In this talk, we show that in the top layer, compact manifolds with \(\mathbf{Riem}(M)={n \choose 2}\) are positive space forms. Then we show that there are no manifolds have their value \(\mathbf {Riem}(M)\) in the interval \((\binom{n}{2}-2, \binom{n}{2})\). The manifold \(S^{n-1}\times S^1\) and arbitrary connected sums of copies of it with connected sums of positive space forms all have \(\mathbf {Riem}=\binom{n-1}{2}\). For \(1\leq p\leq n-2\leq 5\), the manifolds \(S^{n-p}\times T^p\) take the intermediate values \(\mathbf{Riem}=\binom{n-p}{2}\). From the bottom, we prove that simply connected (resp. \(2\)-connected) compact manifolds of dimension \(\geq 5\) (resp. \(\geq 7\)) have \(\mathbf {Riem}\geq 1\) (resp. \(\geq 3\)). The proof of the two last results is based on surgery, in fact we prove that the smooth \(\mathbf {Riem}\) constant doesn't decrease after a surgery on the manifold with adequate codimension.