This is slightly vague as I've not yet come to terms with what I'm actually looking for.
On $S^2$ we may choose charts (stereographic projection) such that the image of a disk (i.e. all points within distance $r$ from a fixed point $p\in S^2$) completely contained inside the domain of the chart , maps onto a disk in $\mathbb{R}^2$.
Let's say $M$ is a connected Riemanninan manifold and $p\in M$. Can I always find such a disk-preserving chart around $p$? Note that for any other $q$ inside the chart, the image of a disk around $q$ contained in the domain should also map to a disk.
edit: I used $S^2$ and $\mathbb{R}^2$ for simplicity. Note that I am asking for $n$-disks on $n$-dimensional manifolds.
Note that stereographic projection maps $B_R(p)$ onto $B_r(0)$ for some $R,\ r$ where $0$ is a origin of ${\bf R}^2$.
In fact any Riemannian manifold $M$ has a chart : $\exp_p : B_r(0)\mapsto B_r(p)$ where $B_r(0)$ is $r$-ball in tangent space and $B_r(p)$ is $r$-ball in $M$