Lifting Dual Connections with the Riemann Extension

• Main Author: Stéphane Puechmorel
• Format: Article
• Language: English
• Published: MDPI AG 2020-11-01
• Series: Mathematics
• Subjects: Information geometry; dual connections; Riemannian extension; cotangent bundle
• Online Access: https://www.mdpi.com/2227-7390/8/11/2079

Let <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> be a Riemannian manifold equipped with a pair of dual connections <inline-formula><math display="inline"><semantics><mrow><mo>(</mo><mo>∇</mo><mo>,</mo><msup><mo>∇</mo><mo>*</mo></msup><mo>)</mo></mrow></semantics></math></inline-formula>. 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 <inline-formula><math display="inline"><semantics><mrow><msup><mi>T</mi><mo>*</mo></msup><mi>M</mi></mrow></semantics></math></inline-formula> using the Riemannian extension of the Levi-Civita connection of <i>M</i>. 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 <inline-formula><math display="inline"><semantics><mrow><mi>T</mi><msup><mi>T</mi><mo>*</mo></msup><mi>M</mi></mrow></semantics></math></inline-formula> 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.