geometry. It starts from scratch and it covers basic topics such as differential and integral calculus on manifolds, connections on vector bundles and their curvatures, basic Riemannian geometry, calculus of variations, DeRham cohomology, integral geometry (tube and Crofton formulas), characteristic classes, elliptic equations on manifolds and Dirac operators. The new edition contains a new chapter on spectral geometry presenting recent results which appear here for the first time in printed form.

Preface vii
1 Manifolds
1(22)
1.1 Preliminaries
1(5)
1.1.1 Space and Coordinatization
1(2)
1.1.2 The implicit function theorem
3(3)
1.2 Smooth manifolds
6(17)
1.2.1 Basic definitions
6(3)
1.2.2 Partitions of unity
9(1)
1.2.3 Examples
10(10)
1.2.4 How many manifolds are there?
20(3)
2 Natural Constructions on Manifolds
23(56)
2.1 The tangent bundle
23(18)
2.1.1 Tangent spaces
23(4)
2.1.2 The tangent bundle
27(2)
2.1.3 Transversality
29(4)
2.1.4 Vector bundles
33(4)
2.1.5 Some examples of vector bundles
37(4)
2.2 A linear algebra interlude
41(26)
2.2.1 Tensor products
41(5)
2.2.2 Symmetric and skew-symmetric tensors
46(7)
2.2.3 The "super" slang
53(3)
2.2.4 Duality
56(8)
2.2.5 Some complex linear algebra
64(3)
2.3 Tensor fields
67(12)
2.3.1 Operations with vector bundles
67(2)
2.3.2 Tensor fields
69(4)
2.3.3 Fiber bundles
73(6)
3 Calculus on Manifolds
79(60)
3.1 The Lie derivative
79(9)
3.1.1 Flows on manifolds
79(2)
3.1.2 The Lie derivative
81(5)
3.1.3 Examples
86(2)
3.2 Derivations of ω (M)
88(6)
3.2.1 The exterior derivative
88(5)
3.2.2 Examples
93(1)
3.3 Connections on vector bundles
94(17)
3.3.1 Covariant derivatives
94(6)
3.3.2 Parallel transport
100(1)
3.3.3 The curvature of a connection
101(4)
3.3.4 Holonomy
105(3)
3.3.5 The Bianchi identities
108(1)
3.3.6 Connections on tangent bundles
109(2)
3.4 Integration on manifolds
111(28)
3.4.1 Integration of 1-densities
111(4)
3.4.2 Orientability and integration of differential forms
115(8)
3.4.3 Stokes' formula
123(4)
3.4.4 Representations and characters of compact Lie groups
127(6)
3.4.5 Fibered calculus
133(6)
4 Riemannian Geometry
139(62)
4.1 Metric properties
139(26)
4.1.1 Definitions and examples
139(4)
4.1.2 The Levi-Civita connection
143(4)
4.1.3 The exponential map and normal coordinates
147(3)
4.1.4 The length minimizing property of geodesies
150(5)
4.1.5 Calculus on Riemann manifolds
155(10)
4.2 The Riemann curvature
165(36)
4.2.1 Definitions and properties
165(4)
4.2.2 Examples
169(3)
4.2.3 Cartan's moving frame method
172(3)
4.2.4 The geometry of submanifolds
175(7)
4.2.5 Correlators and their geometry
182(10)
4.2.6 The Gauss-Bonnet theorem for oriented surfaces
192(9)
5 Elements of the Calculus of Variations
201(26)
5.1 The least action principle
201(9)
5.1.1 The 1-dimensional Euler-Lagrange equations
201(6)
5.1.2 Noether's conservation principle
207(3)
5.2 The variational theory of geodesies
210(17)
5.2.1 Variational formulae
211(3)
5.2.2 Jacobi fields
214(6)
5.2.3 The Hamilton-Jacobi equations
220(7)
6 The Fundamental Group and Covering Spaces
227(14)
6.1 The fundamental group
228(5)
6.1.1 Basic notions
228(4)
6.1.2 Of categories and functors
232(1)
6.2 Covering Spaces
233(8)
6.2.1 Definitions and examples
233(2)
6.2.2 Unique lifting property
235(1)
6.2.3 Homotopy lifting property
236(1)
6.2.4 On the existence of lifts
237(2)
6.2.5 The universal cover and the fundamental group
239(2)
7 Cohomology
241(88)
7.1 DeRham cohomology
241(24)
7.1.1 Speculations around the Poincare" lemma
241(4)
7.1.2 Cech vs. DeRham
245(2)
7.1.3 Very little homological algebra
247(7)
7.1.4 Functorial properties of the DeRham cohomology
254(3)
7.1.5 Some simple examples
257(2)
7.1.6 The Mayer-Vietoris principle
259(3)
7.1.7 The Kunneth formula
262(3)
7.2 The Poincare" duality
265(7)
7.2.1 Cohomology with compact supports
265(3)
7.2.2 The Poincare" duality
268(4)
7.3 Intersection theory
272(19)
7.3.1 Cycles and their duals
272(5)
7.3.2 Intersection theory
277(5)
7.3.3 The topological degree
282(2)
7.3.4 The Thorn isomorphism theorem
284(3)
7.3.5 Gauss-Bonnet revisited
287(4)
7.4 Symmetry and topology
291(21)
7.4.1 Symmetric spaces
291(3)
7.4.2 Symmetry and cohomology
294(4)
7.4.3 The cohomology of compact Lie groups
298(1)
7.4.4 Invariant forms on Grassmannians and Weyl's integral formula
299(7)
7.4.5 The Poincare" polynomial of a complex Grassmannian
306(6)
7.5 Cech cohomology
312(17)
7.5.1 Sheaves and presheaves
313(4)
7.5.2 Cech cohomology
317(12)
8 Characteristic Classes
329(44)
8.1 Chern-Weil theory
329(14)
8.1.1 Connections in principal G-bundles
329(6)
8.1.2 G-vector bundles
335(1)
8.1.3 Invariant polynomials
336(3)
8.1.4 The Chern-Weil Theory
339(4)
8.2 Important examples
343(14)
8.2.1 The invariants of the torus Tn
343(1)
8.2.2 Chern classes
343(3)
8.2.3 Pontryagin classes
346(2)
8.2.4 The Euler class
348(3)
8.2.5 Universal classes
351(6)
8.3 Computing characteristic classes
357(16)
8.3.1 Reductions
358(5)
8.3.2 The Gauss-Bonnet-Chern theorem
363(10)
9 Classical Integral Geometry
373(78)
9.1 The integral geometry of real Grassmannians
373(25)
9.1.1 Co-area formula?
373(13)
9.1.2 Invariant measures on linear Grassmannians
386(9)
9.1.3 Affine Grassmannians
395(3)
9.2 Gauss-Bonnet again?!?
398(11)
9.2.1 The shape operator and the second fundamental form
398(3)
9.2.2 The Gauss-Bonnet theorem for hypersurfaces of an Euclidean space
401(4)
9.2.3 Gauss-Bonnet theorem for domains in an Euclidean space
405(4)
9.3 Curvature measures
409(42)
9.3.1 Tame geometry
409(5)
9.3.2 Invariants of the orthogonal group
414(4)
9.3.3 The tube formula and curvature measures
418(11)
9.3.4 Tube formula ⇒ Gauss-Bonnet formula for arbitrary submanifolds of an Euclidean space
429(2)
9.3.5 Curvature measures of domains in an Euclidean space
431(2)
9.3.6 Crofton formulae for domains of an Euclidean space
433(10)
9.3.7 Crofton formulae for submanifolds of an Euclidean space
443(8)
10 Elliptic Equations on Manifolds
451(72)
10.1 Partial differential operators: algebraic aspects
451(13)
10.1.1 Basic notions
451(6)
10.1.2 Examples
457(2)
10.1.3 Formal adjoints
459(5)
10.2 Functional framework
464(20)
10.2.1 Sobolev spaces in RN
464(7)
10.2.2 Embedding theorems: integrability properties
471(5)
10.2.3 Embedding theorems: differentiability properties
476(4)
10.2.4 Functional spaces on manifolds
480(4)
10.3 Elliptic partial differential operators: analytic aspects
484(20)
10.3.1 Elliptic estimates in RN
485(4)
10.3.2 Elliptic regularity
489(5)
10.3.3 An application: prescribing the curvature of surfaces
494(10)
10.4 Elliptic operators on compact manifolds
504(19)
10.4.1 Fredholm theory
504(9)
10.4.2 Spectral theory
513(5)
10.4.3 Hodge theory
518(5)
11 Spectral Geometry
523(88)
11.1 Generalized functions and currents
523(13)
11.1.1 Generalized functions and operations withy them
523(5)
11.1.2 Currents
528(1)
11.1.3 Temperate distributions and the Fourier transform
529(2)
11.1.4 Linear differential equations with distributional data
531(5)
11.2 Important families of generalized functions
536(12)
11.2.1 Some classical generalized functions on the real axis
536(6)
11.2.2 Homogeneous generalized functions
542(6)
11.3 The wave equation
548(26)
11.3.1 Fundamental solutions of the wave
548(2)
11.3.2 The wave family
550(17)
11.3.3 Local parametrices for the wave equation with variable coefficients
567(7)
11.4 Spectral geometry
574(37)
11.4.1 The spectral function of the Laplacian on a compact manifold
574(6)
11.4.2 Short time asymptotics for the wave kernel
580(8)
11.4.3 Spectral function asymptotics
588(5)
11.4.4 Spectral estimates of smoothing operators
593(5)
11.4.5 Spectral perestroika
598(13)
12 Dirac Operators
611(58)
12.1 The structure of Dirac operators
611(33)
12.1.1 Basic definitions and examples
611(3)
12.1.2 Clifford algebras
614(3)
12.1.3 Clifford modules: the even case
617(5)
12.1.4 Clifford modules: the odd case
622(1)
12.1.5 A look ahead
623(2)
12.1.6 The spin group
625(8)
12.1.7 The complex spin group
633(2)
12.1.8 Low dimensional examples
635(5)
12.1.9 Dirac bundles
640(4)
12.2 Fundamental examples
644(25)
12.2.1 The Hodge-DeRham operator
644(4)
12.2.2 The Hodge-Dolbeault operator
648(6)
12.2.3 The spin Dirac operator
654(6)
12.2.4 The spin0 Dirac operator
660(9)
Bibliography 669(6)
Index 675