Equi-harmonic maps and Plücker formulae for horizontal-holomorphic curves on flag manifolds. (English) Zbl 1296.32005
Summary: We derive Plücker formulae for holomorphic maps into maximal flag manifolds of complex semi-simple Lie groups. Holomorphy is taken with respect to either an invariant complex structure or an invariant almost complex structure that takes part of a \((1,2)\)-symplectic Hermitian structure. The maps are assumed to be horizontal, in the case of a complex structure or to satisfy a generalization of this hypothesis in the \((1,2)\)-symplectic case. We also provide a relationship between holomorphic-horizontal curves and equiharmonic maps.
MSC:
| 32M05 | Complex Lie groups, group actions on complex spaces |
| 30F10 | Compact Riemann surfaces and uniformization |
| 32A99 | Holomorphic functions of several complex variables |
| 32Q15 | Kähler manifolds |
| 32Q60 | Almost complex manifolds |
References:
| [1] | Besse, A.: Einstein Manifolds. Springer, Berlin (1987) · Zbl 0613.53001 · doi:10.1007/978-3-540-74311-8 |
| [2] | Black, M.: Harmonic Maps into Homogeneous Spaces, Pitman Res. Notes in Math, vol. 255. Longman, Harlow (1991) · Zbl 0743.58002 |
| [3] | Borel, A.: Kählerian coset spaces of semi-simple Lie groups. Proc. Nat. Acad. Sci. USA 40, 1147-1151 (1954) · Zbl 0058.16002 · doi:10.1073/pnas.40.12.1147 |
| [4] | Bryant, R.: Lie groups and twistor spaces. Duke Math. J. 52, 223-261 (1985) · Zbl 0582.58011 · doi:10.1215/S0012-7094-85-05213-5 |
| [5] | Burstall, F. E., Rawnsley, J. H.: Twistor Theory for Riemannian Symmetric Spaces. Lect. Notes in Math. vol. 1424. Springer, Berlin (1990) · Zbl 0699.53059 |
| [6] | Calabi, E.: Minimal immersions of surfaces in Euclidean spheres. J. Diff. Geom. 1, 111-125 (1967) · Zbl 0171.20504 |
| [7] | Chern, S.S., Wolfson, J.G.: Minimal surfaces by moving frames. Am. J. Math. 105, 59-83 (1983) · Zbl 0521.53050 · doi:10.2307/2374381 |
| [8] | Cohen, N., Negreiros, C.J.C., Paredes, M., Pinzón, S., San Martin, L.A.\[B.: \cal F-\] structures on the classical flag manifold wich admit (1,2)-simpletic metrics. Tohoku Math. J. 57 (2):261-271 (2005) · Zbl 1088.53018 |
| [9] | Eells, J., Lemaire, L.: Another report on harmonic maps. Bull. Lond. Mat. Soc. 20, 385-524 (1988) · Zbl 0669.58009 · doi:10.1112/blms/20.5.385 |
| [10] | Eells, J., Wood, J.C.: Harmonic maps from surfaces to complex projective spaces. Adv. Math. 49, 217-263 (1983) · Zbl 0528.58007 · doi:10.1016/0001-8708(83)90062-2 |
| [11] | Fels, M., Olver, P.: Moving coframes: II. Regularization and theoretical foundations. Acta App. Math. 55, 127-208 (1999) · Zbl 0937.53013 · doi:10.1023/A:1006195823000 |
| [12] | Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Interscience, New Jersey (1978) · Zbl 0408.14001 |
| [13] | Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, London (1978) · Zbl 0451.53038 |
| [14] | Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 2. Interscience Publishers, Hoboken (1969) · Zbl 0175.48504 |
| [15] | Lichnerowicz, A.: Applications harmoniques et variétés Kählériennes. In: Symposia Mathematica, vol. 3 (Bologna), pp. 341-402 (1970) · Zbl 0193.50101 |
| [16] | Mo, X., Negreiros, C.J.\[C.: (1,2)\]-Symplectic structures on flag manifolds. Tohoku Math. J. 52, 271-282 (2000) · Zbl 0983.53052 · doi:10.2748/tmj/1178224611 |
| [17] | Negreiros, C.J.C.: Some remarks about harmonic maps into flag manifolds. Indiana Univ. Math. J. 37, 617-636 (1988) · Zbl 0662.58016 · doi:10.1512/iumj.1988.37.37030 |
| [18] | Negreiros, C.J.C., Grama, L., San Maritn, L.A.B.: Invariant Hermitian structures and variational aspects of a family of holomorphic curves on flag manifolds. Ann. Glob. Anal. Geom. 40(1), 105-123 (2011) · Zbl 1219.58005 · doi:10.1007/s10455-011-9248-2 |
| [19] | Rawnsley, J.: \[f\]-structures, \[f\]-twistor spaces and harmonic maps, in Geometry Seminar. In: Vesentini, E. (ed.) Luigi Bianchi, II, 1984. Lecture Notes in Math., vol. 1164 (1985) · Zbl 0592.58009 |
| [20] | San Martin, L.A.B., Negreiros, C.J.C.: Invariant almost Hermitian structures on flag manifolds. Adv. Math. 178, 277-310 (2003) · Zbl 1044.53053 · doi:10.1016/S0001-8708(02)00073-7 |
| [21] | Sternberg, S.: Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs (1964) · Zbl 0129.13102 |
| [22] | Yang, K.: Horizontal holomorphic curves in Sp \[(n)\]-flag manifolds. Proc. Am. Math. Soc. 103(1), 265-273 (1988) · Zbl 0659.53043 |
| [23] | Yang, K.: Plücker formulae for the orthogonal group. Bull. Autral. Math. Soc. 40, 447-456 (1989) · Zbl 0676.53063 · doi:10.1017/S0004972700017512 |
| [24] | Yano, K.: On a structure defined by a tensor field of type \[(1,1)\] satisfying \[F^3+F=0\]. Tensor 14, 99-109 (1963) · Zbl 0122.40705 |
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