Geodesic
Harmonic maps into homogeneous Riemannian manifolds: twistor approach
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 59 (2004) no. 6, pp. 1181-1203 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper deals with the twistor approach to the study of harmonic maps $\varphi\colon M\to N$ from Riemann surfaces $M$ to Riemannian manifolds $N$. Let $N$ be a given Riemannian manifold and let $Z$ be an almost complex manifold. The idea of the approach is to construct a so-called twistor bundle $\pi\colon Z\to N$ with the following property: the projection $\pi\circ\psi\colon M\to N$ of any almost holomorphic map $\psi\colon M\to Z$ is a harmonic map. For wide classes of Riemannian manifolds $N$ the twistor approach enables one to construct all harmonic maps $\varphi\colon M\to N$ in this way, thus reducing the original real problem of describing the harmonic maps into Riemannian manifolds to the complex problem of describing the almost holomorphic maps into almost complex manifolds. In this paper a detailed study is made of the following classes of homogeneous Riemannian manifolds $N$ to which the twistor approach can be applied: the compact Lie groups, the loop spaces of such groups, and the Grassmann manifolds, including the Hilbert Grassmannian. 
@article{RM_2004_59_6_a9,
     author = {A. G. Sergeev},
     title = {Harmonic maps into homogeneous {Riemannian} manifolds: twistor approach},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {1181--1203},
     year = {2004},
     volume = {59},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2004_59_6_a9/}
}
TY  - JOUR
AU  - A. G. Sergeev
TI  - Harmonic maps into homogeneous Riemannian manifolds: twistor approach
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2004
SP  - 1181
EP  - 1203
VL  - 59
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/RM_2004_59_6_a9/
LA  - en
ID  - RM_2004_59_6_a9
ER  - 
%0 Journal Article
%A A. G. Sergeev
%T Harmonic maps into homogeneous Riemannian manifolds: twistor approach
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2004
%P 1181-1203
%V 59
%N 6
%U http://geodesic.mathdoc.fr/item/RM_2004_59_6_a9/
%G en
%F RM_2004_59_6_a9
A. G. Sergeev. Harmonic maps into homogeneous Riemannian manifolds: twistor approach. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 59 (2004) no. 6, pp. 1181-1203. http://geodesic.mathdoc.fr/item/RM_2004_59_6_a9/

[1] M. F. Atiyah, “Instantons in two and four dimensions”, Comm. Math. Phys., 93:4 (1984), 437–451 | DOI | MR | Zbl

[2] A. Borel, F. Hirzebruch, “Characteristic classes and homogeneous spaces. I”, Amer. J. Math., 80 (1958), 458–538 | DOI | MR

[3] F. E. Burstall, J. H. Rawnsley, Twistor Theory for Riemannian Symmetric Spaces With Applications to Harmonic Maps of Riemann Surfaces, Lecture Notes in Math., 1424, Springer-Verlag, Berlin, 1990 | MR | Zbl

[4] F. E. Burstall, S. M. Salamon, “Tournaments, flags and harmonic maps”, Math. Ann., 277:2 (1987), 249–265 | DOI | MR | Zbl

[5] F. E. Burstall, J. C. Wood, “The consruction of harmonic maps into complex Grassmannians”, J. Differential Geom., 23:3 (1986), 255–297 | MR | Zbl

[6] I. Davidov, A. G. Sergeev, “Tvistornye prostranstva i garmonicheskie otobrazheniya”, UMN, 48:3 (1993), 3–96 | MR | Zbl

[7] J.-L. Koszul, B. Malgrange, “Sur certaines structures fibrées complexes”, Arch. Math. (Basel), 9 (1958), 102–109 | MR | Zbl

[8] E. Pressli, G. Sigal, Gruppy petel, Mir, M., 1990 | MR

[9] A. G. Sergeev, Kelerova geometriya prostranstv petel, MTsNMO, M., 2001

[10] K. Uhlenbeck, “Harmonic maps into Lie groups: classical solutions of the chiral model”, J. Differential Geom., 30:1 (1989), 1–50 | MR | Zbl

[11] G. Valli, “On the energy spectrum of harmonic 2-spheres in unitary groups”, Topology, 27:2 (1988), 129–136 | DOI | MR | Zbl