Lifting Dual Connections with the Riemann Extension

Let (M,g) be a Riemannian manifold equipped with a pair of dual connections (∇,∇*). Such a structure is known as a statistical manifold since it was defined in the context of information geometry. This paper aims at defining the complete lift of such a structure to the cotangent bundle T*M using the Riemannian extension of the Levi-Civita connection of M. In the first section, common tensors are associated with pairs of dual connections, emphasizing the cyclic symmetry property of the so-called skewness tensor. In a second section, the complete lift of this tensor is obtained, allowing the definition of dual connections on TT*M with respect to the Riemannian extension. This work was motivated by the general problem of finding the projective limit of a sequence of a finite-dimensional statistical manifold.

VIII.—Riemann Extensions which have Kähler Metrics.

SynopsisCertain types of 2n-dimensional Riemannian spaces admitting parallel fields of null n-planes are studied. These are known as Riemann extensions of conformal, projective or other classes of spaces of affine connection. The circumstances under which a 2n-dimensional Riemannian space admits two non-intersecting parallel fields of null n-planes are also discussed. Such spaces satisfy a condition similar to Kähler's condition in the theory of complex manifolds, and hence are called Kähler spaces. Necessary and sufficient conditions are found for a Kähler space to be a Riemann extension with respect to one of the parallel fields of null n-planes, and canonical forms are found for the metrics in the cases of Riemann extensions of conformal and projective spaces.