G-dual Teleparallel Connections in Information Geometry

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.

G-dual Teleparallel Connections in Information Geometry Articles