Given a real, finite-dimensional, smooth parallelizable Riemannian manifold $(\mathcal{N},G)$ endowed with a teleparallel connection $\nabla$ determined by a choice of a global basis of vector fields on $\mathcal{N}$, we show that the $G$-dual connection $\nabla^{*}$ of $\nabla$ in the sense of Information Geometry must be the teleparallel connection determined by the basis of $G$-gradient vector fields associated with a basis of differential one-forms which is (almost) dual to the basis of vector fields determining $\nabla$. We call any such pair $(\nabla,\nabla^{*})$ a $G$-dual teleparallel pair. Then, after defining a covariant $(0,3)$ tensor $T$ uniquely determined by $(\mathcal{N},G,\nabla,\nabla^{*})$, we show that $T$ being symmetric in the first two entries is equivalent to $\nabla$ being torsion-free, that $T$ being symmetric in the first and third entry is equivalent to $\nabla^{*}$ being torsion free, and that $T$ being symmetric in the second and third entries is equivalent to the basis vectors determining $\nabla$ ($\nabla^{*}$) being parallel-transported by $\nabla^{*}$ ($\nabla$). Therefore, $G$-dual teleparallel pairs provide a generalization of the notion of Statistical Manifolds usually employed in Information Geometry, and we present explicit examples of $G$-dual teleparallel pairs arising both in the context of both Classical and Quantum Information Geometry.

翻译：给定一个实、有限维、光滑可平行化的黎曼流形 $(\mathcal{N},G)$，并赋予由 $\mathcal{N}$ 上整体向量场基所确定的远平行联络 $\nabla$。我们证明，在信息几何意义下，$\nabla$ 的 $G$-对偶联络 $\nabla^{*}$ 必然是由与确定 $\nabla$ 的向量场基（几乎）对偶的微分一形式基所关联的 $G$-梯度向量场基来确定的远平行联络。我们将任意这样的对 $(\nabla,\nabla^{*})$ 称为 $G$-对偶远平行对。随后，在定义一个由 $(\mathcal{N},G,\nabla,\nabla^{*})$ 唯一确定的协变 $(0,3)$ 型张量 $T$ 后，我们证明：$T$ 在前两个指标上对称等价于 $\nabla$ 无挠；$T$ 在第一和第三指标上对称等价于 $\nabla^{*}$ 无挠；$T$ 在第二和第三指标上对称等价于确定 $\nabla$（$\nabla^{*}$）的基向量被 $\nabla^{*}$（$\nabla$）平行输运。因此，$G$-对偶远平行对推广了信息几何中通常使用的统计流形概念，并给出了在经典与量子信息几何背景下出现的 $G$-对偶远平行对的具体实例。