Differentiable Connections with Poincaré Groups of Local Transformations. I

Abstract

In the canonical smooth fiber bundles $\pi :\mathbb{R}^{n + 1} \to \mathbb{R}^n$, we study generalized differentiable connections constructed by the author in his previous works. Special emphasis is laid on the investigation of the behavior of these connections under local transformations of the classical Poincaré groups $\mathbb{P}(1,n)$ and extented Poincaré groups $\widetilde {\mathbb{P}}(1,n)$ canonically acting in the given connections. We found all first‐order nonholonomic affine, $\Gamma _1 ,\Gamma _2 {\text{ and }}\Gamma _{1,2}$‐connections with the groups $\mathbb{P}(1,n)$ and $\widetilde {\mathbb{P}}(1,n)$ of local transformations and also constructed classes of the corresponding invariant second‐order connections.