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Concept
Lie group action
Summary
In differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable.
TOC
Definition and first properties
Let \sigma: G \times M \to M, (g, x) \mapsto g \cdot x be a (left) group action of a Lie group G on a smooth manifold M; it is called a Lie group action (or smooth action) if the map \sigma is differentiable. Equivalently, a Lie group action of G on M consists of a Lie group homomorphism G \to \mathrm{Diff}(M). A smooth manifold endowed with a Lie group action is also called a G-manifold.
The fact that the action map \sigma is smooth has a couple of immediate consequences:
- the stabilizers G_x \subseteq G of the group action are closed, thus are Lie subgroups of G
- the orbits G \cdot x \subseteq M of the group action are immersed submanifolds.
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