Let $(M,g)$ be an arbitrary smooth Riemannian manifold of $n$ dimensional and $p_0\in M$. $\nabla$ denotes the Levi Civita connection.
It is well known that there is a coordinate chart around $p_0$ s.t. \begin{eqnarray} & g_{ij}(p_0)=\delta_{ij}\\ & \nabla_{\frac{\partial}{\partial x^i}}\dfrac{\partial}{\partial x^j}(p_0)=0\ \ \ (i,j=1,\cdots, n). \end{eqnarray} "Riemannian normal coordinate chart centered at $p_0$" is one of them.
Can we take a coordinate chart $(U;x^1,\cdots,x^n)$ around $p_0$ s.t. \begin{eqnarray} & g_{ij}(p_0)=\delta_{ij}\\ & \nabla_{\frac{\partial}{\partial x^i}}\dfrac{\partial}{\partial x^j}(p_0)=0\\ & \sum_{k=1}^n\nabla_{\frac{\partial}{\partial x^k}}\nabla_{\frac{\partial}{\partial x^k}}\dfrac{\partial}{\partial x^j}(p_0)=0\\ -(i,j=1,\cdots, n)? \end{eqnarray}