Lifting Dual Connections with the Riemann Extension

Mathematics

Stéphane Puechmorel¹ 

[1] ENAC, Université de Toulouse, 31000 Toulouse, France;

关键词: Information geometry;    dual connections;    Riemannian extension;    cotangent bundle;   

DOI  :  10.3390/math8112079

来源: DOAJ

【 摘 要 】

Let $(M,g)$ be a Riemannian manifold equipped with a pair of dual connections $(\nabla ,{\nabla }^{*})$. 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 $T{T}^{*}M$ 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.

【 授权许可】

Unknown