### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 069, 12 pages      arXiv:1404.0720      https://doi.org/10.3842/SIGMA.2015.069

### Harmonic Maps into Homogeneous Spaces According to a Darboux Homogeneous Derivative

Alexandre J. Santana a and Simão N. Stelmastchuk b
a) Mathematics Department, State University of Maringa (UEM), 87020-900 Maringa, Brazil
b) Mathematics Department, Federal University of Parana (UFPR), 86900-000 Jandaia do Sul, Brazil

Received February 10, 2015, in final form August 24, 2015; Published online August 28, 2015

Abstract
Our purpose is to use a Darboux homogenous derivative to understand the harmonic maps with values in homogeneous space. We present a characterization of these harmonic maps from the geometry of homogeneous space. Furthermore, our work covers all type of invariant geometry in homogeneous space.

Key words: homogeneous space; harmonic maps; Darboux derivative.

pdf (357 kb)   tex (17 kb)

References

1. Abe N., Hasegawa K., An affine submersion with horizontal distribution and its applications, Differential Geom. Appl. 14 (2001), 235-250.
2. Burstall F.E., Rawnsley J.H., Twistor theory for Riemannian symmetric spaces. With applications to harmonic maps of Riemann surfaces, Lecture Notes in Math., Vol. 1424, Springer-Verlag, Berlin, 1990.
3. Catuogno P., A geometric Itô formula, Mat. Contemp. 33 (2007), 85-99.
4. Dai Y.-J., Shoji M., Urakawa H., Harmonic maps into Lie groups and homogeneous spaces, Differential Geom. Appl. 7 (1997), 143-160.
5. Dorfmeister J.F., Inoguchi J.-I., Kobayashi S., A loop group method for harmonic maps into Lie groups, arXiv:1405.0333.
6. Émery M., Stochastic calculus in manifolds, Universitext, Springer-Verlag, Berlin, 1989.
7. Fernandes M.A.N., San Martin L.A.B., Fisher information and $\alpha$-connections for a class of transformational models, Differential Geom. Appl. 12 (2000), 165-184.
8. Fernandes M.A.N., San Martin L.A.B., Geometric proprieties of invariant connections on ${\rm SL}(n,{\mathbb R})/{\rm SO}(n)$, J. Geom. Phys. 47 (2003), 369-377.
9. Helgason S., Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, Vol. 80, Academic Press, Inc., New York - London, 1978.
10. Higaki M., Actions of loop groups on the space of harmonic maps into reductive homogeneous spaces, J. Math. Sci. Univ. Tokyo 5 (1998), 401-421.
11. Khemar I., Elliptic integrable systems: a comprehensive geometric interpretation, Mem. Amer. Math. Soc. 219 (2012), x+217 pages, arXiv:0904.1412.
12. Kobayashi S., Nomizu K., Foundations of differential geometry, Vols. I, II, Interscience Publishers, New York - London, 1963.
13. Nomizu K., Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33-65.
14. Sharpe R.W., Differential geometry. Cartan's generalization of Klein's Erlangen program, Graduate Texts in Mathematics, Vol. 166, Springer-Verlag, New York, 1997.
15. Stelmastchuk S.N., The Itô exponential on Lie groups, Int. J. Contemp. Math. Sci. 8 (2013), 307-326, arXiv:1106.5637.
16. Uhlenbeck K., Harmonic maps into Lie groups: classical solutions of the chiral model, J. Differential Geom. 30 (1989), 1-50.
17. Urakawa H., Biharmonic maps into compact Lie groups and integrable systems, Hokkaido Math. J. 43 (2014), 73-103, arXiv:0910.0692.
18. Urakawa H., Biharmonic maps into symmetric spaces and integrable systems, Hokkaido Math. J. 43 (2014), 105-136, arXiv:1101.3152.