  In this note, we investigate upper bounds of the Neumann eigenvalue problem
for the Laplacian of a bounded domain (with smooth boundary) in a given
complete (not compact a priori) Riemannian manifold with Ricci bounded below .
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 application, we get upper bounds for the Neumann spectrum which is
clearly in agreement with the Weyl law and which is analogous to Buser's upper
bounds of the spectrum of a closed Riemannian manifold with lower bound on the
Ricci curvature.
