Harmonic morphisms from the classical compact semisimple Lie groups. (English) Zbl 1176.58011
An interesting topic is to produce harmonic morphisms between Riemannian or semi-Riemannian manifolds (that is, maps which pull back local harmonic functions to local harmonic functions), see [P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds. London Mathematical Society Monographs. New Series 29. Oxford: Oxford University Press (2003; Zbl 1055.53049)]. The notion of an eigenfamily on a semi-Riemannian manifold can be used to produce a variety of local harmonic morphisms.
In this paper, some results provide eigenfamilies on the Lie groups SL\(_n(\mathbb R)\), SU\(^*(2n)\) and Sp\((n,\mathbb R)\). Here, the first known complex valued harmonic morphisms from the above non-compact Lie groups endowed with their standard Riemannian metrics are given. The notion of eigenfamily is generalized to bi-eigenfamily in order to construct the first known solutions on the non-compact Riemannian SO\(^*(2n)\), SO\((p,q)\), SU\((p,q)\) and Sp\((p,q)\). The authors show how a real analytic bi-eigenfamily on a Riemannian non-compact semi-simple Lie group \(G\) gives rise to a real-analytic bi-eigenfamily on its semi-Riemannian compact dual and vice-versa.
The method of proof is borrowed from the related duality principle for harmonic morphisms from Riemannian symmetric spaces [S. Gudmundsson and M. Svensson, Ann. Global Anal. Geom. 30, No. 4, 313–333 (2006; Zbl 1103.58007)]. This duality principle is here applied to the first known complex valued harmonic morphisms from the compact Lie groups SO\((n)\), SU\((n)\) and Sp\((n)\) equipped with semi-Riemannian metrics.
In this paper, some results provide eigenfamilies on the Lie groups SL\(_n(\mathbb R)\), SU\(^*(2n)\) and Sp\((n,\mathbb R)\). Here, the first known complex valued harmonic morphisms from the above non-compact Lie groups endowed with their standard Riemannian metrics are given. The notion of eigenfamily is generalized to bi-eigenfamily in order to construct the first known solutions on the non-compact Riemannian SO\(^*(2n)\), SO\((p,q)\), SU\((p,q)\) and Sp\((p,q)\). The authors show how a real analytic bi-eigenfamily on a Riemannian non-compact semi-simple Lie group \(G\) gives rise to a real-analytic bi-eigenfamily on its semi-Riemannian compact dual and vice-versa.
The method of proof is borrowed from the related duality principle for harmonic morphisms from Riemannian symmetric spaces [S. Gudmundsson and M. Svensson, Ann. Global Anal. Geom. 30, No. 4, 313–333 (2006; Zbl 1103.58007)]. This duality principle is here applied to the first known complex valued harmonic morphisms from the compact Lie groups SO\((n)\), SU\((n)\) and Sp\((n)\) equipped with semi-Riemannian metrics.
Reviewer: Cornelia-Livia Bejan (Iaşi)
MSC:
| 58E20 | Harmonic maps, etc. |
| 53C43 | Differential geometric aspects of harmonic maps |
| 53C12 | Foliations (differential geometric aspects) |
References:
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| [2] | Baird, P., Wood, J.C.: Harmonic morphisms between Riemannian manifolds. London Math. Soc. Monogr. No. 29, Oxford Univ. Press (2003) · Zbl 1055.53049 |
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| [4] | Fuglede B. (1978). Harmonic morphisms between Riemannian manifolds. Ann. Inst. Fourier 28: 107–144 · Zbl 0339.53026 |
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| [6] | Gudmundsson, S.: The Bibliography of Harmonic Morphisms, http://www.matematik.lu.se/matematiklu/personal/sigma/harmonic/bibliography.html · Zbl 0715.53029 |
| [7] | Gudmundsson S. (1997). On the existence of harmonic morphisms from symmetric spaces of rank one. Manuscripta Math. 93: 421–433 · Zbl 0895.58015 · doi:10.1007/BF02677482 |
| [8] | Gudmundsson S. and Svensson M. (2006). Harmonic morphisms from the Grassmannians and their non-compact duals. Ann. Global Anal. Geom. 30: 313–333 · Zbl 1103.58007 · doi:10.1007/s10455-006-9029-5 |
| [9] | Ishihara T. (1979). A mapping of Riemannian manifolds which preserves harmonic functions. J. Math. Kyoto Univ. 19: 215–229 · Zbl 0421.31006 |
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