  The Freund family of distributions becomes a Riemannian 4-manifold with
Fisher information as metric; we derive the induced $\alpha$-geometry, i.e.,
the $\alpha$-curvature, $\alpha$-Ricci curvature with its eigenvales and
eigenvectors, the $\alpha$-scalar curvature etc. We show that the Freund
manifold has a positive constant 0-scalar curvature, so geometrically it
constitutes part of a sphere. We consider special cases as submanifolds and
discuss their geometrical structures; one submanifold yields 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.
