# Does the Riemann tensor encode all information about the second derivatives of the metric?

In answer to this question I suggested the following as a motivation for the definition of the Riemann tensor:

Let $\mathcal M$ and $\mathcal N$ be two dim-$n$ Lorentzian manifolds with points $x\in\mathcal M$ and $y\in\mathcal N$. Then you can pick local coordinates on $\mathcal M$ and $\mathcal N$ such that the expressions for $g_{\mu\nu}$ agree to second order near $x$ and $y$ iff you can find bases at $x$ and $y$ such that the Riemann tensors are equal.

Is the preceding theorem in fact true? Where can I find a proof of it?

Here is a reference for Riemannian manifolds. The theorem (4.1) is that the Taylor expansion of a Riemannian metric $g$ in coordinates has the form $$g_{ij} = \delta_{ij} - \frac{1}{3}R_{ikjl}x^kx^l - \frac{1}{6}\nabla_m R_{ikjl}x^kx^lx^m + O(x^4).$$