Invariant function

Let $G$ act smoothly on a manifold $M$.

A smooth function $f:M\to \mathbb{R}$ is $G$-invariant if

Equivalently, $f$ is constant on each orbit . In that case, $f$ factors uniquely through the quotient map $\pi:M\to$ as a function $\bar f:M/G\to \mathbb{R}$ with $f=\bar f\circ \pi$.

Examples

1. Radial functions. For the $SO(n)$-action on $\mathbb{R}^n$, the norm $x\mapsto \|x\|$ and any function of $\|x\|$ are invariant.
2. Pullbacks from a quotient. If $\pi:P\to B$ is a principal bundle, then any smooth $h:B\to\mathbb{R}$ gives an invariant function $h\circ \pi$ on $P$.
3. Transitive actions. If the action is transitive (one orbit), for instance $\mathbb{R}^n$ acting on itself by translations, then every invariant smooth function is constant.