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, published 13 January 1997

212 pages
22.9 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]