The Riemann extensions in theory of differential equations and their applications

Анотація

Some properties of the 4-dim Riemannian spaces with the metrics
ds²= 2(za₃-ta₄)dx² +4(za₂-ta₃)dxdy +2(za₁-ta₂)dy² +2dxdz+2dydt connected with the second order nonlinear differential equations
y"+a₁(x, y)y'³+3a₂(x, y)y'²+3a₃(x, y)y'+a₄(x, y) = 0 (*) with arbitrary coefficients a_i(x, y) are studied. The properties of dual equations for the equations (*) are considered. The theory of the invariants of second order ODE`s for investigation of the nonlinear dynamical systems with parameters is used. The property of the eight dimensional extensions of the four-dimensional Riemannian spaces of General Relativity are discussed.