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THE STRUCTURE OF HARMONIC MORPHISMS WITH TOTALLY GEODESIC FIBRES

M. T. MUSTAFA

Department of Mathematical Sciences, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

https://doi.org/10.1142/S0219199704001409Cited by:1 (Source: Crossref)

Abstract

The structure of local and global harmonic morphisms between Riemannian manifolds, with totally fibres, is investigated. It is shown that non-positive curvature of the domain obstructs the existence of global harmonic morphisms with totally geodesic fibres and the only such maps from compact Riemannian manifolds of non-positive curvature are, up to a homothety, totally geodesic Riemannian submersions. Similar results are obtained for local harmonic morphisms with totally geodesic fibres from open subsets of non-negatively curved compact and non-compact manifolds. During the course, we prove non-existence of submersive harmonic morphisms with totally geodesic fibres from some important domains, for instance from compact locally symmetric spaces of non-compact type and open subsets of symmetric spaces of compact type.

Keywords:

AMSC: 58E20

References

P. Baird , Harmonic Maps with Symmetry, Harmonic Morphisms, and Deformation of Metrics , Pitman Research Notes in Mathematics Series 87 ( Pitman , Boston, London, Melbourne , 1983 ) . Google Scholar
P. Baird, Ann. Inst. Fourier 40, 177 (1990). Crossref, Web of Science, Google Scholar
P. Baird and J. Eells, Lecture Notes in Mathematics 894 (1981) pp. 1–25. Google Scholar
P. Baird and J. C. Wood, Math. Ann. 280, 579 (1988), DOI: 10.1007/BF01450078. Crossref, Web of Science, Google Scholar
P. Baird and J. C. Wood, J. Australian Math. Soc. A 51, 118 (1991). Crossref, Web of Science, Google Scholar
P. Baird and J. C. Wood , Harmonic Morphisms between Riemannian Manifolds , London Mathematical Society Monographs New Series ( Oxford University Press , 2003 ) . Crossref, Google Scholar
R. H. Escobales, J. Differential Geom. 10, 253 (1975). Crossref, Google Scholar
B. Fuglede, Ann. Inst. Fourier 28, 107 (1978). Crossref, Google Scholar
S. Gudmundsson, Proc. Edinburgh Math. Soc. 36, 133 (1992). Crossref, Web of Science, Google Scholar
S. Gudmundsson, The geometry of harmonic morphisms, Ph.D. hesis, University of Leeds (1992) . Google Scholar
S. Gudmundsson, Manuscripta Math. 93, 421 (1997). Crossref, Web of Science, Google Scholar
S. Gudmundsson, The Bibliography of Harmonic Morphisms, http://www.maths.lth.se/matematiklu/personal/sigma/harmonic/bibliography.html . Google Scholar
T. Ishihara, J. Math. Kyoto Univ. 19, 215 (1979). Crossref, Google Scholar
A. Kasue and T. Washio, Osaka J. Math. 27, 899 (1990). Web of Science, Google Scholar
J. Lohkamp, Ann. of Math. 140, 655 (1994), DOI: 10.2307/2118620. Crossref, Web of Science, Google Scholar
M. T. Mustafa, J. London Math. Soc. 57(2), 746 (1998), DOI: 10.1112/S0024610798005973. Crossref, Web of Science, Google Scholar
M. T. Mustafa, Conformal Geometry and Dynamics 3, 102 (1999), DOI: 10.1090/S1088-4173-99-00026-0. Crossref, Google Scholar
R. Pantilie, Internat. J. Math. 10, 457 (1999), DOI: 10.1142/S0129167X99000197. Link, Web of Science, Google Scholar
J. C. Wood, Complex Differential Geometry and Nonlinear Partial Differential Equations, Contemp. Math. 49, ed. Y. T. Siu (American Mathematical Society, Providence, R.I., 1986) pp. 145–184. Crossref, Google Scholar