The gauge invariant cosmological Jacobi map from weak lensing at leading
  order
  We compute the weak lensing Jacobi map at first order in perturbation theory
and show that it is both, gauge invariant and symmetric. Linear perturbations
therefore do not induce any rotation. However, vector and tensor perturbations
do induce $B$-modes in the shear. We show that contrary to what is often
claimed in the literature, the shear $B$-mode power spectrum is not fully
determined by the rotation power spectrum. Also the $E$-mode shear power
spectrum is not determined by the convergence power spectrum. While this
difference is small for scalar perturbations, it becomes very relevant for
tensor perturbations, i.e. gravitational waves.
