×

On the existence of harmonic morphisms from symmetric spaces of rank one. (English) Zbl 0895.58015

As the next step towards the understanding of harmonic morphisms, the author constructs the first known examples of harmonic morphisms from the noncompact quaternionic hyperbolic space \(\mathbb{H} H^n\) into \(\mathbb{C}\). The approach is unified in the sense that it also provides a general framework for constructing harmonic morphisms from \(\mathbb{C} H^n\), \(\mathbb{R} H^{2n+1}\), \(\mathbb{H} P^n\), \(\mathbb{C} P^n\), and \(\mathbb{R} P^{2n+1}\) into \(\mathbb{C}\).

MSC:

58E20 Harmonic maps, etc.
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] P. Baird,Riemannian twistors and Hermitian structures on low-dimensional space forms, J. Math. Phys.33 (1992), 3340–3355. · Zbl 0763.32017 · doi:10.1063/1.529935
[2] P. Baird, J. Eells,A conservation law for harmonic maps, in Geometry Symposium Utrecht 1980, Lecture Notes in Mathematics894, 1–25, Springer (1981). · Zbl 0485.58008
[3] P. Baird, S. Gudmundsson,p-Harmonic maps and minimal submanifolds, Math. Ann.294 (1992), 611–624. · Zbl 0757.53031 · doi:10.1007/BF01934344
[4] P. Baird, J. C. Wood,Bernstein theorems for harmonic morphisms from \(\mathbb{R}\) 3 and S3, Math. Ann.280 (1988), 579–603. · Zbl 0621.58011 · doi:10.1007/BF01450078
[5] P. Baird, J. C. Wood,Harmonic morphisms and conformal foliations by geodesics of three-dimensional space forms, J. Austral. Math. Soc. (A)51 (1991), 118–153. · Zbl 0744.53013 · doi:10.1017/S1446788700033358
[6] P. Baird, J. C. Wood,Harmonic morphisms, Seifert fibre spaces and conformal foliations, Proc. London Math. Soc.64 (1992) 170–197. · Zbl 0755.58019 · doi:10.1112/plms/s3-64.1.170
[7] J. Eells, L. Lemaire,Another report on harmonic maps, Bull. London Math. Soc.20 (1988), 385–524. · Zbl 0669.58009 · doi:10.1112/blms/20.5.385
[8] B. Fuglede,Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier28 (1978), 107–144. · Zbl 0339.53026
[9] B. Fuglede,Harmonic morphisms between semi-riemannian manifolds, Ann. Acad. Sci. Fennicae21 (1996), 31–50. · Zbl 0847.53013
[10] S. Gudmundsson,The Bibliography of Harmonic Morphisms, http://www.maths.lth.se/matematiklu/personal/sigma/harmonic/bibliography.html · Zbl 0715.53029
[11] S. Gudmundsson,Harmonic morphisms from complex projective spaces, Geom. Dedicata53 (1994), 155–161. · Zbl 0826.53028 · doi:10.1007/BF01264019
[12] S. Gudmundsson,Non-holomorphic harmonic morphisms from Kähler manifolds, Manuscripta Mathematica85 (1994), 67–78. · Zbl 0826.53029 · doi:10.1007/BF02568184
[13] S. Gudmundsson,Harmonic morphisms from quaternionic projective spaces, Geom. Dedicata56 (1995), 327–332. · Zbl 0834.58016 · doi:10.1007/BF01263573
[14] S. Gudmundsson,Minimal submanifolds of hyperbolic spaces via harmonic morphisms, Geom. Dedicata62 (1996), 269–279. · Zbl 0860.53037 · doi:10.1007/BF00181568
[15] S. Gudmundsson, J. C. Wood,Multivalued harmonic morphisms, Math. Scand.73 (1993), 127–155. · Zbl 0790.58009
[16] S. HelgasonDifferential Geometry, Lie Groups and Symmetric Spaces, Academic Press (1978).
[17] T. Ishihara,A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ.19 (1979), 215–229. · Zbl 0421.31006
[18] M. T. Mustafa,A Bochner technique for harmonic morphisms, J. London Math. Soc. (to appear). · Zbl 0922.58018
[19] B. O’Neill,Semi-Riemannian Geometry, Academic Press (1983).
[20] J. C. Wood,Harmonic morphisms and Hermitian structures on Einstein 4-manifolds, Internat. J. Math.3 (1992), 415–439. · Zbl 0763.53051 · doi:10.1142/S0129167X92000187
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.