On Two Dimensional Flat Hessian Potentials

2025/12/17 by Liu, Hanwen
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2512.15079

openalex publication_date 2025/12/17 · openalex created_date 2025/12/19 · openalex updated_date 2026/07/28

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.

Related

• Riemannian Metrics on Locally Projectively Flat Manifolds
• Hessianizability of surface metrics
• Hessian geometry and Frobenius manifolds with curvature
• Hessian geometry on Lagrange spaces
• Differential geometry in the theory of Hessian operators