8192 (number)

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 $\sigma :G\times M\to M,(g,x)\to g\cdot x$ be a group action. It is a Lie group action if it is differentiable. Thus, in particular, the orbit map $\sigma _{x}:G\to M,g\cdot x$ is differentiable and one can compute its differential at the identity element of G:

${\mathfrak {g}}\to T_{x}M$.

If X is in ${\mathfrak {g}}$, 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 ${\mathfrak {g}}\to \Gamma (TM)$ is a Lie algebra homomorphism.) The kernel of the map can be easily shown to be the Lie algebra ${\mathfrak {g}}_{x}$ of the stabilizer $G_{x}$ (which is closed and thus a Lie subgroup of G.)

Let $P\to M$ be a principal G-bundle. Since G has trivial stabilizers in P, for u in P, $a\mapsto a_{u}^{\#}:{\mathfrak {g}}\to T_{u}P$ is an isomorphism onto a subspace; this subspace is called the vertical subspace.

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, a principal G-bundle.)Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. 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 $EG$ denote the universal bundle, which we can assume to be a manifold since G is compact, and let G act on $EG\times M$ diagonally; the action is free since it is so on the first factor. Thus, one can form the quotient manifold $M_{G}=(EG\times M)/G$. The constriction in particular allows one to define the equivariant cohomology of M; namely, one sets

$H_{G}^{*}(M)=H_{\text{dr}}^{*}(M_{G})$,

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.

See also

• Hamiltonian group action
• Equivariant differential form

References

• Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004