Is the metric in normal coordinates constant for flat manifolds?

Question

Simple question. Let $(M,g)$ be a flat manifold (i.e. a Riemannian manifold with vanishing Riemannian curvature). Is it true that the metric $g_{ij}$ in normal coordinates around each point is constant and equal to the identity, i.e. $g_{ij}\equiv \delta_{ij}$?

It looks the Taylor expansion of $g_{ij}$ can be expressed in terms of the Riemannian curvature, which suggests that this is true, but I am not sure (I think that only a few terms of the Taylor expansion have been calculated).

Answer 1

Working locally throughout, we can regard normal coordinates as maps $\varphi:\mathbb{R}^n\to M$ defined by $$ \varphi(x^1,\cdots,x^n)=\exp_p(x^ie_i) $$ where $e_1,\cdots,e_n$ is an orthonormal basis of $T_pM$. From here, one can prove your claim using the following facts:

1. If $\psi:M\to N$ is a local isometry with $\psi(p)=q$ and $\varphi:\mathbb{R}^n\to M$ is a normal coordinate chart on $M$ centered at $p$, then $\psi\circ\varphi$ is a normal coordinate chart on $N$ centered at $q$.

2. The identity $\operatorname{id}:\mathbb{R}^n\to\mathbb{R}^n$ is a normal coordinate chart on the standard Euclidean $n$-space $(\mathbb{R}^n,\delta)$.

3. A Riemannian $n$-manifold is flat iff it is locally isometric to Euclidean $n$-space.