Differentiable connections with Poincaré groups of local transformations. I (Q1873232)

One considers the canonical fiber bundle \(\pi :{\mathbb R}^{n+1}\rightarrow {\mathbb R}^n\), the natural action of the Lie algebra of the Poincaré group \(P(1,n)\) in the module of vector fields over the fiber bundle and one determines the system of equations satisfied by the local coefficients of a \(P(1,n)\)-invariant connnection on the bundle. It comes out that the coefficients of a \(P(1,n)\)-invariant connection form geometric objects of the given representation of the group \(P(1,n)\) and one studies the relations between certain \(P(1,n)\)-invariant connections. For Part II, see ibid. 42, 81-87 (2002; Zbl 1030.53028) below.