Geodesic
Biharmonic morphisms
Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 1, pp. 145-159.

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Let $(X, \Cal H)$ and $(X',\Cal H')$ be two strong biharmonic spaces in the sense of Smyrnelis whose associated harmonic spaces are Brelot spaces. A biharmonic morphism from $(X,\Cal H)$ to $(X',\Cal H')$ is a continuous map from $X$ to $X'$ which preserves the biharmonic structures of $X$ and $X'$. In the present work we study this notion and characterize in some cases the biharmonic morphisms between $X$ and $X'$ in terms of harmonic morphisms between the harmonic spaces associated with $(X,\Cal H)$ and $(X',\Cal H')$ and the coupling kernels of them.
Classification : 31B30, 31C35, 31D05
Keywords: harmonic space; harmonic morphism; biharmonic space; biharmonic function; biharmonic morphism 
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Chadli, Mustapha; El Kadiri, Mohamed; Haddad, Sabah. Biharmonic morphisms. Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 1, pp. 145-159. http://geodesic.mathdoc.fr/item/CMUC_2005__46_1_a13/