On a Constant Curvature Statistical Manifold

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 $$R=R^*$$ , where $$R^*$$ is the curvature of the dual connection $$\nabla ^*$$ . Moreover, we will show that properly convex structures on a projectively flat compact manifold induces constant curvature $$-1$$ statistical structures and vice versa.