Un-reduction in field theory, with applications

TL;DR

This paper extends the un-reduction technique from Mechanics to covariant Field Theory, applying it to shape matching, non-linear sigma models, and hyperbolic flows of curves, offering new tools for complex geometric problems.

Contribution

It introduces a covariant un-reduction procedure and demonstrates its applications in shape matching, sigma models, and curve flows, expanding the method's scope.

Findings

Successful application to shape matching with multiple variables

Extension of un-reduction to covariant field theory

Potential for new geometric analysis tools

Abstract

The un-reduction procedure introduced previously in the context of Mechanics is extended to covariant Field Theory. The new covariant un-reduction procedure is applied to the problem of shape matching of images which depend on more than one independent variable (for instance, time and an additional labelling parameter). Other possibilities are also explored: non-linear $\sigma$-models and the hyperbolic flows of curves.