Abstract. We generalize the complete lift of real functions introduced by Yano--Ishihara [17], and the complete lift of maps between Euclidean spaces defined by Ou [12], to obtain harmonic maps and morphisms on semi-Riemannian manifolds. A new characterization of harmonic morphisms between (semi-)Riemannian manifolds is obtained. For some quadratic forms which generalize the quadratic maps, we give necessary and sufficient conditions under which they are harmonic maps or morphisms.

AMS Subject Classification (1991): 53C20, 58E20

Keyword(s): Harmonic maps and morphisms, tangent bundle, semi-Riemannian metric, complete lift

Received October 15, 1998, and in revised form June 17, 1999. (Registered under 2741/2009.)