Sub-Finsler geometry and finite propagation speed

TL;DR

This paper explores the geometry induced by first-order differential operators on manifolds, revealing differences from classical Riemannian geometry and analyzing the implications for evolution equations.

Contribution

It introduces and studies a new geometric framework based on distance functions from first-order operators, highlighting differences from traditional Riemannian geometry.

Findings

Distance functions associated to first-order operators can differ significantly from Riemannian distances.

The geometry related to these operators influences the behavior of solutions to evolution equations.

Several results on the properties of these geometries are established.

Abstract

We prove a number of results on the geometry associated to the solutions of evolution equations given by first-order differential operators on manifolds. In particular, we consider distance functions associated to a first-order operator, and discuss the associated geometry, which is sometimes surprisingly different to riemannian geometry.