Differential Geometry and Integrable Systems
About this Title
Martin Guest, Reiko Miyaoka and Yoshihiro Ohnita, Editors
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.
Graduate students and researchers interested in integrable systems and their relations to differential geometry, topology, algebraic geometry, and physics.
Table of Contents
- Naoya Ando – The index of an isolated umbilical point on a surface [MR 1955625]
- John Bolton – The Toda equations and equiharmonic maps of surfaces into flag manifolds [MR 1955626]
- Jean-Marie Burel and Eric Loubeau – $p$-harmonic morphisms: the $1<p<2$ case and some non-trivial examples [MR 1955627]
- Francis Burstall, Franz Pedit and Ulrich Pinkall – Schwarzian derivatives and flows of surfaces [MR 1955628]
- Vivian De Smedt and Simon Salamon – Anti-self-dual metrics on Lie groups [MR 1955629]
- Josef Dorfmeister, Jun-ichi Inoguchi and Magdalena Toda – Weierstraß-type representation of timelike surfaces with constant mean curvature [MR 1955630]
- Norio Ejiri – A differential-geometric Schottky problem, and minimal surfaces in tori [MR 1955631]
- E. V. Ferapontov – Surfaces in 3-space possessing nontrivial deformations which preserve the shape operator [MR 1955632]
- Frédéric Hélein and Pascal Romon – Hamiltonian stationary Lagrangian surfaces in Hermitian symmetric spaces [MR 1955633]
- Hesheng Hu – Line congruences and integrable systems [MR 1955634]
- Xiaoxiang Jiao – Factorizations of harmonic maps of surfaces into Lie groups by singular dressing actions [MR 1955635]
- Hong Jin and Xiaohuan Mo – On submersive $p$-harmonic morphisms and their stability [MR 1955636]
- Kazuyoshi Kiyohara – On Kähler-Liouville manifolds [MR 1955637]
- Masatoshi Kokubu, Masaaki Umehara and Kotaro Yamada – Minimal surfaces that attain equality in the Chern-Osserman inequality [MR 1955638]
- Vladimir S. Matveev – Low dimensional manifolds admitting metrics with the same geodesics [MR 1955639]
- Yoshihiro Ohnita and Seiichi Udagawa – Harmonic maps of finite type into generalized flag manifolds, and twistor fibrations [MR 1955640]
- Joonsang Park – Submanifolds associated to Grassmannian systems [MR 1955641]
- Yusuke Sakane and Takumi Yamada – Harmonic cohomology groups of compact symplectic nilmanifolds [MR 1955642]
- Boris A. Springborn – Bonnet pairs in the 3-sphere [MR 1955643]
- Makiko Sumi Tanaka – Subspaces in the category of symmetric spaces [MR 1955644]
- Hiroyuki Tasaki – Integral geometry of submanifolds of real dimension two and codimension two in complex projective spaces [MR 1955645]
- John C. Wood – Jacobi fields along harmonic maps [MR 1955646]
- Hongyou Wu – Denseness of plain constant mean curvature surfaces in dressing orbits [MR 1955647]