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Musical Isomorphisms and Statistical Manifolds

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Abstract

Let \((M, g, {\nabla }^{(\alpha )})\) be a statistical manifold and \(g^\flat : TM \rightarrow T^*M\) be a musical isomorphism from the tangent bundle onto the cotangent bundle. Using the \(\alpha \)-vertical and \(\alpha \)-horizontal lifts on the tangent bundle of the statistical manifold M, we construct the g-\(\alpha \)-vertical and g-\(\alpha \)-horizontal lifts on the cotangent bundle with the aid of the musical isomorphism \(g^\flat \). We prove that the Lie bracket of the \(\alpha \)-horizontal lifts of vector fields to tangent and cotangent bundles is \(g^\flat \)-related if and only if the \(\alpha \)-curvature tensor is an even function of \(\alpha \). Also, we get statistical structures via the musical isomorphism in the cotangent bundle. Finally, we give the notion of the Schouten–Van Kampen \((\alpha )\)-connection associated with the statistical connection on the cotangent bundles. Furthermore, we provide some non-trivial examples as applications to this study.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Arak University, Arak, 38156-8-8349, Iran

    Esmaeil Peyghan & Leila Nourmohammadifar

  2. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Siraj Uddin

Authors
  1. Esmaeil Peyghan
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  2. Leila Nourmohammadifar
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  3. Siraj Uddin
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Correspondence to Esmaeil Peyghan.

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Peyghan, E., Nourmohammadifar, L. & Uddin, S. Musical Isomorphisms and Statistical Manifolds. Mediterr. J. Math. 19, 225 (2022). https://doi.org/10.1007/s00009-022-02141-z

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