# Constantin Udriste

# Polytechnic University, Bucharest, Romania

## From Integral Manifolds and Metrics to Potential Maps

The main result of our paper is that an integral manifold of a
distribution together with two Riemannian metrics produce a potential
map which is in fact a least squares approximation of the starting
integral manifold.
Section 1 analyses some elements of the geometry produced on jet
bundle of order one by a semi-Riemann Sasaki-like metric.Section 2
describes the maximal integral manifolds of a distribution as
solutions
of a PDEs system of order one.Section 3 studies the Poisson second-
order prolongations of first order PDE systems and formulates the
Lorentz-Udriste World-Force Law on a suitable semi-Riemann-Lagrange
manifold (the base manifold of the jet bundle of order one).Section 4
exploits the idea of least squares Lagrangians, to include the
integral
manifolds of a distribution into a class of extremals. Section 5
refers
to the canonical forms of the vertical metric tensor produced by a
density of energy on jet bundle of order one.

Mathematics Subject Classification:53C43, 31C12, 58E20, 37J99