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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