  In this note we prove an analogue of the Rayleigh-Faber-Krahn inequality,
that is, that the geodesic ball is a maximiser of the first eigenvalue of some
convolution type integral operators, on the sphere $\mathbb{S}^{n}$ and on the
real hyperbolic space $\mathbb{H}^{n}$. It completes the study of such question
for complete, connected, simply connected Riemannian manifolds of constant
sectional curvature. We also discuss an extremum problem for the second
eigenvalue on $\mathbb{H}^{n}$ and prove the Hong-Krahn-Szeg\"{o} type
inequality. The main examples of the considered convolution type operators are
the Riesz transforms with respect to the geodesic distance of the space.
