So I've been computing the Riemann curvature tensor in normal coordinates centered at the point $p\in M$, but I am getting that the curvature is zero at the point $p$. I don't think this is correct, but I can't see what I'm doing wrong. The below calculations are all evaluated at the point $p$.If anyone can point out what I'm doing wrong, I would really appreciate it.
$$\begin{align} R(X,Y)Z &= \nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z\\ &= \nabla_{X}Y^{j}\partial_{j}Z^{i}\partial_{i}-\nabla_{Y}X^{j}\partial_{j}Z^{i}\partial_{i} -\left(X^{k}\partial_{k}Y^{j}-Y^{k}\partial_{k}X^{j}\right)\partial_{j}Z^{i}\partial_{i}\\ &= (X^{k}\partial_{k}\left(Y^{j}\partial_{j}Z^{i}\right)-Y^{k}\partial_{k}\left(X^{j}\partial_{j}Z^{i}\right) -\left(X^{k}\partial_{k}Y^{j}-Y^{k}\partial_{k}X^{j}\right)\partial_{j}Z^{i})\partial_{i}\\ &= (X^{k}\partial_{k}Y^{j}\partial_{j}Z^{i}+X^{k}Y^{j}\partial_{k}\partial_{j}Z^{i}-Y^{k}\partial_{k}X^{j}\partial_{j}Z^{i} -Y^{k}X^{j}\partial_{k}\partial_{j}Z^{i}-\left(X^{k}\partial_{k}Y^{j}-Y^{k}\partial_{k}X^{j}\right)\partial_{j}Z^{i}\partial_{i}\\ &= 0\\ \end{align}$$
Remember that the Christoffel symbols vanish only at $p$, not at nearby points; so while $\nabla_Y Z = Y^j \partial_j Z^i \partial_i$ is true at $p$, $\nabla_X \nabla_Y Z = \nabla_X (Y^j \partial_j Z^i \partial_i)$ is not. Keep track of the Christoffel symbols, expand everything out with the product rule, and then set $\Gamma = 0$ while keeping the $\partial \Gamma$ terms.