  If $(M,g)$ is a compact Riemannian surface then the integrals of
$L^2(M)$-normalized eigenfunctions $e_j$ over geodesic segments of fixed length
are uniformly bounded. Also, if $(M,g)$ has negative curvature and $\gamma(t)$
is a geodesic parameterized by arc length, the measures $e_j(\gamma(t))\, dt$
on $\R$ tend to zero in the sense of distributions as the eigenvalue $\la_j\to
\infty$, and so integrals of eigenfunctions over periodic geodesics tend to
zero as $\la_j\to \infty$. The assumption of negative curvature is necessary
for the latter result.
