The Surface Signature and Rough Surfaces

TL;DR

This paper extends the concept of path signatures to surfaces, introducing the surface signature and rough surface theory, enabling the computation of surface holonomy for irregular surfaces using universal properties.

Contribution

It introduces the surface signature as a universal invariant for surface holonomy and proves a surface extension theorem for rough surfaces, generalizing rough path theory.

Findings

Surface signature generalizes path signature to surfaces.

Surface extension theorem allows computation of rough surface signatures.

Universal property enables analysis of irregular surfaces.

Abstract

Parallel transport, or path development, provides a rich characterization of paths which preserves the underlying algebraic structure of concatenation. The path signature is universal among such maps: any (translation-invariant) parallel transport factors uniquely through the path signature. Furthermore, the path signature is a central object in the theory of rough paths, which provides an integration theory for highly irregular paths. A fundamental result is Lyons' extension theorem, which allows us to compute the signature of rough paths, and in turn provides a way to compute parallel transport of arbitrarily irregular paths. In this article, we consider the notion of surface holonomy, a generalization of parallel transport to the higher dimensional setting of surfaces parametrized by rectangular domains, which preserves the higher algebraic structures of horizontal and vertical concatenation. Building on work of Kapranov, we introduce the surface signature, which is universal among surface holonomy maps with respect to continuous 2-connections. Furthermore, we introduce the notion of a rough surface and prove a surface extension theorem, which allows us to compute the signature of rough surfaces. By exploiting the universal property of the surface signature, this provides a method to compute the surface holonomy of arbitrarily irregular surfaces.