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Lifting Dual Connections with the Riemann Extension

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  • Stéphane Puechmorel

    (ENAC, Université de Toulouse, 31000 Toulouse, France)

Abstract

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 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.

Suggested Citation

  • Stéphane Puechmorel, 2020. "Lifting Dual Connections with the Riemann Extension," Mathematics, MDPI, vol. 8(11), pages 1-15, November.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:2079-:d:448975
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    References listed on IDEAS

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    1. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 4.
    2. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 3.
    3. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 2.
    4. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 2.
    5. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 4.
    6. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 3.
    7. Editorial Article, 0. "The Information for Authors," Economics of Contemporary Russia, Regional Public Organization for Assistance to the Development of Institutions of the Department of Economics of the Russian Academy of Sciences, issue 1.
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    Cited by:

    1. Esmaeil Peyghan & Davood Seifipour & Ion Mihai, 2023. "On the Geometry of Kobayashi–Nomizu Type and Yano Type Connections on the Tangent Bundle with Sasaki Metric," Mathematics, MDPI, vol. 11(18), pages 1-18, September.

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