The first-order nonholonomic connections with the Galilean groups of local transformations (Q1881802)

If the theory of differentiable connections is applied to mathematical physics, then one necessary condition is always preserved: the Lie group of transformations in the physical theory is simultaneously the Lie group of differentiable connections of this theory. However, the classical differentiable connections not always are sufficiently flexible and sometimes do not possess the necessary Lie groups of local transformations, which restricts the scope of their applications. In his papers in [Lith. Math. J. 41, No. 4, 394--400 (2000; Zbl 1030.53028); 42, No. 1, 81--87 (2002; Zbl 1030.53029); and 42, No. 2, 211--218 (2002; Zbl 1030.53030)] the author has constructed some classes of the generalized differentiable connections with the Poincaré groups \(\mathbb{P} (1,n)\) and prolongated Poincaré groups \(\widetilde \mathbb{P}(1,n)\) of local transformations. In the present paper (and in Parts II and III (Zbl 1060.53017 and Zbl 1060.53015), a similar problem is solved for the canonical reprententations of the Galilean group \(\mathbb{G}(1,n)\) and for the corresponding differentiable connections of the first order. Both the regular and special cases are considered. In this first part the same affine nonholonomic \(\Gamma_1,\Gamma_2\), and \(\Gamma_{1,2}\)-connections, as in the previous papers, are studied. Some \(\mathbb{G}(1,n)\)-invariant properties of these connections are established.