# Isometric Immersions and the Waving of Flags

## TL;DR

This paper introduces a geometric model for flag motion using isometric immersions, establishing a manifold structure and deriving equations of motion akin to fluid dynamics, providing a new mathematical framework for studying flags.

## Contribution

It develops a novel geometric framework modeling flags as isometric immersions, with a manifold structure and equations of motion derived from kinetic energy considerations.

## Key findings

- The space of flags forms an infinite-dimensional manifold.
- Equations of motion for flags are derived from a kinetic energy functional.
- The model parallels Arnold's geometric approach to fluid dynamics.

## Abstract

In this article we propose a novel geometric model to study the motion of a physical flag. In our approach a flag is viewed as an isometric immersion from the square with values in $\mathbb R^3$ satisfying certain boundary conditions at the flag pole. Under additional regularity constraints we show that the space of all such flags carries the structure of an infinite dimensional manifold and can be viewed as a submanifold of the space of all immersions. In the second part of the article we equip the space of isometric immersions with its natural kinetic energy and derive the corresponding equations of motion. This approach can be viewed in a similar spirit as Arnold's geometric picture for the motion of an incompressible fluid.