metric on M. We introduce a metric invariant Ein(g)∊[0, n] that

measures in some sense the order of positivity of the scalar curvature

of g. It is positive if and only if g is a positive scalar curvature

metric and it is maximal equal to n if g is an Einstein metric with

positive scalar curvature. We introduce as well a conformal invariant

Ein([g]) and a smooth invariant Ein(M) to be respectively the

supremum of all the Ein(h) where h runs over all Riemannian metrics

in the conformal class of g and the supremum over all Riemannian

metrics on M. In this talk, I will present several examples where one can compute

these invariants and then I will prove some general geometric

properties of these invariants and their interactions with the

topology of the manifold M.