Harmonic $\varphi$-morphisms

C. L. Bejan and M. Benyounes

Abstract: 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), 107--144.

[G] Garcia-Rio, E.; Vanhecke, L.; Vázquez-Abal, M. E.: Harmonic Endomorphism Fields. Illinois J. Math {\bf 40} (1996), 23--30.

[I] Ishihara, T.: A Mapping of Riemannian Manifolds which Preserves Harmonic Functions. J. Math. Kyoto Univ. {\bf 19} (1979), 215--229.

Keywords: $\varphi$-morphism, harmonic maps and morphisms

Classification (MSC2000): 53C20, 58E20, 53C55

Full text of the article:

• Compressed PostScript file (84 kilobytes)
• PDF file (184 kilobytes)