Eigenvalues estimate for the Neumann problem of a bounded domain

In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a domain Omega in a given complete ( not compact a priori) Riemannian manifold ( M, g). For this, we use test functions for the Rayleigh quotient subordinated to a family of open sets constructed in a general metric way, interesting for itself. As applications, we prove that if the Ricci curvature of ( M, g) is bounded below Ric(g) >= -( n - 1) a(2), a >= 0, then there exist constants A(n) > 0, B-n > 0 only depending on the dimension, such that lambda(k)(Omega)