Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds

We give estimates for the $L^p$ norm ($2\le p\le +\mathrm{\infty }$) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere these estimates are sharp. If the curve has non vanishing geodesic curvature, we can improve our results. We also show how our approach apply to higher dimensional manifolds.