Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 37 (1996), No. 1, 31-40.

Complete Lifts of Harmonic Maps and Morphisms between Euclidean Spaces

Ye-Lin Ou

Abstract

In this paper, the concept of
the complete lifts of maps between (real and complex) Euclidean 
spaces is introduced.
 As applications, we are able to use this concept to 
characterize holomorphic maps  $\phi:{\Bbb C}^{m}\supset U\longrightarrow 
{\Bbb  C}^{n}$ and to construct many new examples of 
harmonic morphisms. Finally we show that the  complete lift of the 
quaternion product followed by the complex product is a simple and explicit 
example of a harmonic morphism which does not arise from any K{\"a}hler structure in the sense of Baird and Wood [Internat.\ J.\ Math.\ 6, 161-192 (1995)].

MSC 1991: 53C40, 58E20
Keywords: complete lifts, harmonic morphisms, holomorphic maps