  Cosmological weak lensing has been a highly successful and rapidly developing
research field since the first detection of cosmic shear in 2000. However, it
has recently been pointed out in Yoo et al. that the standard weak lensing
formalism yields gauge-dependent results and, hence, does not meet the level of
accuracy demanded by the next generation of weak lensing surveys. Here, we show
that the Jacobi mapping formalism provides a solid alternative to the standard
formalism, as it accurately describes all the relativistic effects contributing
to the weak lensing observables. We calculate gauge-invariant expressions for
the distortion in the luminosity distance, the cosmic shear components and the
lensing rotation to linear order including scalar, vector and tensor
perturbations. In particular, the Jacobi mapping formalism proves that the
rotation is fully vanishing to linear order. Furthermore, the cosmic shear
components contain an additional term in tensor modes which is absent in the
results obtained with the standard formalism. Our work provides further support
and confirmation of the gauge-invariant lensing formalism needed in the era of
precision cosmology.
