## Restrictions on harmonic morphisms

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- by M. T. Mustafa PDF
- Conform. Geom. Dyn.
**3**(1999), 102-115 Request permission

## 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
- Email: mustafa@giki.edu.pk
- Received by editor(s): December 29, 1997
- Received by editor(s) in revised form: June 8, 1999
- Published electronically: August 16, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**3**(1999), 102-115 - MSC (1991): Primary 58E20, 53C20
- DOI: https://doi.org/10.1090/S1088-4173-99-00026-0
- MathSciNet review: 1716571