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This work is devoted to the study of a class of Poisson-Lie groups endowedwith left invariant metrics. The triples $(G,\pi,<,>)$ are considered, where$G$ is a simply connected Lie group, ?$\pi$ is a multiplicative Poisson tensorand $<,>$ is a left invariant riemannian metric such that Hawkins conditionsare satisfied. Hawkins conditions are necessary conditions for the deformationof the graded algebra of differential forms of a riemannian manifold. Theseconditions come from the deformation of the noncommutative spectral tripledescribing the manifold. The main result of this thesis is the equivalencebetween, on one hand, the geometric problem of classifying riemannianPoisson-Lie groups that satisfy the conditions of Hawkins and, secondly, theproblem of classifying algebraic structures of Lie bialgebras on Milnoralgebras that satisfy certain conditions. Exploiting the fact that thestructures of Lie bialgebras on Milnor algebras, in certain situations, can becalculated, we determine riemannian Poisson-Lie groups that satisfy Hawkins inthe linear case, in the case of Heisenberg in the triangular case and in lowdimensions (up to dimension 5). The general case remains an open problem.

Connexions contravariantes sur les groupes de Lie-Poisson