I remember the following problem back from my undergraduate days:

Suppose that $f\in C^1(\mathbb{R}^n)$ is a map such that for all

p, we have $df(p)\in SO(n)$. Then, $df$ is a constant rotation, or in other words, $f$ is an affine rotation.

There is a clever proof of this fact using local inversion to prove that at any point, your map $f$ must be locally an orientation-preserving isometry.

I can't think of any similar-looking result, so my question is the following:

Is this exercise just an isolated fact, or is it a special case of a more general phenomenon?

Given how often we build undergraduate-level problems from special cases, I am now curious.