On a Torsion/Curvature Analogue of Dual Connections and Statistical Manifolds

TL;DR

This paper introduces a torsion dual connection concept in differential geometry, explores its properties, and establishes theorems relating torsion duality, curvature, and flatness in statistical manifolds, extending dual connection theory.

Contribution

It develops the theory of torsion dual connections, providing geometric interpretations, explicit coefficients, and theorems linking curvature and flatness in the context of statistical manifolds.

Findings

Torsion dual connections preserve parallelogram structures.

Existence of a 3-form measuring deviation from Levi-Civita connection.

Flatness of one connection does not imply flatness of its torsion dual.

Abstract

In analogy to the concept of a non-metric dual connection, which is essential in defining statistical manifolds, we develop that of a torsion dual connection. Consequently, we illustrate the geometrical meaning of such a torsion dual connection and show how the use of both connections preserves the cracking of parallelograms in spaces equipped with a connection and its torsion dual. The coefficients of such a torsion dual connection are essentially computed by demanding a vanishing mutual torsion among the two connections. For this manifold we then prove two basic Theorems. In particular, if both connections are metric-compatible we show that there exists a specific $3$-form measuring how the connection and its torsion dual deviate away from the Levi-Civita one. Furthermore, we prove that for these torsion dual manifolds flatness of one connection does not necessary impose flatness on the other but rather that the curvature tensor of the latter is given by a specific divergence. Finally, we give a self-consistent definition of the mutual curvature tensor of two connections and subsequently define the notion of a curvature dual connection.