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 formula_1 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 formula_2 is differentiable. Equivalently, a Lie group action of G on M consists of a Lie group homomorphism formula_3. A smooth manifold endowed with a Lie group action is also called a "G"-manifold. 
The fact that the action map formula_2 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.
Examples.
For every Lie group G, the following are Lie group actions:
Other examples of Lie group actions include:
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 formula_13 induces an infinitesimal Lie algebra action on M, i.e. a Lie algebra homomorphism formula_14. Intuitively, this is obtained by differentiating at the identity the Lie group homomorphism formula_3, and interpreting the set of vector fields formula_16 as the Lie algebra of the (infinite-dimensional) Lie group formula_17.
More precisely, fixing any formula_18, the orbit map formula_19 is differentiable and one can compute its differential at the identity formula_20. If formula_21, then its image under formula_22 is a tangent vector at "x", and varying "x" one obtains a vector field on "M". The minus of this vector field, denoted by formula_23, is also called the fundamental vector field associated with "X" (the minus sign ensures that formula_24 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 formula_14 is injective if and only if the corresponding global Lie group action is free. This follows from the fact that the kernel of formula_22 is the Lie algebra formula_27 of the stabilizer formula_5. On the other hand, formula_14 in general not surjective. For instance, let formula_30 be a principal "G"-bundle: the image of the infinitesimal action is actually equal to the vertical subbundle formula_31.
Proper actions.
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 formula_7 on G.
Structure of the orbit space.
Given a Lie group action of G on M, the orbit space formula_34 does not admit in general a manifold structure. However, if the action is free and proper, then formula_34 has a unique smooth structure such that the projection formula_38 is a submersion (in fact, formula_38 is a principal "G"-bundle).
The fact that formula_34 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", formula_34 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 formula_42 denote the universal bundle, which we can assume to be a manifold since "G" is compact, and let "G" act on formula_43 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 formula_44 and define the equivariant cohomology of "M" as
where the right-hand side denotes the de Rham cohomology of the manifold formula_46.