Geometry of statistical manifolds

Abstract

A statistical manifold ( M, g, ▿) is a Riemannian manifold ( M, g) equipped with torsion-free affine connections ▿, ▿ ∗ which are dual with respect to g. A point p \ e M is said to be ▿-isotropiv if the sectional curvatures have the same value k( p), and ( M, g, ▿) is said to be ▿-isotropic when M consists entirely of ▿-isotropic points. When the difference tensor α of ▿ and the Levi-Civita connection ▿ 0 of g is “apolar” with respect to g, Kurose has shown that α ≡ 0, and hence ▿ = ▿ ∗ = ▿ 0, provided that k( p) = k(constant). His proof relies on the existence of affine immersion which may no longer hold when k( p) is not constant. One objective of this paper is to show that the above Kurose's result still remains valid when ( M, g, ▿) is assumed only to be ▿-isotropic. We also discuss the case where ( M, g) is complete Riemannian.