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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?

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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 $$ g_{ij} = \delta_{ij} - \frac{1}{3}R_{ikjl}x^kx^l - \frac{1}{6}\nabla_m R_{ikjl}x^kx^lx^m + O(x^4). $$

This is surely also true in the pseudo-Riemannian setting. O'Neill's book probably has this stated and proved somewhere in the first several chapters. I don't have it handy at the moment but will verify when I get home.

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  • $\begingroup$ May be this? Semi-Riemannian Geometry With Applications to Relativity, 103 Barrett O'Neill ,Academic Press, 29-Jul-1983 $\endgroup$ – Narasimham Aug 26 '15 at 11:03

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