Let 800(1, n) denote the connected component of the group of linear transformations of IRn+l preserving the bilinear form Abstract Using techniques of Lie groups and ergodic theory, it can be shown that in a compact manifold of constant negative curvature, the closure of a totally geodesic, complete (immersed) submanifold of dimension atleast 2 is a totally geodesicimmersed submanifold. The mai...

We estimate the $L^2$ norm of restriction to a totally geodesic submanifold eigenfunctions Laplace-Beltrami operator on standard flat torus $\mathbb{T}^d$, $d\ge2$. reduce getting correct bounds counting lattice points in intersection some $\nu$-transverse bands sphere. Moreover, we prove for rational submanifolds arbitrary codimension. In particular, verify conjecture Bourgain-Rudnick $L^2$-re...

We prove that wave maps factor as R1+d??SR??IM, subject to a sign condition, are globally nonlinear stable under small compactly supported perturbations when M is spaceform. The main innovation our assumption on ?S, namely it be semi-Riemannian submersion. This implies the background solution has infinite total energy, making this, best of knowledge, first stability result for factored with ene...