Given a Riemannian manifold $(\mathcal{M}, \{g_x\}_{x \in \mathcal{M}})$ and a fixed point $x \in \mathcal{M}$, does the following procedure yield uniform samples from $\{y \in \mathcal{M} : d_\mathcal{M}(x, y) \le 1 \}$?

1. Sample uniformly from $\{u \in \mathcal{T}_x \mathcal{M} : \lVert u \rVert_x \le 1\}$ (say we know how to do that)
2. $y = \exp_x(u)$

It seems to me that by change of variables the density over $y$ will be constant only if the determinant of the Jacobian of $\exp_x(\cdot)$ is constant. But I'm not sure if this argument is correct mainly because I don't know if the pushforward measure of the metric-induced measure on the set from step 1 via the exponential map is the same as the volume form induced on $\mathcal{M}$ by $g_x$.

(I'm a novice in both Riemannian geometry and measure theory, so I'm sorry if this is trivially true or false, or it doesn't make sense.)