  Given a compact Riemannian manifold $M$ of dimension $m\geq 2$, we study the
space of functions of $L^2(M)$ generated by eigenfunctions of eigenvalues less
than $L\geq 1$ associated to the Laplace-Beltrami operator on $M$. On these
spaces we give a characterization of the Carleson measures and the
Logvinenko-Sereda sets.
