Equiaffine structure and conjugate Ricci-symmetry of a statistical manifold

Abstract

A condition for a statistical manifold to have an equiaffine structure is studied. The facts that dual flatness and conjugate symmetry of a statistical manifold are sufficient conditions for a statistical manifold to have an equiaffine structure were obtained in [2] and [3]. In this paper, a fact that a statistical manifold, which is conjugate Ricci-symmetric, has an equiaffine structure is given, where conjugate Ricci-symmetry is weaker condition than conjugate symmetry. A condition for conjugate symmetry and conjugate Ricci-symmetry to coincide is also given.

Statistical Manifolds with almost Quaternionic Structures and Quaternionic Kähler-like Statistical Submersions

“…On the other hand, the existence of symplectic structures on statistical manifolds was investigated in [12], where the author obtained a duality relation between the Fubini-Study metric on a projective space and the Fisher metric on a statistical model on a finite set. Other interesting results concerning the geometry of statistical manifolds were recently obtained in [13][14][15][16][17][18][19][20][21]. In this paper, we investigate very natural kind of statistical manifold, namely those endowed with almost quaternionic structures, extending the results of K. Takano in a new setting and obtaining new curvature properties of statistical submersions.…”

“…Since T has the symmetry property for vertical vector fields (cf. (8)), using (17) and (30) we derive for all U, V ∈ Γ(V) and α = 1, 2, 3:…”

Gradient solitons on statistical manifolds

“…Recall that a pseudo-Riemannian manifold (M, g) with a pair of dual connections (∇, ∇ * ) is called conjugate Ricci-symmetric [11] if Ric ∇ = Ric ∇ * .…”