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

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

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