Lie group action

Builds on Lie group Smooth manifold Group action Smooth function

Definition

Let $G$ be a Lie group with identity $e$ and let $M$ be a smooth manifold. A (left) Lie group action of $G$ on $M$ is a map $\alpha \colon G \times M \to M$ , written $\alpha(g, p) = g \cdot p$ , such that:

$e \cdot p = p$ for all $p \in M$ ;
$g \cdot (h \cdot p) = (gh) \cdot p$ for all $g, h \in G$ and $p \in M$ ;
$\alpha$ is smooth.

Remarks

Clauses 1 and 2 say that $\alpha$ is a group action; clause 3 is the smoothness condition. Right actions are defined in the same way with $(p \cdot h) \cdot g = p \cdot (hg)$ .

Example: $SO(2)$ acts on $\R^2$ by $A \cdot v = Av$ . The identity matrix fixes every vector, $(AB)v = A(Bv)$ , and the map $(A, v) \mapsto Av$ is polynomial in the entries, hence smooth.

Non-example: let $G = \R$ act on $M = \R$ by $t \cdot x = (x^3 + t)^{1/3}$ , with the real cube root. This is a group action, since $s \cdot (t \cdot x) = (x^3 + t + s)^{1/3}$ , and it is continuous. It is not smooth: the map $t \mapsto t \cdot 0 = t^{1/3}$ is not differentiable at $t = 0$ .

Each $g$ acts by a diffeomorphism of $M$ with inverse given by $g^{-1}$ .

Reading path

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Group action concept
Lie group concept
Smooth function concept
Smooth manifold concept
Lie group action definition

Tag	`GC7T4VW3N1`
URI	`m2://GC7T4VW3N1`
Kind	definition
Writing	W0 stub
MSC	57S20, 22F05, 22E15
Defines	Lie group action
Uses	Lie group, Smooth manifold, Group action, Smooth function
Changed	2 November 2026

Lie group action

Definition

Remarks

Reading path

Further reading

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