Question about Riemannian metrics I was reading a pdf about geodesics on Riemannian manifolds, and I've found this proposition

$\textbf{Proposition :}$ Let $(M,g)$ be a Riemannian manifold. For every point, $p \in M$, in normal coordinates at $p$,
$$g\left(\frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j}\right)_p = \delta_{ij} \quad \text{and} \quad \Gamma^k_{ij}(p)=0.$$

Before this proposition there is the following statement:

The following proposition shows that Riemannian metrics do not admit any local invariants of order one.

What does it mean that a Riemannian metric does not admit any local invariants of some order ?
 A: Let us first take an example of a local invariant: the scalar curvature.
It can be shown that in local coordinates, it reads as
$$
\DeclareMathOperator{\scal}{scal}
\scal = g^{ij}\left( \partial_k\Gamma^k_{ij} - \partial_j\Gamma^k_{ik} + \Gamma^{k}_{ij} \Gamma^{\ell}_{k \ell}  - \Gamma^{k}_{i\ell} \Gamma^{\ell}_{jk}\right).
$$
This expression is a local invariant, in the sense that whatever the coordinate system you choose, this quantity will be the same, and is also invariant under the action of local isometries.
Recall that Christoffel symbols can also be expressed as a polynomial in $g^{ij}$ and $\partial_kg_{\ell m}$.
Therefore, the scalar curvature is a polynomial in the metric, its inverse, and their first and second derivatives.
A natural thing to do then is to try to find other invariants that are functions (and if possible, polynomials) of the different objects
$$
g_{ij}, \partial_kg_{ij}, \partial^2_{k\ell}g_{ij}, \ldots, \partial^{|\alpha|}_{\alpha} g_{ij},\ldots,g^{ij},\partial_k g^{ij},\ldots
$$
where $\alpha$ is a multi index.
The Riemann curvature tensor, the Ricci tensor and the sectional curvature are other examples of objects that can be obtained this way.
More complicated invariants are obtained as sums of contractions of terms of the form $\nabla^{k_1} R\otimes\cdots \otimes \nabla^{k_l}R \otimes  g\otimes \cdots \otimes g \otimes g^{-1}\otimes \cdots \otimes g^{-1}$.
One could ask themself if it is possible to find a non-trivial invariant expression that is obtained only thanks to derivatives of the metric of order up to $1$.
The answer is no.
The reason is that in normal coordinates at a point $p$, we have $g_{ij}(p) = g^{ij}(p) = \delta_{ij}$, and $\partial_kg_{ij}(p) = \partial_kg^{ij}(p)= 0$, and it is therefore useless to try to find an invariant expression in the metric and its first derivatives only.
For instance, if the expression $A=\alpha + \beta^{ij}g_{ij} + \gamma^{ijk} \partial_kg_{ij}$ were invariant, computing it in normal coordinates would show that it is equal to $A=\alpha + \sum_i \beta^{ii}$.
This is what is meant here.
Let me emphasise on the word local here.
All invariants mentioned above are local invariants in the sense that they are invariant under local isometries.
It is somewhat weird that we can compute global invariants thanks to the first derivatives of the metric.
One such invariant is the so-called ADM mass of an asymptotically euclidean manifold.
But this is beyond this discussion.

As comments are not eternal, let me quote @Deane.

Here is a blog entry that shows the first and second order calculations explicitly.
The idea is to choose coordinates that kill off as many components of $g_{ij}$ and its partial derivatives as possible.
The first order term disappears, and the Riemann curvature appears out of the second derivatives.

