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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Restrictions on harmonic morphisms
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by M. T. Mustafa PDF
Conform. Geom. Dyn. 3 (1999), 102-115 Request permission


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:
  • 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:
  • MathSciNet review: 1716571