Lie group action

In differential geometry, a Lie group action is a group action adapted to the smooth setting: $$G$$ is a Lie group, $$M$$ is a smooth manifold, and the action map is differentiable.

Definition and first properties
Let $$\sigma: G \times M \to M, (g, x) \mapsto g \cdot x$$ be a (left) group action of a Lie group $$G$$ on a smooth manifold $$M$$; it is called a Lie group action (or smooth action) if the map $$\sigma$$ is differentiable. Equivalently, a Lie group action of $$G$$ on $$M$$ consists of a Lie group homomorphism $$G \to \mathrm{Diff}(M)$$. A smooth manifold endowed with a Lie group action is also called a $$G$$-manifold.

The fact that the action map $$\sigma$$ is smooth has a couple of immediate consequences:


 * the stabilizers $$G_x \subseteq G$$ of the group action are closed, thus are Lie subgroups of $$G$$
 * the orbits $$G \cdot x \subseteq M$$ of the group action are immersed submanifolds.

Forgetting the smooth structure, a Lie group action is a particular case of a continuous group action.

Examples
For every Lie group $$G$$, the following are Lie group actions:
 * the trivial action of $$G$$ on any manifold
 * the action of $$G$$ on itself by left multiplication, right multiplication or conjugation
 * the action of any Lie subgroup $$H \subseteq G$$ on $$G$$ by left multiplication, right multiplication or conjugation


 * the adjoint action of $$G$$ on its Lie algebra $$\mathfrak{g}$$.

Other examples of Lie group actions include:
 * the action of $$\mathbb{R}$$ on M given by the flow of any complete vector field
 * the actions of the general linear group $$\operatorname{GL}(n,\mathbb{R})$$ and of its Lie subgroups $$G\subseteq\operatorname{GL}(n,\mathbb{R})$$ on $$\mathbb{R}^n$$ by matrix multiplication
 * more generally, any Lie group representation on a vector space
 * any Hamiltonian group action on a symplectic manifold
 * the transitive action underlying any homogeneous space
 * more generally, the group action underlying any principal bundle

Infinitesimal Lie algebra action
Following the spirit of the Lie group-Lie algebra correspondence, Lie group actions can also be studied from the infinitesimal point of view. Indeed, any Lie group action $$\sigma: G \times M \to M$$ induces an infinitesimal Lie algebra action on $$M$$, i.e. a Lie algebra homomorphism $$\mathfrak{g} \to \mathfrak{X}(M)$$. Intuitively, this is obtained by differentiating at the identity the Lie group homomorphism $$G \to \mathrm{Diff}(M)$$, and interpreting the set of vector fields $$\mathfrak{X}(M)$$ as the Lie algebra of the (infinite-dimensional) Lie group $$\mathrm{Diff}(M)$$.

More precisely, fixing any $$x \in M$$, the orbit map $$\sigma_x : G \to M, g \mapsto g \cdot x$$ is differentiable and one can compute its differential at the identity $$e \in G$$. If $$X \in \mathfrak{g}$$, then its image under $$\mathrm{d}_e\sigma_x\colon \mathfrak{g}\to T_xM$$ is a tangent vector at $$x$$, and varying $$x$$ one obtains a vector field on $$M$$. The minus of this vector field, denoted by $$X^\#$$, is also called the fundamental vector field associated with $$X$$ (the minus sign ensures that $$\mathfrak{g} \to \mathfrak{X}(M), X \mapsto X^\#$$ is a Lie algebra homomorphism).

Conversely, by Lie–Palais theorem, any abstract infinitesimal action of a (finite-dimensional) Lie algebra on a compact manifold can be integrated to a Lie group action.

Moreover, an infinitesimal Lie algebra action $$\mathfrak{g} \to \mathfrak{X}(M)$$ is injective if and only if the corresponding global Lie group action is free. This follows from the fact that the kernel of $$\mathrm{d}_e\sigma_x\colon \mathfrak{g}\to T_xM$$ is the Lie algebra $$\mathfrak{g}_x \subseteq \mathfrak{g}$$ of the stabilizer $$G_x \subseteq G$$. On the other hand, $$\mathfrak{g} \to \mathfrak{X}(M)$$ in general not surjective. For instance, let $$\pi: P \to M$$ be a principal $$G$$-bundle: the image of the infinitesimal action is actually equal to the vertical subbundle $$T^\pi P \subset TP$$.

Proper actions
An important (and common) class of Lie group actions is that of proper ones. Indeed, such a topological condition implies that


 * the stabilizers $$G_x \subseteq G$$ are compact
 * the orbits $$G \cdot x \subseteq M$$ are embedded submanifolds
 * the orbit space $$M/G$$ is Hausdorff

In general, if a Lie group $$G$$ is compact, any smooth $$G$$-action is automatically proper. An example of proper action by a not necessarily compact Lie group is given by the action a Lie subgroup $$H \subseteq G$$ on $$G$$.

Structure of the orbit space
Given a Lie group action of $$G$$ on $$M$$, the orbit space $$M/G$$ does not admit in general a manifold structure. However, if the action is free and proper, then $$M/G$$ has a unique smooth structure such that the projection $$M \to M/G$$ is a submersion (in fact, $$M \to M/G$$ is a principal $$G$$-bundle).

The fact that $$M/G$$ is Hausdorff depends only on the properness of the action (as discussed above); the rest of the claim requires freeness and is a consequence of the slice theorem. If the "free action" condition (i.e. "having zero stabilizers") is relaxed to "having finite stabilizers", $$M/G$$ becomes instead an orbifold (or quotient stack).

An application of this principle is the Borel construction from algebraic topology. Assuming that $$G$$ is compact, let $$EG$$ denote the universal bundle, which we can assume to be a manifold since $$G$$ is compact, and let $$G$$ act on $$EG \times M$$ diagonally. The action is free since it is so on the first factor and is proper since $$G$$ is compact; thus, one can form the quotient manifold $$M_G = (EG \times M)/G$$ and define the equivariant cohomology of M as
 * $$H^*_G(M) = H^*_{\text{dr}}(M_G)$$,

where the right-hand side denotes the de Rham cohomology of the manifold $$M_G$$.