On a constant curvature statistical manifold

Abstract

We will show that a statistical manifold $(M,g,\nabla )$ has a constant curvature if and only if it is a projectively flat conjugate symmetric manifold, that is, the affine connection $\nabla$ is projectively flat and the curvatures satisfies $R=R^{\ast }$, where $R^{\ast }$ is the curvature of the dual connection ${\nabla }^{\ast }$. Moreover, we will show that properly convex structures on a projectively flat compact manifold induces constant curvature $-1$ statistical structures and vice versa.