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In differential geometry, a '''Lie group action''' on a manifold ''M'' is a [[group action]] by a [[Lie group]] ''G'' on ''M'' that is a [[differentiable map]]; in particular, it is a [[continuous group action]]. Together with a Lie group action by ''G'', ''M'' is called a '''''G''-manifold'''. The [[orbit type]]s of ''G'' form a stratification of ''M'' and this can be used to understand the geometry of ''M''.

Let <math>\sigma: G \times M \to M, (g, x) \to g \cdot x</math> be a group action. It is a Lie group action if it is differentiable. Thus, in particular, the orbit map <math>\sigma_x : G \to M, g \cdot x</math> is differentiable and one can compute its differential at the identity element of ''G'':
:<math>\mathfrak{g} \to T_x M</math>.
If ''X'' is in <math>\mathfrak{g}</math>, then its image under the above is a tangent vector at ''x'' and, varying ''x'', one obtains a vector field on ''M''; the minus of this vector field is called the [[fundamental vector field]] associated with ''X'' and is denoted by <math>X^\#</math>. (The "minus" ensures that <math>\mathfrak{g} \to \Gamma(TM)</math> is a Lie algebra homomorphism.) The kernel of the map can be easily shown to be the Lie algebra <math>\mathfrak{g}_x</math> of the stabilizer <math>G_x</math> (which is closed and thus a Lie subgroup of ''G''.)<!--disrupting the flow. In partciualr, <math>\mathfrak{g}/\mathfrak{g}_x \to T_x M</math> is injective and it follows that the map induced by the orbit map <math>G/G_x \to M</math> is an injective immersion, whose image is the orbit of ''x''. Hence, the orbits in ''M'' are immersed submanifolds and if ''G'' is compact, the orbits are embedded submanifolds. -->

Let <math>P \to M</math> be a principal ''G''-bundle. Since ''G'' has trivial stabilizers in ''P'', for ''u'' in ''P'', <math>a \mapsto a^\#_u: \mathfrak{g} \to T_u P</math> is an isomorphism onto a subspace; this subspace is called the [[vertical subspace]].

In general, the [[orbit space]] <math>M/G</math> does not admit a manifold structure since, for example, it may not be Hausdorff. However, if ''G'' is compact, then <math>M/G</math> is Hausdorff and if, moreover, the action is free, then <math>M/G</math> is a manifold (in fact, a principal ''G''-bundle.){{citation needed|date=July 2013}} This is a consequence of the [[slice theorem (differential geometry)|slice theorem]]. If the "free action" is relaxed to "finite stabilizer", one instead obtains an [[orbifold]] (or [[quotient stack]].)

A substitute for the construction of the quotient is the [[Borel construction]] from algebraic topology: assume ''G'' is compact and let <math>EG</math> denote the universal bundle, which we can assume to be a manifold since ''G'' is compact, and let ''G'' act on <math>EG \times M</math> diagonally; the action is free since it is so on the first factor. Thus, one can form the ''quotient manifold'' <math>M_G = (EG \times M)/G</math>. The constriction in particular allows one to define the [[equivariant cohomology]] of ''M''; namely, one sets
:<math>H^*_G(M) = H^*_{\text{dr}}(M_G)</math>,
where the right-hand side denotes the de Rham cohomology, which makes sense since <math>M_G</math> has a structure of manifold (thus there is the notion of differential forms.)

If ''G'' is compact, then any ''G''-manifold admits an invariant metric; i.e., a Riemannian metric with respect to which ''G'' acts on ''M'' as isometries.<!--<math>\sigma_x(g)</math> is a linear isomorphism of the vector space <math>T_x M</math> for any ''g'' in <math>G_x</math>. In other words, <math>\sigma_x</math> is a linear representation of <math>G_x</math> on <math>T_x M</math>; called the '''linear isotropy representation'''.-->

== See also ==
*[[Hamiltonian group action]]
*[[Equivariant differential form]]

== References ==
*Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004

[[Category:Group actions]]