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Assuming a-priori a smooth generating vector field, we introduce a generallycovariant measure of the flow geometry called the referential gradient of theflow. The main result is the explicit relation between the referential gradientand the generating vector field, and is provided for from two equivalentperspectives: a Lagrangian specification with respect to a generalizedparameter, and an Eulerian specification making explicit the evolutiondynamics. Furthermore, we provide explicit non-trivial conditions which governthe transformation properties of the referential gradient object.

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