  Thin shells are characterized by a high cost of stretching compared to
bending. As a result isometries of the midsurface of a shell play a crucial
role in their mechanics. In turn, curves with zero normal curvature play a
critical role in determining the number and behavior of isometries. In this
paper, we show how the presence of these curves results in a decrease in the
number of linear isometries. Paradoxically, shells are also known to
continuously fold more easily across these rigidifying curves than other curves
on the surface. We show how including nonlinearities in the strain can explain
this phenomena and demonstrate folding isometries with explicit solutions to
the nonlinear isometry equations. In addition to explicit solutions, exact
geometric arguments are given to validate and guide our analysis in a
coordinate free way.
