  Let $M$ be a closed smooth manifold. In 1999, L. Friedlander and N.
Nadirashvili introduced a new differential invariant $I_1(M)$ using the first
normalized nonzero eigenvalue of the Lalpace-Beltrami operator $\Delta_g$ of a
Riemannian metric $g$. They defined it taking the supremum of this quantity
over all Riemannian metrics in each conformal class, and then taking the
infimum over all conformal classes. By analogy we use $k$-th eigenvalues of
$\Delta_g$ to define the invariants $I_k(M)$ indexed by positive integers $k$.
In the present paper the values of these invariants on surfaces are
investigated. We show that $I_k(M)=I_k(\mathbb{S}^2)$ unless $M$ is a
non-orientable surface of even genus. For orientable surfaces and $k=1$ this
was earlier shown by R. Petrides. In fact L. Friedlander and N. Nadirashvili
suggested that $I_1(M)=I_1(\mathbb{S}^2)$ for any surface $M$ different from
$\mathbb{RP}^2$. We show that, surprisingly enough, this is not true for
non-orientable surfaces of even genus, for such surfaces one has
$I_k(M)>I_k(\mathbb{S}^2)$. We also discuss the connection between the
Friedlander-Nadirashvili invariants and the theory of cobordisms, and
conjecture that $I_k(M)$ is a cobordism invariant.
