Let Riemannian metrics g and ḡ on a connected manifold Mn have the same geodesics (considered as unparameterized curves). Suppose the eigenvalues of one metric with respect to the other are all different at a point. Then, by the famous Levi-Civita’s Theorem, the metrics have a certain standard form near the point. Our main result is a generalization of Levi-Civita’s Theorem for the points where...

Let A be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p > 0. Let G be a connected and reductive algebraic group over K, and let P be a parahoric group scheme over A with generic fiber P/K = G. The special fiber P/k is a linear algebraic group over k. If G splits over an unramified extension of K, we proved in some previous work that the...

We carry out the canonical quantization of the Levi-Cività family of static and cylindrical solutions. The reduced phase space of this family of metrics is proved to coincide with that corresponding to the Kasner model, including the associated symplectic structures, except that the respective domains of definition of one of the phase space variables are not identical. Using this result, we are...