A note on curvature of α-connections of a statistical manifold

The family of α-connections ∇(α) on a statistical manifold M equipped with a pair of conjugate connections ∇ ≡ ∇(1) and ∇∗ ≡ ∇(−1) is given as ∇(α) = 1+α 2 ∇ + 1−α 2 ∇∗. Here, we develop an expression of curvature R for ∇(α) in relation to those for ∇,∇∗. Immediately evident from it is that ∇(α) is equiaffine for any α ∈ R when ∇,∇∗ are dually flat, as previously observed in Takeuchi and Amari (IEEE Transactions on Information Theory 51:1011–1023, 2005). Other related formulae are also developed.