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Abstract and Applied Analysis
Volume 2012, Article ID 315757, 8 pages
Research Article

Harmonic Morphisms Projecting Harmonic Functions to Harmonic Functions

Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Received 30 January 2012; Revised 30 March 2012; Accepted 31 March 2012

Academic Editor: Saminathan Ponnusamy

Copyright © 2012 M. T. Mustafa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For Riemannian manifolds and , admitting a submersion with compact fibres, we introduce the projection of a function via its decomposition into horizontal and vertical components. By comparing the Laplacians on and , we determine conditions under which a harmonic function on projects down, via its horizontal component, to a harmonic function on .