Geometry of contrast functions and conformal geometry
In information geometry, a statistical manifold is a pseudo-Riemannian manifold \((M,h)\) with distinguished torsion-free affine connection \(\nabla\) such that the covariant derivative \(\nabla h\) is symmetric. A contrast function on a smooth manifold \(M\) is a function on \(M\times M\) with specific properties. In particular, each contrast function defines two torsion free connections \(\nabla \) and \(\nabla^*\) and a pseudo-Riemannian metric \(h\) on \(M\). The resulting induced objects \((M,h, \nabla)\) and \((M,h,\nabla^*)\) are called dual statistical manifolds. The Barlett tensors of these structures turn out to be derived from the fourth order derivatives of the contrast function and they are used to express necessary and sufficient conditions for these structures to be projectively flat, dual-projectively flat or conformally flat.
scientific article; zbMATH DE number 1310265