  Harada proposed a modified theory of gravity called Cotton gravity, and
argued that it successfully explains the rotation curves of $84$ galaxies
without the need of dark matter. In this work we use galaxy-galaxy lensing
technique to test whether the modification effect of Cotton gravity can indeed
be a viable substitute for dark matter. Using the spherically symmetric
solution of Cotton gravity, we obtain the deflection angle via Gauss-Bonnet
theorem and the weak lensing shear. We use five galaxy catalogs divided in 5
stellar mass bins from the Sloan Digital Sky Survey Data Release 7 (SDSS DR7),
each of which is further divided into blue star forming galaxy and red passive
galaxy sub-catalogs. We find that Cotton gravity on its own has significant
deviation from the measured galaxy-galaxy lensing signals, thus it cannot
replace the role of dark matter. If we consider the combination of dark matter
and Cotton gravity, the modification is tightly constrained. Our analysis also
applies to other modified gravity theories whose an additional linear term
appears in the Schwarzschild solution.
