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Conformal Geometry and Dynamics

ISSN 1088-4173



Restrictions on harmonic morphisms

Author: M. T. Mustafa
Journal: Conform. Geom. Dyn. 3 (1999), 102-115
MSC (1991): Primary 58E20, 53C20
Published electronically: August 16, 1999
MathSciNet review: 1716571
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Abstract: We consider horizontally (weakly) conformal maps $\phi$ between Riemannian manifolds and calculate a formula for the Laplacian of the dilation of $\phi$, 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.

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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

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