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

The family of α-connections ∇(α) on a statistical manifold $$\mathcal{M}$$ equipped with a pair of conjugate connections $$\nabla \equiv \nabla^{(1)}$$ and $$\nabla^{\ast} \equiv \nabla^{(-1)}$$ is given as $$\nabla^{(\alpha)}=\frac{1+\alpha}{2} \nabla + \frac{1-\alpha}{2} \nabla^{\ast}$$. Here, we develop an expression of curvature R(α) for ∇(α) in relation to those for $$\nabla, \nabla^{\ast}$$. Immediately evident from it is that ∇(α) is equiaffine for any $$\alpha \in \mathbb{R}$$ when $$\nabla, \nabla^{\ast}$$ 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.