The Riemann Extensions in Theory of Ordinary Differential Equations and their Applications

Valery S. Dryuma

paper

The Riemann Extensions in Theory of Ordinary Differential Equations and their Applications

arXiv:nlin/0303023

Abstract

Some properties of the 4-dim Riemannian spaces with the metrics $d s^{2} = 2 (z a_{3} - t a_{4}) d x^{2} + 4 (z a_{2} - t a_{3}) d x d y + 2 (z a_{1} - t a_{2}) d y^{2} + 2 d x d z + 2 d y d t$ associated with the second order nonlinear differential equations $y^{''} + a_{1} (x, y) y^{'}^{3} + 3 a_{2} (x, y) y^{'}^{2} + 3 a_{3} (x, y) y^{'} + a_{4} (x, y) = 0$ with arbitrary coefficients $a_{i} (x, y)$ and 3-dim Einstein-Weyl spaces connected with dual equations $b^{''} = g (a, b, b^{'})$ where the function $g (a, b, b^{'})$ satisfied the partial differential equation $g_{aa cc} + 2 c g_{ab cc} + 2 g g_{a ccc} + c^{2} g_{bb cc} + 2 c g g_{b ccc} + g^{2} g_{cccc} + (g_{a} + c g_{b}) g_{ccc} - 4 g_{ab c} -$ $- 4 c g_{bb c} - c g_{c} g_{b cc} - 3 g g_{b cc} - g_{c} g_{a cc} + 4 g_{c} g_{b c} - 3 g_{b} g_{cc} + 6 g_{bb} = 0$ are considered. Some applications to the studying of the nonlinear dynamical systems and the Riemann manifolds in General Relativity are discussed.

37 pages, Latex