Call a lie group representation $G \subset GL(n, \mathbb{R})$ rigid if any infinitely differentiable function $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ whose derivative is always in $G$, is of the form $f(x) = mx + b$, $m \in G$.
I am not sure if there is already a word for this.
Is there an easy way to tell if a representation is rigid? Maybe some necessary or sufficient conditions?
For example:
$GL(n, \mathbb{R})$ is rigid iff $n = 0$.
$O(n, \mathbb{R})$ is rigid because $f$ preserves the length of curves and angles between them, so it preserves the shortest path: lines.
$\mathbb{C}^* \subset GL(2, \mathbb{R})$ is not rigid because there are holomorphic functions other than $mx + b$.
$SL(n, \mathbb{R})$ is not rigid when $n > 1$ because there are many volume-preserving transformations, for example by Cavalieri's principle. It is rigid when $n \leq 1$.
$\mathbb{R}^* \subset GL(n, \mathbb{R})$ preserves lines and angles, so drawing a parallelogram shows it's rigid when $n \neq 1$.
Bear in mind that I don't know much about lie groups.