Confusion regarding an expression that relates the metric tensor and Riemann Tensor

Question

I found an expression that connects the metric tensor and Riemann Tensor in the following way:-

$g_{\mu\nu}~=~\eta_{\mu\nu}~+~\frac{1}{3}R_{\mu\alpha\nu\beta}x^\alpha x^\beta~+~.....$

Is this expression correct? How can one either prove/derive it?

Reference to any source is welcomed.

PS: This is the original question. It is not structured well and it led to confusion. So, I am asking it again.

Answer 1

This expansion is not true in every coordinate system, but it is true in Riemann normal coordinates. See this MathOverflow question and its answers for its derivation.