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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 \(g = \frac{{\partial^{2} f}}{{\partial x^{i} \partial x^{j} }}{\text{d}}x^{i} \circ {\text{d}}x^{j}\) In this article, I show that, when g is nondegenerate and smooth, it is always smoothly locally Hessianizable.

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Hessianizability of Surface Metrics

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  • 发表日期：

    2025年

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  • Robert L. Bryant

摘要

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 \(g = \frac{{\partial^{2} f}}{{\partial x^{i} \partial x^{j} }}{\text{d}}x^{i} \circ {\text{d}}x^{j}\) In this article, I show that, when g is nondegenerate and smooth, it is always smoothly locally Hessianizable.