  Mixing, and coherence are fundamental issues at the heart of understanding
transport in fluid dynamics and other non-autonomous dynamical systems.
Recently, the notion of coherence has come to a more rigorous footing, and
particularly within the recent advances of finite-time studies of
non-autonomous dynamical systems. Here we define shape coherent sets as a means
to emphasize the intuitive notion of ensembles which "hold together" for some
period of time, and we contrast this notion to other recent perspectives of
coherence, notably "coherent pairs", and likewise also to the geodesic theory
of material lines. We will relate shape coherence to the differential geometry
concept of curve congruence through matching curvatures. We show that points in
phase space where there is a zero-splitting between stable and unstable
manifolds locally correspond to points where curvature will evolve only slowly
in time. Then we develop curves of points with zero-angle, meaning
non-hyperbolic splitting, by continuation methods in terms of the implicit
function theorem. From this follows a simple ODE description of the boundaries
of shape coherent sets. We will illustrate our methods with popular benchmark
examples, and further investigate the intricate structure of foliations
geometry.
