Let $(M^n,g)$ be a Riemannian manifold with a fixed point $p$. Can we find a local coordinate system $(x\_1,x\_2,\cdots, x\_n)$ around $p$ satisfying the following two at the same time:


(1) $(x\_1,x\_2,\cdots, x\_n)$ are harmonic coordinates, i.e., $\Delta\_g x\_i=0$;


(2) $(x\_1,x\_2,\cdots, x\_n)$ are normal coordinates at $p$, i.e., $\partial\_k g\_{ij}=0$ at $p$?



Existence of normal and harmonic coordinates around a point
Yes. This follows from a standard fact about the Laplacian $\Delta\_g$, (because it is a second-order elliptic operator): The fact is this:


If $u$ is a smooth function on an open neighborhood $U$ of $p$ such that $\Delta\_gu$ vanishes to order $k$ at $p$, then there is a (possiblly smaller) $p$-neighborhood $V\subseteq U$ on which there exists a smooth function $v$ that satisfies $\Delta\_gv=0$ and $u-v$ vanishes to order $k{+}2$ at $p$.


Now fix $p$ and choose normal coordinates $y=(y^1,\ldots,y^n)$ centered at $p$. Because these are normal coordinates, $\displaystyle\frac{\partial g\_{ij}}{\partial y^k}$ vanishes at $p$ for all $i,j,k$, and hence $\Delta\_g y^i$ vanishes to order $1$ at $p$. Hence there exist functions $x^i$ on an open neighborhood of $p$ such that $\Delta\_g x^i=0$ and $y^i-x^i$ vanishes to order $3$ at $p$ for all $1\le i\le n$.


By shrinking $V$, one can arrange that $x = (x^1,\ldots,x^n)$ is a coordinate system on a neighborhood of $p$ as well. Since $y^i = x^i + O(|x|^3)$, and since
$$
g = g\_{ij}(y)\,\mathrm{d}y^i\mathrm{d}y^j 
$$
where $g\_{ij}(y)-g\_{ij}(0)$ and $\mathrm{d}y^i - \mathrm{d}x^i$ all vanish to order at least $2$ at $p$, it follows that
$g - g\_{ij}(0)\,\mathrm{d}x^i\mathrm{d}x^j$ vanishes to order at least $2$ at $p$.


Hence, if $g = {\bar g}\_{ij}(x)\,\mathrm{d}x^i\mathrm{d}x^j$, then $\displaystyle\frac{\partial {\bar g}\_{ij}}{\partial x^k}$ vanishes at $p$ for all $i,j,k$, as desired.


