Geometry associated with normal distributions. (Q1581892)

The notion of a Hessian structure was introduced by the first author [Ann. Inst. Fourier 30, 91--128 (1980; Zbl 0424.53023)]. In the paper under review, for a linear mapping \(\rho\) of a domain \(\Omega\) into the space of positive definite symmetric matrices, the authors construct an exponential family of probability distributions on \({\mathbb R}^n\) and study a Hessian structure on \({\mathbb R}^n\times\Omega\) given by the exponential family. Such families contain \(n\)-dimensional normal distributions and a family of constant negative curvature. Next, it is proved that the cohomology groups of the complex of \(E(\Gamma\backslash\Omega,\rho)\)-valued forms on a compact hyperbolic affine manifold \(\Gamma\backslash\Omega\) vanish, where \(\Gamma\) is a discrete subgroup of \(\text{GL}(m,{\mathbb R})\) acting properly discontinuously and freely on \(\Omega\) and \(E(\Gamma\backslash\Omega)\) is the vector bundle associated with the universal covering \(\pi :\Omega\to\Gamma\backslash\Omega\) and \(\rho\). As a consequence, a generalization of \textit{J.-L. Koszul}'s theorem [Ann. Inst. Fourier 18, 103--114 (1968; Zbl 0167.50103)] is derived.