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

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.