From integral manifolds and metrics to potential maps

Abstract

Our paper contains two main results: (1) the integral manifolds of a distribution together with two Riemann metrics produce potential maps which are in fact least squares approximations of the starting integral manifolds; (2) the least squares energy admits extremals satisfying periodic boundary conditions. Section 1 contains historical and bibliographical notes. Section 2 analyses some elements of the geometry produced on the jet bundle of order one by a semi-Riemann Sasaki-like metric. Section 3 describes the maximal integral manifolds of a distribution as solutions of a PDEs system of order one. Section 4 studies Poisson-like second-order prolongations of first order PDE systems and formulates the Lorentz-Udri