I was wondering whether the metric tensor in normal coordinates can be expressed somehow in terms of the exponential map. Cause I just don't see how the metric in normal coordinates is actually defined.
My intuition tells me that since $exp$ is a parametrisation of the manifold (locally), the metric is defined by $(g_{i,j}):=(Dexp)^T Dexp$ is this correct?
Let $(M,g)$ be a Riemannian manifold and $p \in M$. Recall that normal coordinates about $p$ map a neighbourhood of the identity in $T_{p}M$ to a neighbourhood of $p$ via the exponential map $exp|_{p}$. The metric in normal coordinates is defined (e.g. in the context of the Gauss lemma) by taking polar coordinates on $T_{p}M$ and noting that for $\epsilon$ sufficiently small there is a diffeomorphism
$\begin{align} f: (0,\epsilon)\times S^n & \to B \\ (r,v) & \mapsto exp|_{p}(rv) \end{align}$
The resulting chart defines so-called geodesic polar coordinates about $p$. We can also define the geodesic spheres $\Sigma_{r} := f(S^n(r))$. Gauss's Lemma states that locally, geodesic spheres are orthogonal to geodesics, and it follows that the metric near p can be written as $$ g = dr^2 + h(r,v)$$ where $h$ is the restriction of $g$ to $\Sigma_r$. Note that since h is positive definite it is immediately clear that geodesics (which are purely radial in normal coordinates) locally minimise path length.
There is of course no unique definition of normal coordinates; they are defined up to a change of basis on $T_{p}M$. Physicists like to fix local inertial coordinates at $p$ by choosing normal coordinates with the additional condition $g_{ab}|_{p} = \eta_{ab}$, but this is a matter of taste. It is worth noting that in the Lorentzian case, Gauss's lemma no longer applies; for example, geodesics can locally maximise their length.