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Differential Geometry and Integrable Systems
Edited by: Martin Guest, Tokyo Metropolitan University, Japan, Reiko Miyaoka, Sophia University, Tokyo, Japan, and Yoshihiro Ohnita, Tokyo Metropolitan University, Japan
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Contemporary Mathematics
2002; 349 pp; softcover
Volume: 308
ISBN-10: 0-8218-2938-6
ISBN-13: 978-0-8218-2938-7
List Price: US$98
Member Price: US$78.40
Order Code: CONM/308
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Ideas and techniques from the theory of integrable systems are playing an increasingly important role in geometry. Thanks to the development of tools from Lie theory, algebraic geometry, symplectic geometry, and topology, classical problems are investigated more systematically. New problems are also arising in mathematical physics. A major international conference was held at the University of Tokyo in July 2000. It brought together scientists in all of the areas influenced by integrable systems. This book is the first of three collections of expository and research articles.

This volume focuses on differential geometry. It is remarkable that many classical objects in surface theory and submanifold theory are described as integrable systems. Having such a description generally reveals previously unnoticed symmetries and can lead to surprisingly explicit solutions. Surfaces of constant curvature in Euclidean space, harmonic maps from surfaces to symmetric spaces, and analogous structures on higher-dimensional manifolds are some of the examples that have broadened the horizons of differential geometry, bringing a rich supply of concrete examples into the theory of integrable systems.

Many of the articles in this volume are written by prominent researchers and will serve as introductions to the topics. It is intended for graduate students and researchers interested in integrable systems and their relations to differential geometry, topology, algebraic geometry, and physics.

The second volume from this conference, also available from the AMS, is Integrable Systems, Topology, and Physics, Volume 309 in the Contemporary Mathematics series. The forthcoming third volume will be published by the Mathematical Society of Japan and will be available outside of Japan from the AMS in the Advanced Studies in Pure Mathematics series.

Readership

Graduate students and researchers interested in integrable systems and their relations to differential geometry, topology, algebraic geometry, and physics.

Table of Contents

  • N. Ando -- The index of an isolated umbilical point on a surface
  • J. Bolton -- The Toda equations and equiharmonic maps of surfaces into flag manifolds
  • J.-M. Burel and E. Loubeau -- \(p\)-harmonic morphisms: The \(1<p<2\) case and some non-trivial examples
  • F. Burstall, F. Pedit, and U. Pinkall -- Schwarzian derivatives and flows of surfaces
  • V. De Smedt and S. Salamon -- Anti-self-dual metrics on Lie groups
  • J. Dorfmeister, J.-i. Inoguchi, and M. Toda -- Weierstraß-type representation of timelike surfaces with constant mean curvature
  • N. Ejiri -- A differential-geometric Schottky problem, and minimal surfaces in tori
  • E. V. Ferapontov -- Surfaces in 3-space possessing nontrivial deformations which preserve the shape operator
  • F. Hélein and P. Romon -- Hamiltonian stationary Lagrangian surfaces in Hermitian symmetric spaces
  • H. Hu -- Line congruences and integrable systems
  • X. Jiao -- Factorizations of harmonic maps of surfaces into Lie groups by singular dressing actions
  • H. Jin and X. Mo -- On submersive \(p\)-harmonic morphisms and their stability
  • K. Kiyohara -- On Kähler-Liouville manifolds
  • M. Kokubu, M. Umehara, and K. Yamada -- Minimal surfaces that attain equality in the Chern-Osserman inequality
  • V. S. Matveev -- Low dimensional manifolds admitting metrics with the same geodesics
  • Y. Ohnita and S. Udagawa -- Harmonic maps of finite type into generalized flag manifolds, and twistor fibrations
  • J. Park -- Submanifolds associated to Grassmannian systems
  • Y. Sakane and T. Yamada -- Harmonic cohomology groups of compact symplectic nilmanifolds
  • B. A. Springborn -- Bonnet pairs in the 3-sphere
  • M. S. Tanaka -- Subspaces in the category of symmetric spaces
  • H. Tasaki -- Integral geometry of submanifolds of real dimension two and codimension two in complex projective spaces
  • J. C. Wood -- Jacobi fields along harmonic maps
  • H. Wu -- Denseness of plain constant mean curvature surfaces in dressing orbits
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