The classification of paths in the Galilean geometry

V. I. Chilin, K. K. Muminov

National University of Uzbekistan named after Mirzo Ulugbek

Abstract: Let $X$ be $n$-dimensional linear space over field $\mathbb R$ of real numbers and let $GL(n, \mathbb {R})$ be the group of all invertible linear transformations of the space $ X $. Two paths $x(t), \ y(t)\subset X, \ t \in (0,1), $ are called $G$-equivalent with respect to the action of the subgroup $ G $ of the group $ GL(n, \mathbb {R}) $ if $ g (x(t)) =y(t) $ for some $ g \in G $ and all $t \in (0,1)$. One of the important problems of differential geometry is finding necessary and sufficient conditions such that the paths $x(t), \ y(t)$ are $G$-equivalent. The solutions of this problem use methods of the theory of differential invariants, giving a description of finite rational bases of differential fields of $ G $-invariant differential rational functions. These bases provide effective criteria for $ G $-equivalence of paths. This approach was used in for solving the problem of the equivalence of paths with respect to the action of the symplectic, orthogonal and pseudo-orthogonal groups.
An important example of a non-Euclidean geometry is the Galileo geometry. The group $\Gamma (n, \mathbb {R}) $ of all invertible linear transformations of the space $ X $, preserving the Galilean metric, are called Galileo's group. We give the following description of a finite rational basis in the differential field $\mathbb{R} \langle x_1,\dots,x_n \rangle^{\Gamma(n,\mathbb{R})}$ of all $\Gamma(n,\mathbb R)$-invariant differential rational functions.
In the field $\mathbb{R} \langle x_1,\dots,x_n \rangle^{\Gamma(n,\mathbb{R})}$ the following differential polynomials form its a rational basis:
\begin{gather*} \varphi_k(x_1,\dots,x_n) = \sum_{i=2}^n(x_i^{(k)})^2, \ k=0,\dots, n-2;\\ \psi(x_1,\dots,x_n) = x_1. \end{gather*}
Using this rational basis the following necessary and sufficient conditions for the $\Gamma(n,\mathbb R)$-equivalence of two regular paths are established. Two regular paths $x(t)=\{x_i(t)\}_{i=1}^n$ and $y(t)=\{y_i(t)\}_{i=1}^n$ are $\Gamma(n,\mathbb{R})$-equivalent if and only if $y_1(t)=\pm x_1(t)$ and $\sum\limits^n_{i=2}\left(x^{(m)}_i(t)\right)^2= \sum\limits^n_{i=2}\left(y^{(m)}_i(t)\right)^2$ for all $t\in(0,1)$ and $m=0,1,\dots,n-2$.

Keywords: Galileo space, a group of movements, differential invariant.

UDC: 512.745

MSC: 53A15, 53A55, 53B30