On the theory of integration of tensor differential equations by the absolute-integral method (Q1063880)

The author raises the problem of solving differential equations in non- Euclidean spaces using tensorial integration. The given example, namely \[ g_{ij}\frac{dv^ j}{dt}+\Gamma_{kj,i} v^ k\frac{dx^ j}{dt}=G_ i+H_{ij} f^ j(t) \] where \(G_ i\) and \(H_{ij}\) are covariant-constant tensors, is considered nevertheless, in Euclidean space. The author concludes, examining the solution of the above mentioned equation, that the generalization for Riemannian spaces raises difficulties. But, this was clear at the beginning, because the existence of a covariant-constant vector \(G_ i\) is not ensured in Riemannian space (it exists in parallelizable spaces). - At the end the integration of the free particle motion over a surface of revolution is studied.