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.

‘… an accessible introduction to some of the fundamental connections beween differental geometry, Lie groups and integrable Hamiltonian systems.’

Source: L’Enseignement Mathématique

‘It is very rare to find a book that can take a student from the very basics of a subject to the frontiers of active research. The author is to be congratulated on having produced just such a rarity!’

Source: Bulletin of the London Mathematics Society

‘The book will certainly be appreciated by mathematicians as well as theoretical physics interested in the subject.’

Source: European Mathematical Society