G-dual teleparallel connections in Information Geometry
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Given a real, finite-dimensional, smooth parallelizable Riemannian manifold (𝒩,G) endowed with a teleparallel connection ∇ determined by a choice of a global basis of vector fields on 𝒩, we show that the G-dual connection ∇^* of ∇ 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 ∇. We call any such pair (∇,∇^*) a G-dual teleparallel pair. Then, after defining a covariant (0,3) tensor T uniquely determined by (𝒩,G,∇,∇^*), we show that T being symmetric in the first two entries is equivalent to ∇ being torsion-free, that T being symmetric in the first and third entry is equivalent to ∇^* being torsion free, and that T being symmetric in the second and third entries is equivalent to the basis vectors determining ∇ (∇^*) being parallel-transported by ∇^* (∇). 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.
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