Twistor Theory for Riemannian Symmetric Spaces: With Applications to Harmonic Maps of Riemann Surfaces

Francis E. Burstall, John H. Rawnsley (auth.)9780387526027, 0387526021

In this monograph on twistor theory and its applications to harmonic map theory, a central theme is the interplay between the complex homogeneous geometry of flag manifolds and the real homogeneous geometry of symmetric spaces. In particular, flag manifolds are shown to arise as twistor spaces of Riemannian symmetric spaces. Applications of this theory include a complete classification of stable harmonic 2-spheres in Riemannian symmetric spaces and a Bäcklund transform for harmonic 2-spheres in Lie groups which, in many cases, provides a factorisation theorem for such spheres as well as gap phenomena. The main methods used are those of homogeneous geometry and Lie theory together with some algebraic geometry of Riemann surfaces. The work addresses differential geometers, especially those with interests in minimal surfaces and homogeneous manifolds.

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Table of contents :
Introduction….Pages 1-5
Homogeneous geometry….Pages 6-14
Harmonic maps and twistor spaces….Pages 15-21
Symmetric spaces….Pages 22-38
Flag manifolds….Pages 39-62
The twistor space of a Riemannian symmetric space….Pages 63-70
Twistor lifts over Riemannian symmetric spaces….Pages 71-80
Stable Harmonic 2-spheres….Pages 81-89
Factorisation of harmonic spheres in Lie groups….Pages 90-105