Hessianizability of surface metrics

TL;DR

This paper proves that any smooth, nondegenerate surface metric can always be locally expressed as the Hessian of a function, establishing a fundamental property of surface metrics.

Contribution

It demonstrates that all smooth, nondegenerate surface metrics are smoothly locally Hessianizable, a previously unconfirmed property.

Findings

Any smooth, nondegenerate surface metric is locally Hessianizable.

The result applies to all smooth, nondegenerate metrics on surfaces.

This confirms a fundamental property of surface metrics in differential geometry.

Abstract

A symmetric quadratic form $g$ on a surface~$M$ is said to be locally Hessianizable if each $p\in M$ has an open neighborhood~$U$ on which there exists a local coordinate chart $(x^1,x^2):U\to\mathbb{R}^2$ and a function $f:U\to\mathbb{R}$ such that, on $U$, we have $$ g = \frac{\partial^2 f}{\partial x^i\partial x^j}\,\mathrm{d} x^i\circ\mathrm{d} x^j. $$ In this article, I show that, when $g$ is nondegenerate and smooth, it is always smoothly locally Hessianizable.