A Transcendence Basis in the Differential Field of Invariants of Pseudo-Galilean Group

Abstract

Let G be a subgroup in the group of all invertible linear transformations of a finite-dimensional real space X. One of the problems in differential geometry is that of finding easily verified necessary and sufficient conditions for G-equivalence of paths in X. In solving this problem, we use methods of the theory of differential invariants which give descriptions of transcendence bases of differential fields of G-invariant differential rational functions. Having explicit forms of transcendence bases, we can obtain efficient criteria for G-equivalence of paths with respect to actions of the special linear, orthogonal, pseudoorthogonal, and symplectic groups. We present a description of one finite transcendence basis in the differential field of differential rational functions invariant with respect to the action of the pseudo-Galilean group ΓO. Based on this, we establish necessary and sufficient conditions of ΓO-equivalence of paths.