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**On Hessian Riemannian structures.**
*(English)*
Zbl 1021.53021

Let \(Q\) be an open subset of \(\mathbb{R}^n\) and \(f: Q\to \mathbb{R}\) a strongly convex function. Then the Hessian of \(f\) determines a Riemannian metric \(g\) on \(Q\) by \(g_{ij}(x)= {\partial^2 f(x)\over\partial x_i\partial x_j}\) and this is called a Hessian Riemannian structure on \(Q\). To motivate his work, the author starts with the computation of the Riemanian curvature tensor of this \((Q,g)\) for general dimension \(n\) and for \(n= 2\). It turns out that surprisingly, the expression only involves derivatives of order two and three and not of order four as one could expect a priori. The understanding of this phenomenon is the real motivation of the author’s research explained in this paper.

This understanding comes from a nicely developed theory which leads to a characterization result of a Hessian Riemannian structure \((Q,g)\) on a connected open subset \(Q\) of \(\mathbb{R}^n\) such that \(H^1(Q,\mathbb{R})= 0\) in terms of a natural connection in the general linear group \(\text{GL}(n,\mathbb{R})^+\). This group is viewed as the total space of a principal \(\text{SO}(n)\)-bundle over the space \(P\) of positive definite symmetric matrices of type \((n\times n)\) and its pullback over \(Q\) under the map \(g: Q\to P\).

After the treatment of the general case, the author considers the case \(n=2\) and provides an interpretation of the curvature of a Hessian Riemannian structure at a given point in terms of an umbilic point of a surface in \(\mathbb{R}^3\), namely the graph of the function \(f\) viewed as a surface in \(\mathbb{R}^3\).

This understanding comes from a nicely developed theory which leads to a characterization result of a Hessian Riemannian structure \((Q,g)\) on a connected open subset \(Q\) of \(\mathbb{R}^n\) such that \(H^1(Q,\mathbb{R})= 0\) in terms of a natural connection in the general linear group \(\text{GL}(n,\mathbb{R})^+\). This group is viewed as the total space of a principal \(\text{SO}(n)\)-bundle over the space \(P\) of positive definite symmetric matrices of type \((n\times n)\) and its pullback over \(Q\) under the map \(g: Q\to P\).

After the treatment of the general case, the author considers the case \(n=2\) and provides an interpretation of the curvature of a Hessian Riemannian structure at a given point in terms of an umbilic point of a surface in \(\mathbb{R}^3\), namely the graph of the function \(f\) viewed as a surface in \(\mathbb{R}^3\).

Reviewer: L.Vanhecke (Leuven)