On the Geometric Structure of Flows I: The Referential Gradient. A Generally Covariant Measure of Flow Geometry

TL;DR

This paper introduces the referential gradient, a covariant measure of flow geometry, relating it explicitly to the generating vector field through both Lagrangian and Eulerian perspectives, with conditions on its transformation properties.

Contribution

It provides the first explicit relation between the referential gradient and the generating vector field, advancing the understanding of flow geometry in a covariant framework.

Findings

Explicit relation between referential gradient and vector field

Dual Lagrangian and Eulerian descriptions of flow

Conditions governing transformation properties of the gradient

Abstract

Assuming a-priori a smooth generating vector field, we introduce a generally covariant measure of the flow geometry called the referential gradient of the flow. The main result is the explicit relation between the referential gradient and the generating vector field, and is provided for from two equivalent perspectives: a Lagrangian specification with respect to a generalized parameter, and an Eulerian specification making explicit the evolution dynamics. Furthermore, we provide explicit non-trivial conditions which govern the transformation properties of the referential gradient object.