Isometric Immersions and the Waving of Flags

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 spirit similar to Arnold’s geometric picture for the motion of an incompressible fluid.