On Hessian structures on the Euclidean space and the hyperbolic space (Q1282249)

For a manifold \(M\) with flat affine connection \(D\), one considers a Riemannian metric \(g\) which stands as the real version of a Kähler metric and calls \((M,g)\) a Hessian structure. Since the set of almost complex structures making \(g\) a Kähler metric is finite dimensional, the question arises whether a set of flat affine connections making \(g\) a Hessian metric is finite dimensional or not. In this paper, it is shown that if \((M,g)\) is Euclidean space \((\mathbb{R}^n,g_0)\) or hyperbolic space \((H^n,g_0)\), then the former has at least the freedom of \(n\) functions on \(\mathbb{R}\) and the latter of \(n-1\) functions. Accordlingly, the set of all flat connections \(D\) admitting these Hessian structures is infinite dimensional, for both cases.