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 symmetric. Linear perturbations therefore do not induce any rotation. However, vector tensor $B$-modes shear. contrary to what often claimed literature, shear $B$-mode power spectrum fully determined by rotation spectrum. Also $E$-mode convergence While this difference small for scalar perturbations, becomes very relevant i.e. gravitational waves.