On Hessian Riemannian Structures

In Proposition 4.1 a characterization is given of Hessian Rieman-nian structures in terms of a natural connection in the general linear group GL(n; R) + , which is viewed as a principal SO(n)-bundle over the space of positive deenite symmetric n n-matrices. For n = 2, Proposition 5.3 contains an interpretation of the curvature of a Hessian Riemannian structure at a given point, in terms of an umbilic point of a related surface in R 3. 0 Introduction In convex programming, one makes use of a so-called self-concordant barrier function f on an open convex subset Q of R n , cf. the book 7] of Nesterov and Nemirovskii, and one is interested in the behaviour of the geodesics of the Riemannian structure deened by the Hessian of f. I got acquainted with the subject when I was asked to give an introduction to Riemannian geometry at the conference HPOPT'99 at the Erasmus University Rotterdam, in June 1999. The formula for the curvature tensor of a Hessian Riemannian structure, cf. (1.7) below, involves only second and third order derivatives of the function f, and no fourth order ones as one would a priori expect. In my attempt to understand this, I arrived at the characterization of Hessian Riemannian structures in Proposition 4.1. In the case n = 2 there is also an interpretation of the curvature of a Hessian Riemannian structure in terms of umbilic points of surfaces, see Section 5. The study of Hessian Riemannian structures on convex domains goes back at least to Koszul 6] and Vinberg 11], who were inspired by the theory of bounded domains in C n with its Bergmann metric. Closely related to our subject is Shima's theory of Hessian manifolds, cf. 10]. Ruuska 8] characterized Hessian Riemannian structures as those which admit an abelian Lie algebra of gradient vector elds, where the local action is simply transitive. Hitchin 4] characterized Hessian Riemannian structures in term of a Lagrangean sub-manifolds of the cotangent bundle. I am grateful to Nigel Hitchin and Lieven Vanhecke for getting me started with the literature on Hessian Riemannian structures. As a general reference on diierential geometry one may use 5].