A characterization of the alpha-connections on the statistical manifold of multivariate normal distributions

We study a statistical manifold $(\mathcal{N},g^F,{\mathrm{\nabla }}^A,{\mathrm{\nabla }}^{A\ast })$ of multivariate normal distributions, where $g^F$ is the Fisher metric and ${\mathrm{\nabla }}^A$ is the Amari-Chentsov connection and ${\mathrm{\nabla }}^{A\ast }$ is its conjugate connection. We will show that it admits a solvable Lie group structure and moreover the Amari-Chentsov connection ${\mathrm{\nabla }}^A$ on $(\mathcal{N},g^F)$ will be characterized by the conjugate symmetry, i.e., a curvatures identity $R=R^{\ast }$ of a connection $\mathrm{\nabla }$ and its conjugate connection ${\mathrm{\nabla }}^{\ast }$.