  In this note we present upper bounds for the variational eigenvalues of the
$p$-Laplacian on smooth domains of complete $n$-dimensional Riemannian
manifolds and Neumann boundary conditions, and on compact (boundaryless)
Riemannian manifolds. In particular, we provide upper bounds in the conformal
class of a given manifold $(M,g)$ for $1<p\leq n$, and upper bounds for all
$p>1$ when we fix a metric $g$. To do so, we use a metric approach for the
construction of suitable test functions for the variational characterization of
the eigenvalues. The upper bounds agree with the well-known asymptotic estimate
of the eigenvalues due to Friedlander. We also present upper bounds for the
variational eigenvalues on hypersurfaces bounding smooth domains in a
Riemannian manifold in terms of the isoperimetric ratio.
