Consider components of metric tensor [itex]g'[/itex] in a coordinate system

$$ g'=

\begin{pmatrix}

xy & 1 \\

1 & xy \\

\end{pmatrix}

$$

We can find a transformation rule which brings [itex]g'[/itex] to euclidean metric [itex] g=\begin{pmatrix}

1 & 0 \\

0 & 1\\

\end{pmatrix}[/itex], namely

$$A^T*g'*A=g$$

where [itex]A=\begin{pmatrix}

-\frac{1}{\sqrt{xy}} & 1 \\

1 & -\frac{1}{\sqrt{xy}}\\

\end{pmatrix}[/itex] .

Levi-Civita connection for [itex]g[/itex] has all components as zero but not all components are vanishing for [itex]g'[/itex].

**So if I want to find geodseics given [itex]g'[/itex] I could find appropriate transformation where**

**components of**[itex]g'[/itex] looks like [itex]g[/itex] but in this case geodesics are going to be straight lines given ANY [itex]g'[/itex].Is this a wrong statement?