The geometry of embedded pseudo-Riemannian surfaces in terms of Poisson brackets

TL;DR

This paper extends the expression of curvature for embedded surfaces from Riemannian to pseudo-Riemannian manifolds using Poisson brackets, providing explicit formulas for indefinite metrics.

Contribution

It generalizes the Poisson bracket approach to curvature from Riemannian to pseudo-Riemannian surfaces, including explicit formulas for indefinite metrics.

Findings

Derived formulas for Gauss and mean curvature in pseudo-Riemannian settings

Extended Poisson bracket expressions to indefinite metric spaces

Provided explicit formulas for surfaces in 1^m with indefinite metrics

Abstract

Arnlind, Hoppe and Huisken showed how to express the Gauss and mean curvature of a surface embedded in a Riemannian manifold in terms of Poisson brackets of the embedding coordinates. We generalize these expressions to the pseudo-Riemannian setting and derive explicit formulas for the case of surfaces embedded in $\R^m$ with indefinite metric.