Let 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 is differentiable. Equivalently, a Lie group action of G on M consists of a Lie group homomorphism . A smooth manifold endowed with a Lie group action is also called a G-manifold.
The fact that the action map is smooth has a couple of immediate consequences:
Forgetting the smooth structure, a Lie group action is a particular case of a continuous group action.
For every Lie group G, the following are Lie group actions:
Other examples of Lie group actions include:
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 induces an infinitesimal Lie algebra action on M, i.e. a Lie algebra homomorphism . Intuitively, this is obtained by differentiating at the identity the Lie group homomorphism , and interpreting the set of vector fields as the Lie algebra of the (infinite-dimensional) Lie group .
More precisely, fixing any , the orbit map is differentiable and one can compute its differential at the identity . If , then its image under is a tangent vector at x, and varying x one obtains a vector field on M. The minus of this vector field, denoted by , is also called the fundamental vector field associated with X (the minus sign ensures that is a Lie algebra homomorphism).
Moreover, an infinitesimal Lie algebra action is injective if and only if the corresponding global Lie group action is free. This follows from the fact that the kernel of is the Lie algebra of the stabilizer . On the other hand, in general not surjective. For instance, let be a principal G-bundle: the image of the infinitesimal action is actually equal to the vertical subbundle .
An important (and common) class of Lie group actions is that of proper ones. Indeed, such a topological condition implies that
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 on G.
Given a Lie group action of G on M, the orbit space does not admit in general a manifold structure. However, if the action is free and proper, then has a unique smooth structure such that the projection is a submersion (in fact, is a principal G-bundle).
The fact that 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", 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 denote the universal bundle, which we can assume to be a manifold since G is compact, and let G act on 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 and define the equivariant cohomology of M as
where the right-hand side denotes the de Rham cohomology of the manifold .