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

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

2003-03-11

arxiv.org/abs/nlin/0303023v1

Theoretical and Mathematical Physics, v.128 (1), (2001), 15-26

Nonlinear Sciences

Exactly Solvable and Integrable Systems

37 pages, Latex

Scientific paper

Some properties of the 4-dim Riemannian spaces with the metrics $$ ds^2=2(za_3-ta_4)dx^2+4(za_2-ta_3)dxdy+2(za_1-ta_2)dy^2+2dxdz+2dydt $$ associated with the second order nonlinear differential equations $$ y''+a_{1}(x,y){y'}^3+3a_{2}(x,y){y'}^2+3a_{3}(x,y)y'+a_{4}(x,y)=0 $$ 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 $$ g_{aacc}+2cg_{abcc}+2gg_{accc}+c^2g_{bbcc}+2cgg_{bccc}+ g^2g_{cccc}+(g_a+cg_b)g_{ccc}-4g_{abc}- $$ $$ -4cg_{bbc} -cg_{c}g_{bcc}- 3gg_{bcc}-g_cg_{acc}+ 4g_cg_{bc}-3g_bg_{cc}+6g_{bb} =0 $$ are considered. Some applications to the studying of the nonlinear dynamical systems and the Riemann manifolds in General Relativity are discussed.

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