Some properties of the 4-dim Riemannian spaces with 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.
Categories: gr-qc