Any statistical manifold has a contrast function. On the \(C^ 3\)- functions taking the minimum at the diagonal of the product manifold
A statistical manifold is a Riemannian manifold \((M,g)\) with a symmetric covariant tensor \(T\) of order 3. Definition. A contrast function of a differentiable manifold \(M\) is a real-valued smooth function \(\rho\) on \(M\times M\) such that \(\rho(x,y)\geq 0\) with equality if and only if \(x= y\) and \(X_ x X_ x\rho(x,y)_{| x=y}>0\) for any smooth vector field \(X\) on \(M\) which is non-zero at \(x\). We use the following notation for a function on \(M\) defined by the value of partial derivative in \(M\times M\) with respect to the smooth vector fields \(X_ 1,\dots,X_ n\), \(Y_ 1,\dots,Y_ m\) on \(M\): \[ \rho(X_ 1,\dots,X_ n| Y_ 1,\dots,Y_ m)(z)= (X_ 1)_ x\cdots (X_ n)_ x(Y_ 1)_ y\cdots (Y_ m)_ y\rho(x,y)_{| x=z,y=z}. \] The Riemannian metric tensor \(g\) is defined by \(g(X, Y)= -\rho(X| Y)\). The trilinear form \(T\) is defined by \(T(X,Y,Z)= -\rho(XY| Z)+ \rho(Z| XY)\). The main result of the paper is follows: Theorem. For any statistical manifold \((M,g,T)\) we can find a contrast function \(\rho\) of \(M\) which gives \(g\) and \(T\) by the above formulas.
scientific article; zbMATH DE number 482174