Restriction of toral eigenfunctions to totally geodesic submanifolds

We estimate the $L^2$ norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace-Beltrami operator on the standard flat torus ${\mathbb{T}}^d$, $d\ge 2$. We reduce getting correct bounds to counting lattice points in the intersection of some $\nu$-transverse bands on the sphere. Moreover, we prove the correct bounds for rational totally geodesic submanifolds of arbitrary codimension. In particular, we verify the conjecture of Bourgain-Rudnick on $L^2$-restriction estimates for rational hyperplanes. On ${\mathbb{T}}^2$, we prove the uniform $L^2$ restriction bounds for closed geodesics. On ${\mathbb{T}}^3$, we obtain explicit $L^2$ restriction estimates for the totally geodesic submanifolds, which improve the corresponding results by Burq-G'erard-Tzvetkov, Hu, Chen-Sogge.