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The theory of statistical manifolds w.r.t. a conformal structure is reviewed in a creative manner and developed. By analogy, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>-manifolds are introduced. New conformal invariant tools are defined. A necessary condition for the <i>f</i>-conformal equivalence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>-manifolds is found, extending that for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-conformal equivalence for statistical manifolds. Certain examples of these new defined geometrical objects are given in the theory of Iinformation.