On the α-connections and the α-conformal equivalence on statistical manifolds

Purpose In 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mover accent="true"><m:mrow><m:mi>R</m:mi></m:mrow><m:mo>¯</m:mo></m:mover></m:mrow></m:math> for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mover accent="true"><m:mrow><m:mo>∇</m:mo></m:mrow><m:mo>¯</m:mo></m:mover></m:mrow></m:math> in relation to those for ∇ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mover accent="true"><m:mrow><m:mo>∇</m:mo></m:mrow><m:mo stretchy="false">^</m:mo></m:mover></m:mrow></m:math> . Design/methodology/approach In 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. Findings 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, we give the relations between curvature tensors and we construct some examples. Originality/value 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, we give the relations between curvature tensors and we construct some examples.