Abstract
PurposeIn this paper, we give some properties of the α-connections on statistical manifolds and we study the α-conformal equivalence where we develop an expression of curvature R¯ for ∇¯ in relation to those for ∇ and ∇^.Design/methodology/approachIn the first section of this paper, we prove some results about the α-connections of a statistical manifold where we give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds treated in [1, 3], and we construct some examples.FindingsWe give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples.Originality/valueWe give some properties of the difference tensor K and we determine a relation between the curvature tensors; this relation is a generalization of the results obtained in [1]. In the second section, we introduce the notion of α-conformal equivalence of statistical manifolds, we give the relations between curvature tensors and we construct some examples.
Highlights
Let ðM m; gÞ be a Riemannian manifold and ∇ a torsion free linear connection on M
The triple ðM m; ∇; gÞ is called a statistical manifold if ∇g is symmetric and the pair ð∇; gÞ is called a statistical structure
∇ is called the dual connection of ∇!, the triple M m; ∇; g is called the dual statistical manifold of ðM m; ∇; gÞ and
Summary
Let ðM m; gÞ be a Riemannian manifold and ∇ a torsion free linear connection on M. The triple ðM m; ∇; gÞ is called a statistical manifold if ∇g is symmetric and the pair ð∇; gÞ is called a statistical structure. For a statistical manifold ðM m; ∇; gÞ, let ∇* be an affine connection on M such that,. The affine connection ∇ is torsion!free, and ∇ g symmetric. ∇ is called the dual connection of ∇!, the triple M m; ∇; g is called the dual statistical manifold of ðM m; ∇; gÞ and. JEL Classification — 53A15, 53B05, 53C42 © Khadidja Addad and Seddik Ouakkas. Published in the Arab Journal of Mathematical Sciences. The authors would like to thank the referee for some useful comments and their helpful suggestions that have improved the quality of this paper
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