The blending of algebra, geometry, and differential equations has a long and distinguished history, dating back to the work of Sophus Lie and Elie Cartan. Overviewing the depth of their influence over the past 100 years presents a formidable challenge. A conference was held on the centennial of Lie's death to reflect upon and celebrate his pursuits, later developments, and what the future may hold. This volume showcases the contents, atmosphere, and results of that conference. Of particular importance are two survey articles: Morimoto develops a synthetic study of Lie groups, geometric structures, and differential equations from a unified viewpoint of nilpotent geometry. Yamaguchi and Yatsui discuss the geometry of higher order differential equations of finite type. Contributed research articles cover a wide range of disciplines, from geometry of differential equations, CR-geometry, and differential geometry to topics in mathematical physics. This volume is intended for graduate students studying differential geometry and analyis and advanced graduate students and researchers interested in an overview of the most recent progress in these fields. Information for our distributors: Published for the Mathematical Society of Japan by Kinokuniya, Tokyo, and distributed worldwide, except in Japan, by the AMS. All commercial channel discounts apply.

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lgli/[Advanced Studies in Pure Mathematics 37]Lie Groups, Geometric Structures and Differential Equations ―― One Hundred Years after Sophus Lie ――, (ed.) Tohru Morimoto, Hajime Sato, Keizo Yamaguchi, 2002, 501p.pdf

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Lie groups, geometric structures and differential equations one hundred years after Sophus Lie ; [International Conference Entitled "Lie Groups, Geometric Structures, and Differential Equations - One Hundred Years after Sophus Lie", which took place ... in Kyoto and Nara in December 1999

Tōru Morimoto; Hajime Satō; Keizō Yamaguchi; International Conference Entitled Lie Groups, Geometric Structures, and Differential Equations - One Hundred Years after Sophus Lie

Tohru Morimoto; Hajime Satō; Keizo Yamaguchi; 肇 佐藤; 英一 坂内; 佳三 山口; 徹 森本; Nihon Sūgakkai

Tohru Morimoto; Hajime Satō; Keizō Yamaguchi; Nihon Sugakkai

Tohru Morimoto, Hajime Sato, Keizo Yamaguchi(Eds.)

Kyoto Daigaku Suri Kaiseki Kenkyujo

Advanced studies in pure mathematics (Tokyo, Japón), 37, Tokyo, [Providence, RI, ©2002

Advanced studies in pure mathematics, Tōkyō, 2002

June 2002

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Includes bibliographical references.
"The present issue ... is an outgrowth of the international conference ... which took place, on the occasion of the centennial after the death of Sophus Lie (1842-1899), in Kyoto and Nara in December 1999."--Pref.

Preface 3
Contents 5
Robert L. Bryant, Levi-Flat Minimal Hypersurfaces in Two-dimensional Complex Space Forms 6
Introduction 6
§1. Two-Dimensional Complex Space Forms 8
1.1. The group G_R 9
1.2. The complex space form P2_R 9
§2. Real Hypersurfaces 10
2.1. First invariants 10
2.2. Differential consequences of the structure equations 11
§3. Existence of Solutions 15
3.1. Solutions of type 1 15
3.2. Solutions of type 2 18
3.3. Solutions of type 3 26
References 48
[He] 49
Andreas Čap and Gerd Schmalz, Partially Integrable Almost CR Manifolds of CR Dimension and Codimension Two 50
§1. Introduction 50
§2. Partially integrable almost CR manifolds of CR dimensionand codimension two 53
2.1. almost CR manifolds 53
2.2. Embedded almost CR manifolds 54
2.3. Non-degeneracy 56
2.4. The case of CR dimension and codimension two 57
2.5. The hyperbolic case 59
2.6. Hyperbolic almost CR manifolds as parabolic geometries 60
2.7. The elliptic case 64
2.8. An equivalence of categories 66
2.9. Real parabolic geometries of type (PSL(3,C), B) 68
§3. Interpretations of torsions in the elliptic case 70
3.1. The curvature of the normal Cartan connection 70
3.2. Harmonic curvature components 72
3.3. The Nijenhuis tensor 74
3.4. The other torsions of type (1,1) 75
3.5. Torsions of type (0,2) 76
3.6. Torsion free elliptic CR manifolds 79
References 81
[10] 82
Jacques Gasqui and Hubert Goldschmidt, Some Remarks on the Infinitesimal Rigidity of the Complex Quadric 84
Introduction 84
§1. Symmetric spaces 85
§2. Criteria for infinitesimal rigidity 88
§3. The complex quadric 91
§4. Totally geodesic submanifolds of the quadric 95
§5. Infinitesimal rigidity of the quadric 98
References 100
[7] 101
Sergei lgonin and Joseph Krasil'shchik, On One-Parametric Families of Bäcklund Transformations 104
Introduction 104
§1. Equations and coverings 105
§2. C-complex and deformations 109
§3. Bäcklund transformations and the main result 112
References 118
[5] 119
Goo Ishikawa, Makoto Kimura and Reiko Miyaoka, Submanifolds with Degenerate Gauss Mappings in Spheres 120
§1. Introduction 120
§2. Ferns inequality for submanifolds with degenerate Gauss mapping 122
§3. Examples related to isoparametric hypersurfaces 124
§4. Stiefel manifolds and complex quadrics 126
§5. Complex submanifolds in complex quadrics 128
§6. Submanifolds with degenerate Gauss mapping in spheres 129
§7. Examples 133
§8. Austere submanifolds in spheres 136
§9. Examples satisfying Ferns equality in spheres 138
§10. Hypersurfaces with degenerate Gauss mappings in the four dimensional sphere 144
§11. Appendix 146
References 151
[3] 152
[22] 153
[41] 154
Paul Kersten and Joseph Krasil'shchik, Complete Integrabilityof the Coupled KdV-mKdV System 156
Introduction 156
§1. Geometrical and algebraic background 157
§2. Basic computations and results 165
2.1. Conservation laws and nonlocal variables 165
2.2. Local and nonlocal symmetries 168
2.3. Recursion operator 169
§3. Conclusion 172
Appendix 172
References 175
[9] 176
Masatake Kuranishi, An Approach to the Cartan Geometry I: Conformal Riemann Manifolds 178
Introduction 178
§1. The Homogeous Conformal Space 181
References 200
Bernard Malgrange, Differential Algebra and Differential Geometry 202
§1. Introduction 202
§2. D-varieties 203
§3. Formal integrability and generic involutiveness 205
§4. General morphisms 206
Literature 208
Tohru Morimoto, Lie Algebras, Geometric Structures and Differential Equations on Filtered Manifolds 210
Introduction 210
§0. Filtered manifolds 214
§1. Transitive Lie algebras on filtered manifolds 218
§2. Geometric structures on filtered manifolds 224
§3. Differential equations on filtered manifolds 241
References 254
[Gol67b] 255
[Mor0x] 256
[Yat92] 257
Takaaki Nomura, Cayley Transforms and Symmetry Conditions for Homogeneous Siegel Domains 258
Introduction 258
§1. Preliminaries 259
§2. Family of Cayley transforms 261
§3. Norm equalities 263
§4. Berezin transforms 266
§5. Poisson kernel 268
References 269
[22] 270
Peter J. Olver, The Canonical Contact Form 272
§1. Introduction 272
§2. Contact Forms on Jet Bundles 273
§3. The Prolonged General Linear Group 275
§4. The Leibniz Group 280
§5. The Contact Group 283
References 289
[10] 290
Hideki Omori, Associativity Breaks Down in Deformation Quantization 292
§1. Introduction 292
§2. Extensions of product formula 295
2.1. Intertwiner, or coordinate transformations 297
§3. Vacuums, half-inverses and the break down of the associativity 300
3.1. Star-exponentials of quadratic forms in the Weyl ordering expression 300
3.2. Horizontal distributions 302
3.3. *-exponentials and vacuums 302
3.4. Anomarous phenomena 306
3.5. Several product formulas 307
§4. Star exponential functions in the normal ordering expression 310
4.1. The case b=0 as the simplest case 311
4.2. Several facts, concluded from the case a=0 312
4.3. The case ab≠0, Proof of the first half of Theorem 2 315
§5. Proof of Theorem 2 317
References 319
[O,el.2] 320
Arkadi L. Onishchik, Lifting of Holomorphic Actions on Complex Supermanifolds 322
Introduction 322
§1. Preliminaries on superrnanifolds 323
1.1. Split and non-split superrnanifolds 323
1.2. The tangent sheaf 324
1.3. Sheaves of automorphisms 326
§2. Cochain complexes and classification theorems 327
2.1. The Čech complex 327
2.2. A resolution of the automorphism sheaf 329
2.3. Non-abelian cohomology of groups 331
§3. Actions of Lie groups on supermanifolds 333
3.1. Actions and the lifting problem 333
3.2. The 1-cohomology of compact Lie groups 336
3.3. Lifting and invariant cocycles 337
References 339
Tetsuya Ozawa and Hajime Sato, Contact Transformations and Their Schwarzian Derivatives 342
§1. Introduction 342
§2. Contact Schwarzian derivative 346
§3. Some remarks on contact transformations 346
§4. Fundamental equation 350
§5. PDE systems related to contact transformation 353
§6. Construction of contact transformation via solutions of PDE system 357
§7. Integrability conditions 362
§8. Connection formula of contact Schwarzian derivatives 365
§9. Contact transformations with vanishing contact Schwarzian derivatives 367
References 370
Per Tomter, Isometric Immersions into Complex Projective Space 372
§1. 373
§2. 374
§3. 375
§4. 379
§5. 384
§6. 389
§7. 397
§8. 398
References 401
Keizo Yamaguchi and Tomoaki Yatsui, Geometry of Higher Order Differential Equations of Finite Type Associated with Symmetric Spaces 402
Introduction 402
§1. Differential Systems and Pseudo-Product Structures 408
1.1. Differential Equations of Finite Type 408
1.2. Geometry of Differential Systems (Tanaka Theory) 410
1.3. Symbol Algebra of (J^k, C^k) 413
§2. Pseudo-product GLA g=⊕_{p∈Z} g_p of type (l,S) 414
2.1. Pseudo-projective GLA of order k of bidegree (n, m) 415
2.2. Pseudo-product GLA of type (l,S) 418
§3. Se-ashi's Theory for Linear Equations of Finite Type 423
3.1. Model equations for the typical symbol of type (l,S) 423
3.2. Plücker embedding equations 428
§4. Generalized Spencer Cohomology 431
4.1. Cohomology of Lie algebras 432
4.2. Generalized Spencer cohomology H*(G) and H*(b_,g) 432
4.3. Calculation of the cohomology H^p(G)_{r,s} 433
4.4. Gradations of semisimple Lie algebras and Kostant's theorem 437
4.5. Parameterization of pseudo-product GLA of type (l,S) 438
§5. First cohomology of pseudo-product graded Lie algebras 440
§6. Second cohomology of pseudo-product graded Lie algebras 445
6.1. Computation of H2(G)_{r,-1} 445
6.2. Computation of H2(G)_{r,0} 447
6.3. Computation of H2(G)_{r,s} (s=1,2) 449
§7. The symbol algebras of the Plücker embedding equations 454
7.1. The case i=1 or 2 (the case \breve{b1}≠0) 454
7.2. The case i≧3 (the case \breve{b1}=0) 455
7.3. The computation of H2(G)_{r,2} when i=3 and l≧5 457
References 461
[Tan70] 462
Akira Yoshioka, Contact Weyl Manifold over a Symplectic Manifold 464
§1. Introduction 464
§2. Weyl manifold 467
2.1. Deformation quantization 467
2.2. Review of Weyl manifold 469
2.3. Poincaré-Cartan class 475
§3. Contact algebra bundle and connection 481
3.1. Construction of C_M; proof of Theorem A 482
3.2. Construction of connection ∂: proof of Thoerem B 484
§4. Weyl charts and Classical charts 491
4.1. Basic Lemma 491
4.2. Section \hat{τ} ∈ Γ(C_M) and classical charts 492
4.3. Expression of ∂ in the classical charts 493
4.4. Expression of ∂ in classical charts 495
References 497
[DL] 498

The blending of algebra, geometry, and differential equations has a long history dating back to the work of Sophus Lie and Elie Cartan. A conference was held on the centennial of Lie's death to reflect upon his pursuits, later developments, and what the future may hold. This title showcases the contents, atmosphere, and results of that conference.

Edited By Tohru Morimoto, Hajime Sato, Keizo Yamaguchi. The Present Issue ... Is An Outgrowth Of The International Conference ... Which Took Place, On The Occasion Of The Centennial After The Death Of Sophus Lie (1842-1899), In Kyoto And Nara In December 1999.--pref. Includes Bibliographical References.

2021-12-27