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Let, we are in a 2d metric where $g_{xx}=1, g_{yy}=x^2$, therefore $|e_x|=1$, $|e_y|=x$. If we try to draw the metric in a grid - it looks something like the image I uploaded. Note that, along the X axis the $e_x$ basis is always 1 unit long, but $e_y$ grows as we move along the X axis. And that causes a problem. $g_{xx}=1$, therefore a constant, and should not grow as we move along the Y axis. But as we can see, $g_{xx}$, which is the squared length of $e_x$ grows, $CD >AB$. What is the reason for this inconsistency? What am I missing? Use Christoffel Symbols in answer if needed.

enter image description here

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First of all note, that the metric you wrote actually is the metric for polar coordinates, where x=r and y=$\varphi$. The problem in your picture is, that you drew the lines from A to C and from B to D as straight lines, which they are not, as they go along circles around the origin (constant r). Although this answer is very short I think it contains all the info you need, I hope it helped you.

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  • $\begingroup$ Thanks, I can visualise it now. $\endgroup$
    – Nayeem1
    Commented Jun 7, 2023 at 7:41

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