Research article

Statistical connections on decomposable Riemann manifold

  • Received: 09 February 2020 Accepted: 11 May 2020 Published: 29 May 2020
  • MSC : 53B05, 53C07, 53C25

  • Let $(M, g, \varphi)$ be an $n$-dimensional locally decomposable Riemann manifold, that is, $g(\varphi X, Y) = g(X, \varphi Y)$ and $\nabla \varphi = 0$, where $\nabla $ is Riemann (Levi-Civita) connection of metric $g$. In this paper, we construct a new connection on locally decomposable Riemann manifold, whose name is statistical ($\alpha, \varphi)$-connection. A statistical $\alpha $-connection is a torsion-free connection such that $% \overline{\nabla }g = \alpha C$, where $C$ is a completely symmetric $(0, 3)$% -type cubic form. The aim of this article is to use connection $\overline{% \nabla }$ and product structure $\varphi $ in the same equation, which is possible by writing the cubic form $C$ in terms of the product structure $% \varphi $. We examine some curvature properties of the new connection and give examples of it.

    Citation: Cagri Karaman. Statistical connections on decomposable Riemann manifold[J]. AIMS Mathematics, 2020, 5(5): 4722-4733. doi: 10.3934/math.2020302

    Related Papers:

  • Let $(M, g, \varphi)$ be an $n$-dimensional locally decomposable Riemann manifold, that is, $g(\varphi X, Y) = g(X, \varphi Y)$ and $\nabla \varphi = 0$, where $\nabla $ is Riemann (Levi-Civita) connection of metric $g$. In this paper, we construct a new connection on locally decomposable Riemann manifold, whose name is statistical ($\alpha, \varphi)$-connection. A statistical $\alpha $-connection is a torsion-free connection such that $% \overline{\nabla }g = \alpha C$, where $C$ is a completely symmetric $(0, 3)$% -type cubic form. The aim of this article is to use connection $\overline{% \nabla }$ and product structure $\varphi $ in the same equation, which is possible by writing the cubic form $C$ in terms of the product structure $% \varphi $. We examine some curvature properties of the new connection and give examples of it.


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  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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