Restrictions on harmonic morphisms

Author:
M. T. Mustafa

Journal:
Conform. Geom. Dyn. **3** (1999), 102-115

MSC (1991):
Primary 58E20, 53C20

DOI:
https://doi.org/10.1090/S1088-4173-99-00026-0

Published electronically:
August 16, 1999

MathSciNet review:
1716571

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider horizontally (weakly) conformal maps between Riemannian manifolds and calculate a formula for the Laplacian of the dilation of , using the language of moving frames. Applying this formula to harmonic horizontally (weakly) conformal maps or equivalently to harmonic morphisms we obtain a Weitzenböck formula similar to an earlier result of the author (J. London Math. Soc. (2) **57** (1998), 746-756), and hence vanishing results for harmonic morphisms from compact manifolds of positive curvature. Further, a method is developed to obtain restrictions on harmonic morphisms from some *non-compact* and *non-positively curved* domains. Finally, a discussion of restrictions on harmonic morphisms between simply connected space forms is given.

**1.**Baird, P.:*Harmonic maps with symmetry, harmonic morphisms, and deformation of metrics*, Pitman Research Notes in Mathematics Series**87**, Pitman, Boston, London, Melbourne, 1983. MR**85i:58038****2.**Baird, P. and Wood, J. C.: Harmonic morphisms and conformal foliations by geodesics of three-dimensional space forms,*J. Austral. Math. Soc. Ser. A***51**(1991) 118-153. MR**92k:53048****3.**Bryant, R. L.: Harmonic morphisms with fibres of dimension one,*Communications in Analysis and Geometry*(to appear).**4.**Eells, J. and Lemaire, L.: A report on harmonic maps,*Bull. London Math. Soc.***10**(1978) 1-68. MR**82b:58033****5.**Eells, J. and Lemaire, L.: Another report on harmonic maps,*Bull. London Math. Soc.***20**(1988) 385-524. MR**89i:58027****6.**Eells, J. and Sampson, J. H.: Harmonic mappings of Riemannian manifolds,*Amer. J. Math.***86**(1964) 109-160. MR**29:1603****7.**Eells, J. and Yiu, P.: Polynomial harmonic morphisms between Euclidean spheres,*Proc. Amer. Math. Soc.***123**(1995) 2921-2925. MR**95k:58048****8.**Fuglede, B.: Harmonic morphisms between Riemannian manifolds,*Ann. Inst. Fourier (Grenoble)***28**(1978) 107-144. MR**80h:58023****9.**Goldberg, S. I., Ishihara, T. and Petridis N. C.: Mappings of bounded dilatation of Riemannian manifolds,*J. Diff. Geometry***10**(1975) 619-630. MR**52:11787****10.**Gudmundsson, S.: Harmonic morphisms between spaces of constant curvature,*Proc. Edinburgh Math. Soc.***36**(1993) 133-143. MR**93j:58034****11.**Gudmundsson, S.: Harmonic morphisms from complex projective spaces,*Geom. Dedicata***53**(1994) 155-161. MR**95j:58034****12.**Gudmundsson, S.:*The Bibliography of Harmonic Morphisms*,

http://www.maths.lth.se/matematiklu/personal/sigma/harmonic/bibliography.html.**13.**Ishihara, T.: A mapping of Riemannian manifolds which preserves harmonic functions,*J. Math. Kyoto Univ.***19**(1979) 215-229. MR**80k:58045****14.**Kasue, A. and Washio, T.: Growth of equivariant harmonic maps and harmonic morphisms,*Osaka J. Math.***29**(1992) 899-928. MR**92d:58043**; MR**93e:58038****15.**Lohkamp, J.: Metrics of negative Ricci curvature,*Ann. of Math.***140**(1994) 655-683. MR**95i:53042****16.**Montaldo, S.: Harmonic maps and morphisms via moving frames, Lecture notes, University of Leeds (1997).**17.**Mustafa, M. T.: A Bochner technique for harmonic morphisms,*J. London Math. Soc.*(2)**57**(1998) 746-756. CMP**99:05****18.**Mustafa, M. T. and Wood, J. C.: Harmonic morphisms from three-dimensional Euclidean and spherical space forms,*Algebras, Groups and Geometries***15**(1998) 155-172. CMP**99:09****19.**Ou, Y. L. and Wood, J. C.: On the classification of quadratic harmonic morphisms between Euclidean spaces,*Algebras, Groups and Geometries***13**(1996) 41-53. MR**97d:58063****20.**Willmore, T. J.:*Riemannian geometry*, Oxford University Press (1993). MR**95e:53002****21.**Wood, J. C.: Harmonic morphisms, foliations and Gauss maps, Complex differential geometry and nonlinear partial differential equations (Providence, R.I.) (Y.T. Siu, ed.),*Contemp. Math.***49**, Amer. Math. Soc., Providence, R.I., 1986, 145-184. MR**87i:58045****22.**Wood, J. C.: Harmonic maps and morphisms in 4 dimensions,*Geometry, Topology and Physics*, Proceedings of the First Brazil-USA Workshop, Campinas, Brazil, June 30-July 7, 1996, B. N. Apanasov, S. B. Bradlow, K. K. Uhlenbeck (Editors), Walter de Gruyter Co., Berlin, New York (1997), 317-333. MR**99b:58067**

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Additional Information

**M. T. Mustafa**

Affiliation:
Assistant Professor, Faculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology, Topi, Distt. Swabi, N.W.F.P., Pakistan

Email:
mustafa@giki.edu.pk

DOI:
https://doi.org/10.1090/S1088-4173-99-00026-0

Keywords:
Harmonic morphisms,
harmonic maps,
Bochner technique

Received by editor(s):
December 29, 1997

Received by editor(s) in revised form:
June 8, 1999

Published electronically:
August 16, 1999

Article copyright:
© Copyright 1999
American Mathematical Society