Equivalence of paths in some non-euclidean geometry

Let $G$ be a subgroup of the group of all reversible linear transformations of a finitedimensional real space $R^n$. One of the problems of differential geometry is to find easily verifiable necessary and sufficient conditions that ensure that $G$ is the equivalence of paths lying in $R^n$. The article establishes the necessary and sufficient conditions for the equivalence of paths in some non-Euclidean geometry.