I'm trying to understand some proof written in local coordinates. The situation is as follows:
We have a $2$-dim Riemannian manifold $(M,g)$ and a conformal diffeomorphism $f:M\rightarrow M$ such that $\tilde{g}:=f^*g=e^{2\lambda}g$. The claim in the proof is that
$$
e^{2\lambda}g^{ij}= \sum_{k,l=1}^2g^{kl}\frac{\partial f_i}{\partial x_k}\frac{\partial f_j}{\partial x_l}
$$
but I don't see this. By calculation I got
$$
e^{2\lambda}g_{ij}=\tilde{g}_{ij}:=\tilde{g}(\partial_i,\partial_j)=\sum_{kl=1}^2 g_{kl}\frac{\partial f_k}{\partial x_i}\frac{\partial f_l}{\partial x_j}
$$
and don't now how to continue.
I'd be thankfull for some help!
Best Answer
Let $G,D$ be the matrices given by $G_{ij}=g_{ij}$ and $D_{ij}=\frac{\partial f_i}{\partial x_j}$. Then what you have calculated is
$$ e^{2\lambda}G=D^tGD $$ Multiplying with $(D^tG)^{-1}$ from the left and $G^{-1}D^t$ from the right yields
$$ e^{2\lambda}G^{-1}=DG^{-1}D^t $$ and writing out the matrix multiplication explicitly, yields exactly what you wanted to show.