Membrane geometry with auxiliary variables and quadratic constraints

TL;DR

This paper introduces a method to analyze membrane geometry by treating metric and curvature as auxiliary variables with quadratic constraints, linking their variations to conserved stress tensors.

Contribution

It presents a novel approach using auxiliary variables and quadratic constraints to study membrane geometries and their stress responses.

Findings

Established relationship between constraint multipliers and stress tensor

Analyzed Hamiltonian response to variable deformations

Provided a framework for membrane geometry analysis

Abstract

Consider a surface described by a Hamiltonian which depends only on the metric and extrinsic curvature induced on the surface. The metric and the curvature, along with the basis vectors which connect them to the embedding functions defining the surface, are introduced as auxiliary variables by adding appropriate constraints, all of them quadratic. The response of the Hamiltonian to a deformation in each of the variables is examined and the relationship between the multipliers implementing the constraints and the conserved stress tensor of the theory established.