Conformal implementation of the projective equivalence over statistical manifolds (Q1359379)

A statistical manifold is a triple \((S,g,t)\) where S is a Riemannian manifold, \(g\) is a metric tensor and \(t\) is a symmetric 3-covariant tensor. Let \(\alpha\) be any real number. Statistical manifolds \((S,g,t)\) and \((S',g',t')\) are \(\alpha\)-c\(\cup\)p-related if, for some exact 1-form \(\psi\) over \(S, g'=\eta g, t'=\eta(t+\text{sym}(g\otimes \psi)),\) where \(\eta\) is a positive smooth function over \(S\) that satisfies \(d(\text{Log} \eta)=-\alpha\psi.\) The authors prove that the families of \(\alpha\)-connections of two \(\alpha\)-c\(\cup\)p-related statistical manifolds are projectively related. A relation between the curvature 2-forms of two related statistical manifolds is obtained. The problem of similarity between statistical manifolds and their classification are treated from the point of view of the notion of ``\(\alpha\)-c\(\cup\)p-related statistical manifolds''. Two statistical manifolds \((S,g,t)\) and \((S',g',t')\) are \(\alpha\)-c\(\cup\)p-equivalent if there exists a diffeomorphism \(\mathcal F\) such that \(g'=\eta\mathcal F^*g, t'=\eta\mathcal F^*(t+\text{sym}(g\otimes\psi)).\) The set of such diffeomorphisms constitutes the group of automorphisms Aut\((S,g,t)\) of the statistical manifold. The elements \(\mathcal F\in \text{Aut}(S,g,t)\) are in one-to-one correspondence with the automorphisms \(\overline{\mathcal F}\) of \(Gl(S)\) which map the bundle of conformal frames \(CO(S,g)\) into itself and give rise to the projective equivalence of the family of \(\alpha\)-connection forms. Examples of exact statistical manifolds with automorphism group of maximal dimension are given.