Abstract : We state the precise value of the best constant in Sobolev inequalities on an arbitrary Riemannian manifold when the functions are invariant by an arbitrary isometry group. We also state the precise value of the best constant in the exceptional case of Sobolev inequalities in the presence of symetries. Knowing the precise value of these constants allow us to get results on the existence of solutions of PDE 's of a higher exponant, relatively to the case where there is no symmetry. Then we wonder about the existence of the second best constant, and we state that this constant exists under conditions, which are not very restrictive since they allow us to give an answer to questions left open by different authors, and in fact give answer to any case that we can build. The proof of this theorem leads to the development of very subtle analysis technics, especially the concentration phenomenon of a sequence of solutions of a PDE. We also prove results on the geometry of orbits.