Preface ......................................................... v
Contents ...................................................... vii
Global aspects of Finsler geometry .............................. 1
Tadashi Aikou and Laszlo Kozma
1. Finsler metrics and connections .............................. 1
2. Geodesies in Finsler manifolds .............................. 16
3. Comparison theorems: Cartan-Hadamard theorem, Bonnet-
Myers theorem, Laplacian and volume comparison .............. 27
4. Rigidity theorems: Finsler manifolds of scalar curvature
and locally symmetric Finsler metrics ....................... 31
5. Closed geodesies on Finsler manifolds, sphere theorem and
the Gauss-Bonnet formula .................................... 33
Morse theory and nonlinear differential equations .............. 41
Thomas Bartsch, Andrzej Szulkin and Michel Willem
1. Introduction ................................................ 41
2. Global theory ............................................... 43
3. Local theory ................................................ 51
4. Applications ................................................ 53
5. Strongly indefinite Morse theory ............................ 58
6. Strongly indefinite variational problems .................... 63
Index theory ................................................... 75
David Bleecker
1. Introduction and some history ............................... 75
2. Fredholm operators - theory and examples .................... 77
3. The space of Fredholm operators and K-theory ................ 86
4. Elliptic operators and Sobolev spaces ....................... 95
5. The Atiyah-Singer Index Theorem ............................ 104
6. Generalizations ............................................ 139
Partial differential equations on closed and open manifolds ... 147
Jürgen Eichhorn
1. Introduction ............................................... 147
2. Sobolev spaces ............................................. 148
3. Non-linear Sobolev structures .............................. 157
4. Self-adjoint linear differential operators on manifolds
and their spectral theory .................................. 167
5. The spectral value zero .................................... 177
6. The heat equation, the heat kernel and the heat flow ....... 189
7. The wave equation, its Hamiltonian approach and
completeness ............................................... 204
8. Index theory on open manifolds ............................. 211
9. The continuity method for non-linear PDEs on open
manifolds .................................................. 235
10.Teichmüller theory ......................................... 237
11.Harmonic maps .............................................. 248
12.Non-linear field theories .................................. 253
13.Gauge theory ............................................... 258
14.Fluid dynamics ............................................. 268
15.The Ricci flow ............................................. 277
Spectral geometry ............................................. 289
P. Gilkey
1. Introduction ............................................... 289
2. The geometry of operators of Laplace and Dirac type ........ 290
3. Heat trace asymptotics for closed manifolds ................ 291
4. Hearing the shape of a drum ................................ 294
5. Heat trace asymptotics of manifolds with boundary .......... 296
6. Heat trace asymptotics and index theory .................... 303
7. Heat content asymptotics ................................... 306
8. Heat content with source terms ............................. 312
9. Time dependent phenomena ................................... 314
10.Spectral boundary conditions ............................... 315
11.Operators which are not of Laplace type .................... 316
12.The spectral geometry of Riemannian submersions ............ 317
Lagrangian formalism on Grassmann manifolds ................... 327
D.R. Grigore
1. Introduction ............................................... 327
2. Grassmann manifolds ........................................ 328
3. The Lagrangian formalism on a Grassmann manifold ........... 344
4. Lagrangian formalism on second order Grassmann bundles ..... 357
5. Some applications .......................................... 362
Sobolev spaces on manifolds ................................... 375
Emmanuel Hebey and Frederic Robert
1. Motivations ................................................ 375
2. Sobolev spaces on manifolds: definition and first
properties ................................................. 376
3. Equality and density issues ................................ 380
4. Embedding theorems, Part I ................................. 381
5. Euclidean type inequalities ................................ 388
6. Embedding theorems, Part II ................................ 389
7. Embedding theorems, Part III ............................... 392
8. Compact embeddings ......................................... 394
9. Best constants ............................................. 394
10.Explicit sharp inequalities ................................ 405
11.The Cartan-Hadamard conjecture ............................. 406
Harmonic maps ................................................. 417
Frederic Helein and John C. Wood
1. Harmonic functions on Euclidean spaces ..................... 418
2. Harmonic maps between Riemannian manifolds ................. 421
3. Weakly harmonic maps and Sobolev spaces between
manifolds .................................................. 429
4. Regularity ................................................. 442
5. Existence methods .......................................... 453
6. Other analytical properties ................................ 468
7. Twistor theory and completely integrable systems ........... 473
Topology of differentiable mappings ........................... 493
Kevin Houston
1. Introduction ............................................... 493
2. Manifolds and singularities ................................ 495
3. Milnor fibre ............................................... 498
4. Monodromy .................................................. 504
5. Stratifications of spaces .................................. 507
6. Stratified Morse theory .................................... 512
7. Rectified homotopical depth ................................ 517
8. Relative stratified Morse theory ........................... 520
9. Topology of images and multiple point spaces ............... 521
10.The image computing spectral sequence ...................... 525
Group actions and Hilbert's fifth problem ..................... 533
Sören Illman
1. Hilbert's fifth problem .................................... 534
2. Lie groups and manifolds ................................... 539
3. Group actions .............................................. 540
4. Cartan and proper actions of Lie groups .................... 543
5. Non-paracompact Cartan G-manifolds ......................... 548
6. Homogeneous spaces of Lie groups ........................... 550
7. Twisted products ........................................... 552
8. Slices ..................................................... 555
9. The strong Cr topologies, 1 ≤ r < ∞, and the very-strong
C∞ topology ................................................ 559
10.Continuity of induced maps in the strong Cr topologies,
1 ≤ r < ∞, and in the very-strong C∞ topology .............. 563
11.The product theorem ........................................ 568
12.The equivariant glueing lemma .............................. 569
13.Whitney approximation ...................................... 570
14.Haar integrals of Cs maps, 1 ≤ s ≤ ∞,, and of real
analytic maps .............................................. 572
15.Continuity of the averaging maps in the strong Cr
topologies, 1 ≤ r < ∞, and in the very-strong C∞
topology ................................................... 575
16.Approximation of if-equivariant Cs maps, 1 ≤ s ≤ ∞,,
by K-equivariant real analytic maps, in the strong Cs
topologies, 1 ≤ s < ∞,, and in the very-strong C∞
topology ................................................... 578
17.Approximation of Cs if-slices, 1 ≤ s ≤ ∞, ................... 580
18.Proof of the main theorem .................................. 582
Exterior differential systems ................................. 591
Niky Kamran
1. Introduction ............................................... 591
2. Exterior differential systems .............................. 593
3. Basic existence theorems for integral manifolds of C∞,
systems .................................................... 595
4. Involutive analytic systems and the Cartan-Kahler
Theorem .................................................... 600
5. Prolongation and the Cartan-Kuranishi Theorem .............. 607
6. A Cartan-Kahler Theorem for C∞ Pfaffian systems ............ 608
7. Characteristic cohomology .................................. 612
8. Topological obstructions ................................... 613
9. Applications to second-order scalar hyperbolic partial
differential equations in the plane ........................ 614
10.Some applications to differential geometry ................. 616
Weil bundles as generalized jet spaces ........................ 625
Ivan Kolar
1. Weil algebras .............................................. 627
2. Weil bundles ............................................... 632
3. On the geometry of TA -prolongations ....................... 641
4. Fiber product preserving bundle functors ................... 650
5. Some applications .......................................... 656
Distributions, vector distributions, and immersions of
manifolds in Euclidean spaces ................................. 665
Julius Korbas
1. Introduction ............................................... 665
2. Distributions on Euclidean spaces .......................... 666
3. Distributions and related concepts on manifolds ............ 674
4. Vector distributions or plane fields on manifolds .......... 680
5. Immersions and embeddings of manifolds in Euclidean
spaces ..................................................... 706
Geometry of differential equations ............................ 725
Boris Kruglikov and Valentin Lychagin
1. Geometry of jet spaces ..................................... 726
2. Algebra of differential operators .......................... 733
3. Formal theory of PDEs ...................................... 741
4. Local and global aspects ................................... 751
Global variational theory in fibred spaces .................... 773
D. Krupka
1. Introduction ............................................... 773
2. Prolongations of fibred manifolds .......................... 775
3. Differential forms on prolongations of fibred manifolds .... 782
4. Lagrange structures ........................................ 790
5. The structure of the Euler-Lagrange mapping ................ 806
6. Invariant variational principles ........................... 816
7. Remarks .................................................... 820
Second Order Ordinary Differential Equations in Jet Bundles
and the Inverse Problem of the Calculus of Variations ......... 837
O. Krupkova and G.E. Prince
1. Introduction ............................................... 837
2. Second-order differential equations on fibred manifolds .... 839
3. Variational structures in the theory of SODEs .............. 850
4. Symmetries and first integrals ............................. 863
5. Geometry of regular SODEs on R × TM ........................ 879
6. The inverse problem for semisprays ......................... 884
Elements of noncommutative geometry ........................... 905
Giovanni Landi
1. Introduction ............................................... 905
2. Algebras instead of spaces ................................. 907
3. Modules as bundles ......................................... 908
4. Homology and cohomology .................................... 910
5. The Chern characters ....................................... 915
6. Connections and gauge transformations ...................... 919
7. Noncommutative manifolds ................................... 923
8. Toric noncommutative manifolds ............................. 935
9. The spectral geometry of the quantum group SUq (2) ......... 938
De Rham cohomology ............................................ 953
M.A. Malakhaltsev
1. De Rham complex ............................................ 954
2. Integration and de Rham cohomology. De Rham currents.
Harmonic forms ............................................. 959
3. Generalizations of the de Rham complex ..................... 961
4. Equivariant de Rham cohomology ............................. 965
5. Complexes of differential forms associated to
differential geometric structures .......................... 967
Topology of manifolds with corners ............................ 983
J. Margalef-Roig and E. Outerelo Dominguez
1. Introduction ............................................... 983
2. Quadrants .................................................. 984
3. Differentiation theories ................................... 996
4. Manifolds with corners .................................... 1019
5. Manifolds with generalized boundary ....................... 1031
Jet manifolds and natural bundles ............................ 1035
D.J. Saunders
1. Introduction .............................................. 1035
2. Jets ...................................................... 1036
3. Differential equations .................................... 1050
4. The calculus of variations ................................ 1056
5. Natural bundles ........................................... 1063
Some aspects of differential theories ........................ 1069
Jozsef Szilasi and Rezso L. Lovas
1. Background ................................................ 1072
2. Calculus in topological vector spaces and beyond .......... 1077
3. The Chern - Rund derivative ............................... 1101
Variational sequences ........................................ 1115
R. Vitolo
1. Preliminaries ............................................. 1118
2. Contact forms ............................................. 1120
3. Variational bicomplex and variational sequence ............ 1128
4. C-spectral sequence and variational sequence .............. 1136
5. Finite order variational sequence ......................... 1143
6. Special topics ............................................ 1146
7. Notes on the development of the subject ................... 1153
The Oka-Grauert-Gromov principle for holomorphic bundles ..... 1165
Pit-Mann Wong
1. Stein manifolds and Stein spaces .......................... 1166
2. Oka's theorem ............................................. 1170
3. Grauert's Oka principle ................................... 1173
4. Gromov's Oka principle .................................... 1176
5. The case of Riemann surfaces .............................. 1178
6. Complete intersections .................................... 1180
7. Embedding dimensions of Stein spaces ...................... 1184
8. Oka principle with growth condition ....................... 1188
9. Oka's principle and the Moving Lemma in hyperbolic
geometry .................................................. 1194
10.The algebraic version of Oka's principle .................. 1206
Abstracts .................................................... 1211
Index ........................................................ 1219
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