On Two Dimensional Flat Hessian Potentials

TL;DR

This paper characterizes and explicitly constructs potentials for flat Hessian metrics on surfaces, combining differential geometry and integrable systems techniques.

Contribution

It provides a theoretical description and explicit construction of potentials for flat Hessian metrics on surfaces under mild conditions.

Findings

Characterization of potentials for flat Hessian metrics on surfaces

Explicit construction methods using integrable systems

Theoretical description under mild technical conditions

Abstract

A Riemannian metric is termed a Hessian metric if in some coordinate system it can be locally represented as the Hessian quadratic form of some locally defined smooth potential function. Under very mild extra technical conditions, we first theoretically describe the potentials of flat Hessian metrics on surfaces, and then construct these potentials explicitly using methods from integrable systems.