  We prove a couple of new endpoint geodesic restriction estimates for
eigenfunctions. In the case of general 3-dimensional compact manifolds, after a
$TT^*$ argument, simply by using the $L^2$-boundedness of the Hilbert transform
on $\R$, we are able to improve the corresponding $L^2$-restriction bounds of
Burq, G\'erard and Tzvetkov and Hu. Also, in the case of 2-dimensional compact
manifolds with nonpositive curvature, we obtain improved $L^4$-estimates for
restrictions to geodesics, which, by H\"older's inequality and interpolation,
implies improved $L^p$-bounds for all exponents $p\ge 2$. We do this by using
oscillatory integral theorems of H\"ormander, Greenleaf and Seeger, and Phong
and Stein, along with a simple geometric lemma (Lemma \ref{lemma3.2}) about
properties of the mixed-Hessian of the Riemannian distance function restricted
to pairs of geodesics in Riemannian surfaces. We are also able to get further
improvements beyond our new results in three dimensions under the assumption of
constant nonpositive curvature by exploiting the fact that in this case there
are many totally geodesic submanifolds.
