text: FTLE ridges have proven to be a simple and efficient tool for the visualize hyperbolic LCSs in a number of physical problems, yielding intriguing images of initial positions of hyperbolic LCSs in different applications (see, e.g., Figs. 5a-b). However, FTLE ridges obtained over sliding time windows  do not form material surfaces. Thus, ridges of  under varying  cannot be used to define Lagrangian objects, such as hyperbolic LCSs. Indeed, a locally strongest repelling material surface over  will generally not play the same role over  and hence its evolving position at time  will not be a ridge for . Nonetheless, evolving second-derivative FTLE ridges computed over sliding intervals of the form  have been identified by some authors broadly with LCSs. In support of this identification, it is also often argued that the material flux over such sliding-window FTLE ridges should necessarily be small.
  The "FTLE ridge=LCS" identification, however, suffers form the following conceptual and mathematical problems:
  * Second-derivative FTLE ridges are necessarily straight lines and hence do not exist in physical problems.
  * FTLE ridges computed over sliding time windows  with a varying  are generally not Lagrangian and the flux through them is generally not small.
  * In particular, a broadly referenced material flux formula for FTLE ridges is incorrect, even for straight FTLE ridges
  * FTLE ridges mark hyperbolic LCS positions, but also highlight surfaces of high shear. A convoluted mixture of both types of surfaces often arises in applications (see Fig. 6 for an example).
  * There are several other types LCSs (elliptic and parabolic) beyond the hyperbolic LCSs highlighted by FTLE ridges