G-dual Teleparallel Connections in Information Geometry

Abstract

Given a real, finite-dimensional, smooth parallelizable Riemannian manifold $$(\mathcal {N},G)$$ ( N , G ) endowed with a teleparallel connection $$\nabla $$ ∇ determined by a choice of a global basis of vector fields on $$\mathcal {N}$$ 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 $$(\mathcal {N},G,\nabla ,\nabla ^{*})$$ ( N , G , ∇ , ∇ ∗ ) , 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.