Let  be two Riemannian manifolds and V be an open subset in  Then we construct local harmonic morphisms    For this, we first construct a semi-conformal map, the semi-conformality being invariant by deformation of domain and codomain, we deform metric g on M by  making f harmonic. We obtain a local existence theorem for harmonic and semi-conformal map, thus a harmonic morphism on the subset open V onto a surface.

Riemannian manifold, harmonic morphism, equivariant maps.