= Projective transformation weights of the Schwarzian derivative
For $V_1=\partial_x$, $V_2=x\partial_x$, and $V_3=x^2\partial_x$, direct application of their third <prolongations of a vector field>[prolongations] gives
$$
\operatorname{pr}^{(3)}V_1(S)=0,
\qquad
\operatorname{pr}^{(3)}V_2(S)=-2S,
\qquad
\operatorname{pr}^{(3)}V_3(S)=-4xS.
$$
Thus the Schwarzian is invariant under translation and has weight minus two under dilation; the third identity is its infinitesimal projective transformation law.
Back to article page
