# Orientation Asymmetric Surface Model for Membranes: Finsler Geometry Modeling

## TL;DR

This paper introduces a Finsler geometry-based discrete surface model for membranes, allowing non-Euclidean metrics and capturing orientation asymmetries, which may better represent real physical membranes.

## Contribution

It demonstrates that Finsler geometry enables well-defined discrete membrane models with nontrivial metrics and highlights their orientation asymmetry, a novel feature for membrane modeling.

## Key findings

- Discrete Finsler geometry model is well-defined for membranes with non-Euclidean metrics.
- The model exhibits orientation asymmetry on invertible surfaces.
- Potential advantage for modeling real membranes with asymmetries.

## Abstract

We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping ${\bf r}$ from a two dimensional parameter space $M$ to the three dimensional Euclidean space ${\bf R}^3$. The metric variable $g_{ab}$, which is always fixed to the Euclidean metric $\delta_{ab}$, can be extended to a more general non-Euclidean metric on $M$ in the continuous model. The problem we focus on in this paper is whether such an extension is well-defined or not in the discrete model. We find that a discrete surface model with nontrivial metric becomes well-defined if it is treated in the context of Finsler geometry (FG) modeling, where triangle edge length in $M$ depends on the direction. It is also shown that the discrete FG model is orientation assymetric on invertible surfaces in general, and for this reason, the FG model has a potential advantage for describing real physical membranes, which are expected to have some assymetries for orientation changing transformations.