Let (M, g) be a smooth compact Riemannian N-manifold, N is an element of 2, let p is an element of(1, N) real, and let H-1(p)(M) be the Sobolev space of order p involving first derivatives of the functions. By the Sobolev embedding theorem, H-1(p)(M) subset of L-P* (M) where p(*) = N-p/(N-p). Classically, this leads to some Sobolev inequality (I-p(1)), and then to some Sobolev inequality (I-p(p)) where each term in (I-1(p)) is elevated to the power p Long-standing questions were to know if the optimal versions with respect to the first constant of (I-p(1)) and (I-p(p)) do hold. Such questions received an affirmative answer by Hebey-Vangon for p = 2. We prove here that for p > 2, and p(2) < N, the optimal version of (I-p(p)) is false if the scalar curvature of g is positive somewhere. In particular, there exist manifolds for which the optimal versions of (I-p(1)) are true, while the optimal versions of (I-p(p)) are false. Among other results, we prove also that the assumption on the sign of the scalar curvature is sharp by showing that for any p is an element of (1, N), the optimal Version of (I-p(p)) holds on flat tori. (C) 1998 Academic Press.