Let \((M,g)\) be a Riemannian manifold. In normal coordinates in any point \(p\in M\), we obtain the following approximation of the metric tensor
\[ g_{ij}(x) = \delta_{ij} - \frac{1}{3} R_{ijkl}(0) x^kx^l + O(\lvert x\rvert^3) \]where \(R_{ijkl}\) are the components of the curvature tensor.