Extrinsic geometry and Hamiltonian analysis of symmetric teleparallel gravity

We analyze the properties of foliations in presence of non-metricity, deriving the generalized Gauss-Codazzi relations in full generality. These results are employed to study the teleparallel framework of non-metric geometry, obtaining constraints on the extrinsic and intrinsic tensors. In particular, an extrinsic symmetric two-tensor plays the role of the extrinsic curvature in Riemannian geometry, whereas no other geometric object can induce new dynamical degrees of freedom. Furthermore, we analyze the variational principle in presence of non-metricity, obtaining the boundary terms for the well-posed and well-defined Cauchy problem. Finally, we exploit the previous results to construct the Hamiltonian of the symmetric teleparallel equivalent of General Relativity, providing a proof that this theory shares the same number of degrees of freedom with its Riemannian counterpart.