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