A general material surface experiences shear and strain in its deformation, both of which depend continuously on initial conditions by the continuity of the map .
The averaged strain and shear within a strip of -close material lines, therefore, typically show  variation within such a strip.
The two-dimensional geodesic theory of LCSs seeks exceptionally coherent locations where this general trend fails, resulting in an order of magnitude smaller variability in shear or strain than what is normally expected across an  strip. Specifically, the geodesic theory searches for LCSs as special material lines around which  material strips show no  variability either in the material-line
averaged shear (Shearless LCSs) or in the material-line averaged strain (Strainless or Elliptic LCSs). Such LCSs turn out to be null-geodesics of appropriate metric tensors defined by the deformation field—hence the name of this theory.
Shearless LCSs are found to be null-geodesics of a Lorentzian metric tensor  defined as
Such null-geodesics can be proven to be tensorlines of the Cauchy–Green strain tensor, i.e., are tangent to the direction field formed by the strain eigenvector fields . Specifically, repelling LCSs are trajectories of  starting from local maxima of the  eigenvalue field. Similarly, attracting LCSs are trajectories of  starting from local minims of the  eigenvalue field. This agrees with the conclusion of the local variational theory of LCSs. The geodesic approach, however, also sheds more light on the robustness of hyperbolic LCSs: hyperbolic LCSs only prevail as stationary curves of the averaged shear functional under variations that leave their endpoints fixed. This is to be contrasted with parabolic LCSs (see below), which are also shearless LCSs but prevail as stationary curves to the shear functional even under arbitrary variations. As a consequence, individual trajectories are objective, and statements about the coherent structures they form should also be objective.
A sample application is shown in Fig. 9, where the sudden appearance of a hyperbolic core (strongest attracting part of a stretchline) within the oil spill caused the notable Tiger-Tail instability in the shape of the oil spill.