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Harmonic maps, loop groups, and integrable systems. (English) Zbl 0898.58010
London Mathematical Society Student Texts. 38. Cambridge: Cambridge University Press. xiii, 194 p. (1997).
The main goal of the book is to provide a good explanation of how the theory of loop groups can be used to study harmonic maps. It gives an excellent account of the interaction that exists between the fundamental notions, theories and problems of differential geometry, Lie groups, and integrable Hamiltonian systems. The reader is referred to the two survey papers of J. Eells and L. Lemaire [Bull. Lond. Math. Soc. 10, 1-68 (1978; Zbl 0401.58003); ibid. 20, 385-524 (1988; Zbl 0669.58009)] for the history of harmonic maps and its problems, including the progress that had been achieved on the classification problem before 1988.
The motivation for the writing of the book was inspired by some recent advances in the theory of harmonic maps, which are based upon the theory of integrable systems.
It is an important contribution for both graduate students and mathematicians.

MSC:
58E20 Harmonic maps, etc.
22E67 Loop groups and related constructions, group-theoretic treatment
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
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