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Harmonic mappings into homogeneous Riemannian manifolds: the twistor approach. (English. Russian original) Zbl 1075.53043

Russ. Math. Surv. 59, No. 6, 1181-1203 (2004); translation from Usp. Mat. Nauk 59, No. 6, 177-200 (2004).
The author uses the twistor approach in order to study the harmonic maps \(\varphi:M\to N\) from Riemann surfaces into Riemannian manifolds. By this approach he transforms the problem of finding a harmonic map into a problem of finding the almost holomorphic maps into almost complex manifolds. First, he presents a short survey on harmonic maps of Riemannian manifolds and on the problem of the harmonic maps into compact Lie groups (Uhlenbeck construction). The harmonic equation is represented in the form of a zero-curvature equation for a family of connections depending on a spectral parameter \(\lambda\). From this construction one can construct a Bäcklund-type transformation on the space of harmonic maps which enables one to produce harmonic maps. Next, the author presents some aspects and results from the theory of loop spaces of compact Lie groups and flag manifolds. He defines a series of twistor bundles over Grassmann manifolds. He obtains a twistor interpretations of the Uhlenbeck construction and considers harmonic maps into Grassmannian of a Hilbert space, constructing analogues of flag bundles over the Hilbert Grassmannian.

MSC:

53C28 Twistor methods in differential geometry
53C43 Differential geometric aspects of harmonic maps
58E20 Harmonic maps, etc.
32L25 Twistor theory, double fibrations (complex-analytic aspects)
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