Harmonic Maps, Loop Groups, and Integrable Systems University-level introduction that leads to topics of current research in the theory of harmonic maps. Martin A. Guest (Author) 9780521589321, Cambridge University Press Paperback / softback, published 13 January 1997 212 pages22.9 x 15.3 x 1.3 cm, 0.29 kg 'The book will certainly be appreciated by mathematicians as well as theoretical physics interested in the subject.' European Mathematical Society Harmonic maps are generalisations of the concept of geodesics. They encompass many fundamental examples in differential geometry and have recently become of widespread use in many areas of mathematics and mathematical physics. This is an accessible introduction to some of the fundamental connections between differential geometry, Lie groups, and integrable Hamiltonian systems. The specific goal of the book is to show how the theory of loop groups can be used to study harmonic maps. By concentrating on the main ideas and examples, the author leads up to topics of current research. The book is suitable for students who are beginning to study manifolds and Lie groups, and should be of interest both to mathematicians and to theoretical physicists. Preface Acknowledgements Part I. One-Dimensional Integrable Systems: 1. Lie groups 2. Lie algebras 3. Factorizations and homogeneous spaces 4. Hamilton's equations and Hamiltonian systems 5. Lax equations 6. Adler-Kostant-Symes 7. Adler-Kostant-Symes (continued) 8. Concluding remarks on one-dimensional Lax equations Part II. Two-Dimensional Integrable Systems: 9. Zero-curvature equations 10. Some solutions of zero-curvature equations 11. Loop groups and loop algebras 12. Factorizations and homogeneous spaces 13. The two-dimensional Toda lattice 14. T-functions and the Bruhat decomposition 15. Solutions of the two-dimensional Toda lattice 16. Harmonic maps from C to a Lie group G 17. Harmonic maps from C to a Lie group (continued) 18. Harmonic maps from C to a symmetric space 19. Harmonic maps from C to a symmetric space (continued) 20. Application: harmonic maps from S2 to CPn 21. Primitive maps 22. Weierstrass formulae for harmonic maps Part III. One-Dimensional and Two-Dimensional Integrable Systems: 23. From 2 Lax equations to 1 zero-curvature equation 24. Harmonic maps of finite type 25. Application: harmonic maps from T2 to S2 26. Epilogue References Index. Subject Areas: Differential calculus & equations [PBKJ]
