Consider components of metric tensor $g'$ in a coordinate system $$g'= \begin{pmatrix} xy & 1 \\ 1 & xy \\ \end{pmatrix} $$ We can transformation rule which brings $g'$ to euclidean metric $g=\begin{pmatrix} 1 & 0 \\ 0 & 1\\ \end{pmatrix}$, namely $$A^T*g'*A=g$$ where $A=\begin{pmatrix} -\frac{1}{\sqrt{xy}} & 1 \\ 1 & -\frac{1}{\sqrt{xy}}\\ \end{pmatrix}$ .
Levi-Civita connection for $g$ has all components as zero but not all components are vanishing for $g'$. So if I want to find geodseics given $g'$ I could find appropriate transformation where $g'$ looks like $g$ but in this case geodesics are going to be straight lines given ANY $g'$.
Is this a wrong statement?