Hessianizability of surface metrics

2024/05/11 by Bryant, Robert L. · 1 citation
#53B05 #58A15 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2405.06998

Abstract

A symmetric quadratic form g on a surface~M is said to be locally Hessianizable if each p∈ M has an open neighborhood~U on which there exists a local coordinate chart (x¹,x²):U→ℝ² and a function f:U→ℝ such that, on U, we have g = (∂² f)/(∂ xⁱ∂ x^j) d xⁱ\circd x^j. In this article, I show that, when g is nondegenerate and smooth, it is always smoothly locally Hessianizable.

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