Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 44, No. 2, pp. 309321 (2003) 

Harmonic $\varphi$morphismsC. L. Bejan and M. BenyounesUniversitatea Al. I. Cuza, Seminar Matematic, 6600 Iasi, Romania, email: bejan@math.tuiasi.ro; Université de Brest, Mathématiques, 6 av. Le Gorgeu, 29200 Brest, France, email: Michele.Benyounes@univbrest.frAbstract: By extending the main result of [G], we characterize the harmonicity of any $\varphi$morphism $\Phi:TM\rightarrow TN,$ covering a map $\varphi:M\rightarrow N,$ between Riemannian manifolds, when the tangent bundles carry the complete lift metric. By following the pattern of (classical) harmonic morphisms [B], [E], we introduce in a natural way the notion of harmonic $\varphi$morphism and give a characterization that corresponds to the one obtained in [F], [I]. One of the properties is that $\varphi$ is a harmonic morphism if and only if ${\rm d}\varphi$ is a harmonic $\varphi$morphism. We end with some examples and applications to (1,1)tensor fields. [B] Baird, P.; Wood, J. C.: Harmonic Morphisms between Riemannian Manifolds. LMS Monograph Series, Oxford University Press. [E] Eells, J.; Lemaire, L.: Selected Topics in Harmonic Maps. CBMS Regional Conf. Ser. in Math. {\bf 50}, AMS Providence 1983. [F] Fuglede, B.: Harmonic Morphisms between Riemannian Manifolds. Ann. Inst Fourier Grenoble {\bf 28} (1978), 107144. [G] GarciaRio, E.; Vanhecke, L.; VázquezAbal, M. E.: Harmonic Endomorphism Fields. Illinois J. Math {\bf 40} (1996), 2330. [I] Ishihara, T.: A Mapping of Riemannian Manifolds which Preserves Harmonic Functions. J. Math. Kyoto Univ. {\bf 19} (1979), 215229. Keywords: $\varphi$morphism, harmonic maps and morphisms Classification (MSC2000): 53C20, 58E20, 53C55 Full text of the article:
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