We show that the Freund manifold has a positive constant 0-scalar curvature, so geometrically it constitutes part of a sphere.

The Mathematica journal

In this article we consider the Freund family of distributions as a 4-manifold equipped with Fisher information as Riemannian metric and derive the induced a-geometry, that is, the a-Ricci curvature, the a-scalar curvature, and so forth. We show that the Freund manifold has a positive constant 0-scalar curvature, so geometrically it constitutes part of a sphere. We examine special cases as submanifolds and discuss their geometrical structures; via one submanifold we provide examples of neighbourhoods of the independent case for bivariate distributions having identical exponential marginals. Thus, since exponential distributions complement Poisson point processes, we obtain a means to discuss the neighbourhood of independence for random processes. Our approach using Mathematica handles the a-geometry calculations and graphics effectively and could be transferred to other distribution families.

We show that the Freund manifold has a positive constant 0-scalar curvature, so geometrically it constitutes part of a sphere.

Neighbourhoods of Independence for Random Processes via Information Geometry