Elsevier S & T

Handbook of Global Analysis

Krupka, Demeter; Saunders, David

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Table of contents

Cover
Title pageiii
Copyright Pageiv
Prefacev
Table of Contentsvii
Chapter 1 Global aspects of Finsler geometry1
1 Finsler metrics and connections1
2 Geodesics in Finsler manifolds16
3 Comparison theorems: Cartan–Hadamard theorem, Bonnet–Myers theorem, Laplacian and volume compa27
4 Rigidity theorems: Finsler manifolds of scalar curvature and locally symmetric Finsler metrics31
5 Closed geodesics on Finsler manifolds, sphere theorem and the Gauss–Bonnet formula33
Chapter 2 Morse theory and nonlinear differential equations41
1 Introduction41
2 Global theory43
3 Local theory51
4 Applications53
5 Strongly indefinite Morse theory58
6 Strongly indefinite variational problems63
Chapter 3 Index theory75
1 Introduction and some history75
2 Fredholm operators – theory and examples77
3 The space of Fredholm operators and K-theory86
4 Elliptic operators and Sobolev spaces95
5 The Atiyah-Singer Index Theorem104
6 Generalizations139
Chapter 4 Partial differential equations on closed and open manifolds147
1 Introduction147
2 Sobolev spaces148
3 Non-linear Sobolev structures157
4 Self-adjoint linear differential operators on manifolds and their spectral theory167
5 The spectral value zero177
6 The heat equation, the heat kernel and the heat flow189
7 The wave equation, its Hamiltonian approach and completeness204
8 Index theory on open manifolds211
9 The continuity method for non-linear PDEs on open manifolds235
10 Teichmüller theory237
11 Harmonic maps248
12 Non-linear field theories253
13 Gauge theory258
14 Fluid dynamics268
15 The Ricci flow277
Chapter 5 Spectral geometry289
1 Introduction289
2 The geometry of operators of Laplace and Dirac type290
3 Heat trace asymptotics for closed manifolds291
4 Hearing the shape of a drum294
5 Heat trace asymptotics of manifolds with boundary296
6 Heat trace asymptotics and index theory303
7 Heat content asymptotics306
8 Heat content with source terms312
9 Time dependent phenomena314
10 Spectral boundary conditions315
11 Operators which are not of Laplace type316
12 The spectral geometry of Riemannian submersions317
Chapter 6 Lagrangian formalism on Grassmann manifolds327
1 Introduction327
2 Grassmann manifolds328
3 The Lagrangian formalism on a Grassmann manifold344
4 Lagrangian formalism on second order Grassmann bundles357
5 Some applications362
Chapter 7 Sobolev spaces on manifolds375
1 Motivations375
2 Sobolev spaces on manifolds: definition and first properties376
3 Equality and density issues380
4 Embedding theorems, Part I381
5 Euclidean type inequalities388
6 Embedding theorems, Part II389
7 Embedding theorems, Part III392
8 Compact embeddings394
9 Best constants394
10 Explicit sharp inequalities405
11 The Cartan-Hadamard conjecture406
Chapter 8 Harmonic maps417
1 Harmonic functions on Euclidean spaces418
2 Harmonic maps between Riemannian manifolds421
3 Weakly harmonic maps and Sobolev spaces between manifolds429
4 Regularity442
5 Existence methods453
6 Other analytical properties468
7 Twistor theory and completely integrable systems473
Chapter 9 Topology of differentiable mappings493
1 Introduction493
2 Manifolds and singularities495
3 Milnor fibre498
4 Monodromy504
5 Stratifications of spaces507
6 Stratified Morse theory512
7 Rectified homotopical depth517
8 Relative stratified Morse theory520
9 Topology of images and multiple point spaces521
10 The image computing spectral sequence525
Chapter 10 Group actions and Hilbert’s fifth problem533
1 Hilbert’s fifth problem534
2 Lie groups and manifolds539
3 Group actions540
4 Cartan and proper actions of Lie groups543
5 Non-paracompact Cartan G-manifolds548
6 Homogeneous spaces of Lie groups550
7 Twisted products552
8 Slices555
9 The strong Cr topologies, 1 ≤ r < infinity, and the very-strong Cinfinity topology559
10 Continuity of induced maps in the strong Cr topologies, 1 ≤ r < infinity, and in the very-stron563
11 The product theorem568
12 The equivariant glueing lemma569
13 Whitney approximation570
14 Haar integrals of Cs maps, 1 ≤ s ≤ infinity, and of real analytic maps572
15 Continuity of the averaging maps in the strong Cr topologies, 1 ≤ r < infinity, and in the very575
16 Approximation of K-equivariant Cs maps, 1 ≤ s ≤ infinity, by K-equivariant real analytic maps578
17 Approximation of Cs K-slices, 1 ≤ s ≤ infinity580
18 Proof of the main theorem582
Chapter 11 Exterior differential systems591
1 Introduction591
2 Exterior differential systems593
3 Basic existence theorems for integral manifolds of Cinfinity systems595
4 Involutive analytic systems and the Cartan-Kähler Theorem600
5 Prolongation and the Cartan-Kuranishi Theorem607
6 A Cartan-Kähler Theorem for Cinfinity Pfaffian systems608
7 Characteristic cohomology612
8 Topological obstructions613
9 Applications to second-order scalar hyperbolic partial differential equations in the plane614
10 Some applications to differential geometry616
Chapter 12 Weil bundles as generalized jet spaces625
1 Weil algebras627
2 Weil bundles632
3 On the geometry of TA-prolongations641
4 Fiber product preserving bundle functors650
5 Some applications656
Chapter 13 Distributions, vector distributions, and immersions of manifolds in Euclidean spaces665
1 Introduction665
2 Distributions on Euclidean spaces666
3 Distributions and related concepts on manifolds674
4 Vector distributions or plane fields on manifolds680
5 Immersions and embeddings of manifolds in Euclidean spaces706
Chapter 14 Geometry of differential equations725
1 Geometry of jet spaces726
2 Algebra of differential operators733
3 Formal theory of PDEs741
4 Local and global aspects751
Chapter 15 Global variational theory in fibred spaces773
1 Introduction773
2 Prolongations of fibred manifolds775
3 Differential forms on prolongations of fibred manifolds782
4 Lagrange structures790
5 The structure of the Euler-Lagrange mapping806
6 Invariant variational principles816
7 Remarks820
Chapter 16 Second Order Ordinary Differential Equations in Jet Bundles and the Inverse Problem of th837
1 Introduction837
2 Second-order differential equations on fibred manifolds839
3 Variational structures in the theory of SODEs850
4 Symmetries and first integrals863
5 Geometry of regular SODEs on R X TM879
6 The inverse problem for semisprays884
Chapter 17 Elements of noncommutative geometry905
1 Introduction905
2 Algebras instead of spaces907
3 Modules as bundles908
4 Homology and cohomology910
5 The Chern characters915
6 Connections and gauge transformations919
7 Noncommutative manifolds923
8 Toric noncommutative manifolds935
9 The spectral geometry of the quantum group SUq(2)938
Chapter 18 De Rham cohomology953
1 De Rham complex954
2 Integration and de Rham cohomology. De Rham currents. Harmonic forms959
3 Generalizations of the de Rham complex961
4 Equivariant de Rham cohomology965
5 Complexes of differential forms associated to differential geometric structures967
Chapter 19 Topology of manifolds with corners983
1 Introduction983
2 Quadrants984
3 Differentiation theories996
4 Manifolds with corners1019
5 Manifolds with generalized boundary1031
Chapter 20 Jet manifolds and natural bundles1035
1 Introduction1035
2 Jets1036
3 Differential equations1050
4 The calculus of variations1056
5 Natural bundles1063
Chapter 21 Some aspects of differential theories1069
1 Background1072
2 Calculus in topological vector spaces and beyond1077
3 The Chern – Rund derivative1101
Chapter 22 Variational sequences1115
1 Preliminaries1118
2 Contact forms1120
3 Variational bicomplex and variational sequence1128
4 C-spectral sequence and variational sequence1136
5 Finite order variational sequence1143
6 Special topics1146
7 Notes on the development of the subject1153
Chapter 23 The Oka-Grauert-Gromov principle for holomorphic bundles1165
1 Stein manifolds and Stein spaces1166
2 Oka’s theorem1170
3 Grauert’s Oka principle1173
4 Gromov’s Oka principle1176
5 The case of Riemann surfaces1178
6 Complete intersections1180
7 Embedding dimensions of Stein spaces1184
8 Oka principle with growth condition1188
9 Oka’s principle and the Moving Lemma in hyperbolic geometry1194
10 The algebraic version of Oka’s principle1206
Abstracts1211
Index1219

Book details

Vendor Elsevier S & T
SKU 9780444528339
ISBN-13 9780080556734
Author Krupka, Demeter; Saunders, David
Category Mathematics
Subject Mathematical Analysis

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This is a comprehensive exposition of topics covered by the American Mathematical Society’s classification “Global Analysis, dealing with modern developments in calculus expressed using abstract terminology. It will be invaluable for graduate students and researchers embarking on advanced studies in mathematics and mathematical physics.

This book provides a comprehensive coverage of modern global analysis and geometrical mathematical physics, dealing with topics such as; structures on manifolds, pseudogroups, Lie groupoids, and global Finsler geometry; the topology of manifolds and differentiable mappings; differential equations (including ODEs, differential systems and distributions, and spectral theory); variational theory on manifolds, with applications to physics; function spaces on manifolds; jets, natural bundles and generalizations; and non-commutative geometry.

- Comprehensive coverage of modern global analysis and geometrical mathematical physics
- Written by world-experts in the field
- Up-to-date contents

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