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We compute the weak lensing Jacobi map at first order in perturbation theoryand show that it is both, gauge invariant and symmetric. Linear perturbationstherefore do not induce any rotation. However, vector and tensor perturbationsdo induce $B$-modes in the shear. We show that contrary to what is oftenclaimed in the literature, the shear $B$-mode power spectrum is not fullydetermined by the rotation power spectrum. Also the $E$-mode shear powerspectrum is not determined by the convergence power spectrum. While thisdifference is small for scalar perturbations, it becomes very relevant fortensor perturbations, i.e. gravitational waves.

The gauge invariant cosmological Jacobi map from weak lensing at leading order