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.

lgrsnf/Morimoto T., et al. (eds.) Lie groups, geometric structures and differential equations.. 100 years after Sophus Lie (ASPM37, MS Japan, 2002)(ISBN 4931469213)(T)(O)(501s)_MD_.djvu

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

edited by Tohru Morimoto, Hajime Sato, Keizo Yamaguchi

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

Advanced studies in pure mathematics, Tōkyō, 2002

June 2002

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
Contents
Robert L. Bryant, Levi-Flat Minimal Hypersurfaces in Two-dimensional Complex Space Forms
Introduction
§1. Two-Dimensional Complex Space Forms
1.1. The group G_.
1.2. The complex space form .._.
§2. Real Hypersurfaces
2.1. First invariants
2.2. Differential consequences of the structure equations
§3. Existence of Solutions
3.1. Solutions of type 1
3.2. Solutions of type 2
3.3. Solutions of type 3
References
[He]
Andreas Cap and Gerd Schmalz, Partially Integrable Almost CR Manifolds of CR Dimension and Codimension Two
§1. Introduction
§2. Partially integrable almost CR manifolds of CR dimensionand codimension two
2.1. almost CR manifolds
2.2. Embedded almost CR manifolds
2.3. Non-degeneracy
2.4. The case of CR dimension and codimension two
2.5. The hyperbolic case
2.6. Hyperbolic almost CR manifolds as parabolic geometries
2.7. The elliptic case
2.8. An equivalence of categories
2.9. Real parabolic geometries of type (PSL(3,.), B)
§3. Interpretations of torsions in the elliptic case
3.1. The curvature of the normal Cartan connection
3.2. Harmonic curvature components
3.3. The Nijenhuis tensor
3.4. The other torsions of type (1,1)
3.5. Torsions of type (0,2)
3.6. Torsion free elliptic CR manifolds
References
[10]
Jacques Gasqui and Hubert Goldschmidt, Some Remarks on the Infinitesimal Rigidity of the Complex Quadric
Introduction
§1. Symmetric spaces
§2. Criteria for infinitesimal rigidity
§3. The complex quadric
§4. Totally geodesic submanifolds of the quadric
§5. Infinitesimal rigidity of the quadric
References
[7]
Sergei lgonin and Joseph Krasil'shchik, On One-Parametric Families of Bäcklund Transformations
Introduction
§1. Equations and coverings
§2. C-complex and deformations
§3. Bäcklund transformations and the main result
References
[5]
Goo Ishikawa, Makoto Kimura and Reiko Miyaoka, Submanifolds with Degenerate Gauss Mappings in Spheres
§1. Introduction
§2. Ferns inequality for submanifolds with degenerate Gauss mapping
§3. Examples related to isoparametric hypersurfaces
§4. Stiefel manifolds and complex quadrics
§5. Complex submanifolds in complex quadrics
§6. Submanifolds with degenerate Gauss mapping in spheres
§7. Examples
§8. Austere submanifolds in spheres
§9. Examples satisfying Ferns equality in spheres
§10. Hypersurfaces with degenerate Gauss mappings in the four dimensional sphere
§11. Appendix
References
[3]
[22]
[41]
Paul Kersten and Joseph Krasil'shchik, Complete Integrabilityof the Coupled KdV-mKdV System
Introduction
§1. Geometrical and algebraic background
§2. Basic computations and results
2.1. Conservation laws and nonlocal variables
2.2. Local and nonlocal symmetries
2.3. Recursion operator
§3. Conclusion
Appendix
References
[9]
Masatake Kuranishi, An Approach to the Cartan Geometry I: Conformal Riemann Manifolds
Introduction
§1. The Homogeous Conformal Space
References
Bernard Malgrange, Differential Algebra and Differential Geometry
§1. Introduction
§2. D-varieties
§3. Formal integrability and generic involutiveness
§4. General morphisms
Literature
Tohru Morimoto, Lie Algebras, Geometric Structures and Differential Equations on Filtered Manifolds
Introduction
§0. Filtered manifolds
§1. Transitive Lie algebras on filtered manifolds
§2. Geometric structures on filtered manifolds
§3. Differential equations on filtered manifolds
References
[Gol67b]
[Mor0x]
[Yat92]
Takaaki Nomura, Cayley Transforms and Symmetry Conditions for Homogeneous Siegel Domains
Introduction
§1. Preliminaries
§2. Family of Cayley transforms
§3. Norm equalities
§4. Berezin transforms
§5. Poisson kernel
References
[22]
Peter J. Olver, The Canonical Contact Form
§1. Introduction
§2. Contact Forms on Jet Bundles
§3. The Prolonged General Linear Group
§4. The Leibniz Group
§5. The Contact Group
References
[10]
Hideki Omori, Associativity Breaks Down in Deformation Quantization
§1. Introduction
§2. Extensions of product formula
2.1. Intertwiner, or coordinate transformations
§3. Vacuums, half-inverses and the break down of the associativity
3.1. Star-exponentials of quadratic forms in the Weyl ordering expression
3.2. Horizontal distributions
3.3. *-exponentials and vacuums
3.4. Anomarous phenomena
3.5. Several product formulas
§4. Star exponential functions in the normal ordering expression
4.1. The case b=0 as the simplest case
4.2. Several facts, concluded from the case a=0
4.3. The case ab.0, Proof of the first half of Theorem 2
§5. Proof of Theorem 2
References
[O,el.2]
Arkadi L. Onishchik, Lifting of Holomorphic Actions on Complex Supermanifolds
Introduction
§1. Preliminaries on superrnanifolds
1.1. Split and non-split superrnanifolds
1.2. The tangent sheaf
1.3. Sheaves of automorphisms
§2. Cochain complexes and classification theorems
2.1. The Cech complex
2.2. A resolution of the automorphism sheaf
2.3. Non-abelian cohomology of groups
§3. Actions of Lie groups on supermanifolds
3.1. Actions and the lifting problem
3.2. The 1-cohomology of compact Lie groups
3.3. Lifting and invariant cocycles
References
Tetsuya Ozawa and Hajime Sato, Contact Transformations and Their Schwarzian Derivatives
§1. Introduction
§2. Contact Schwarzian derivative
§3. Some remarks on contact transformations
§4. Fundamental equation
§5. PDE systems related to contact transformation
§6. Construction of contact transformation via solutions of PDE system
§7. Integrability conditions
§8. Connection formula of contact Schwarzian derivatives
§9. Contact transformations with vanishing contact Schwarzian derivatives
References
Per Tomter, Isometric Immersions into Complex Projective Space
§1.
§2.
§3.
§4.
§5.
§6.
§7.
§8.
References
Keizo Yamaguchi and Tomoaki Yatsui, Geometry of Higher Order Differential Equations of Finite Type Associated with Symmetric Spaces
Introduction
§1. Differential Systems and Pseudo-Product Structures
1.1. Differential Equations of Finite Type
1.2. Geometry of Differential Systems (Tanaka Theory)
1.3. Symbol Algebra of (J^k, C^k)
§2. Pseudo-product GLA ..=._{p..} .._p of type (..,S)
2.1. Pseudo-projective GLA of order k of bidegree (n, m)
2.2. Pseudo-product GLA of type (..,S)
§3. Se-ashi's Theory for Linear Equations of Finite Type
3.1. Model equations for the typical symbol of type (..,S)
3.2. Plücker embedding equations
§4. Generalized Spencer Cohomology
4.1. Cohomology of Lie algebras
4.2. Generalized Spencer cohomology H*(..) and H*(.._,..)
4.3. Calculation of the cohomology H^p(..)_{r,s}
4.4. Gradations of semisimple Lie algebras and Kostant's theorem
4.5. Parameterization of pseudo-product GLA of type (..,S)
§5. First cohomology of pseudo-product graded Lie algebras
§6. Second cohomology of pseudo-product graded Lie algebras
6.1. Computation of H.(..)_{r,-1}
6.2. Computation of H.(..)_{r,0}
6.3. Computation of H.(..)_{r,s} (s=1,2)
§7. The symbol algebras of the Plücker embedding equations
7.1. The case i=1 or 2 (the case reve{...}.0)
7.2. The case i.3 (the case reve{...}=0)
7.3. The computation of H.(..)_{r,2} when i=3 and l.5
References
[Tan70]
Akira Yoshioka, Contact Weyl Manifold over a Symplectic Manifold
§1. Introduction
§2. Weyl manifold
2.1. Deformation quantization
2.2. Review of Weyl manifold
2.3. Poincaré-Cartan class
§3. Contact algebra bundle and connection
3.1. Construction of C_M; proof of Theorem A
3.2. Construction of connection .: proof of Thoerem B
§4. Weyl charts and Classical charts
4.1. Basic Lemma
4.2. Section hat{.} . .(C_M) and classical charts
4.3. Expression of . in the classical charts
4.4. Expression of . in classical charts
References
[DL]

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.

2024-08-02