Attracting Lagrangian coherent structures on Riemannian manifolds.

It is a wide-spread convention to identify repelling Lagrangian Coherent Structures (LCSs) with ridges of the forward finite-time Lyapunov exponent (FTLE) field and to identify attracting LCSs with ridges of the backward FTLE. However, we show that, in two-dimensional incompressible flows, also attracting LCSs appear as ridges of the forward FTLE field. This raises the issue of the characterization of attracting LCSs using a forward finite-time Lyapunov analysis. To this end, we extend recent results regarding the relationship between forward and backward maximal and minimal FTLEs, to both the whole finite-time Lyapunov spectrum and to stretch directions. This is accomplished by considering the singular value decomposition (SVD) of the linearized flow map. By virtue of geometrical insights from the SVD, we provide characterizations of attracting LCSs in forward time for two geometric approaches to hyperbolic LCSs. We apply these results to the attracting FTLE ridge of the incompressible saddle flow.