In differential geometry, a Lie group action on a manifoldM 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 types of G form a stratification of M and this can be used to understand the geometry of M. Let be a group action. It is a Lie group action if it is differentiable. Thus, in particular, the orbit map is differentiable and one can compute its differential at the identity element of G: If X is in, 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. The kernel of the map can be easily shown to be the Lie algebra of the stabilizer Let be a principal G-bundle. Since G has trivial stabilizers in P, for u in P, is an isomorphism onto a subspace; this subspace is called the vertical subspace. A fundamental vector field on P is thus vertical. In general, the orbit space does not admit a manifold structure since, for example, it may not be Hausdorff. However, if G is compact, then is Hausdorff and if, moreover, the action is free, then is a manifold This is a consequence of the slice theorem. If the "free action" is relaxed to "finite stabilizer", one instead obtains an orbifold A substitute for the construction of the quotient is the Borel construction from algebraic topology: assume G is compact and 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. Thus, one can form the quotient manifold. The constriction in particular allows one to define the equivariant cohomology of M; namely, one sets where the right-hand side denotes the de Rham cohomology, which makes sense since has a structure of manifold If G is compact, then any G-manifold admits an invariant metric; i.e., a Riemannian metricwith respect to which G acts on M as isometries.
