Physvillain

Let $P$ is an arbitrary zero point at which we expand around. Then expanding metric around $P$ is : $$g_{ij} = \left. g_{ij} \right\vert_{P} + \frac{1}{2!} \left. \left( \partial_k g_{ij} \right) \right\vert_{P} x^k + \frac{1}{3!} \left. \left( \partial_{k_2}\partial_{k_1} g_{ij} \right) \right\vert_{P} x^{k_1} x^{k_2} + \cdots$$ Curvature tensor generally describes $\nabla_{[i}\nabla_{j]}$. Exa..