Lie group action

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 types of G form a stratification of M and this can be used to understand the geometry of M.

Let σ : G × M → M , ( g , x ) → g ⋅ x be a group action. It is a Lie group action if it is differentiable. Thus, in particular, the orbit map σ x : G → M , g ⋅ x is differentiable and one can compute its differential at the identity element of G:

If X is in g , 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 X # . (The "minus" ensures that g → Γ ( T M ) is a Lie algebra homomorphism.) The kernel of the map can be easily shown (cf. Lie correspondence) to be the Lie algebra g x of the stabilizer G x (which is closed and thus a Lie subgroup of G.)

Let P → M be a principal G-bundle. Since G has trivial stabilizers in P, for u in P, a ↦ a u # : g → T u 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 M / G does not admit a manifold structure since, for example, it may not be Hausdorff. However, if G is compact, then M / G is Hausdorff and if, moreover, the action is free, then M / G is a manifold (in fact, M → M / G is a principal G-bundle.) This is a consequence of the 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 E G denote the universal bundle, which we can assume to be a manifold since G is compact, and let G act on E G × M diagonally; the action is free since it is so on the first factor. Thus, one can form the quotient manifold M G = ( E G × M ) / G . 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 M G 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.