The complex metric on the Riemann sphere is given in the Wikipedia article to be
$$ds^2=\frac{4}{(1+\zeta\bar \zeta)^2}d\zeta d\bar \zeta$$
while the sphere should be mapped to itself under $SL(2,\mathbb{C})$ Moebius transformations
$$\zeta\to\frac{a \zeta + b}{c \zeta + d}~,\qquad \bar \zeta\to\frac{\bar a \bar \zeta + \bar b}{\bar c \bar \zeta + \bar d}$$
with $$a d-b c=\bar a\bar d-\bar c\bar b=1.$$ However, when I explicitly plug these transformations into $ds^2$ it does not reduce back to itself, as it should under an isometry. Is there a mistake somewhere, or did I get myself confused? What is the correct Riemann sphere metric with complex coordinates that is isometric to itself under Moebius transformations?