Invariant function - Knowlpedia

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

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

f(g\cdot x)=f(x)\qquad\text{for all }g\in G,\ x\in M.

Equivalent characterizations

Equivalently, $f$ is constant on each orbit. It therefore factors uniquely as a set map through the quotient:

f=\bar f\circ\pi,\qquad \pi:M\to M/G,\quad \bar f:M/G\to\mathbb R.

When $M/G$ has a smooth structure for which $\pi$ is a submersion, $\bar f$ is smooth.

Examples

Radial functions. For the $SO(n)$ -action on $\mathbb{R}^n$ , the smooth function $x\mapsto \|x\|^2$ and every smooth function of $\|x\|^2$ are invariant. The norm itself is invariant as a set function but is not smooth at the origin for $n\ge1$ .
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 $P$ .
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.