a smooth group action

A group is abstract until it does something. A group action is the group 'acting' on a space — permuting its points in a way that respects the group law. When the group is a Lie group and the space is a smooth manifold, we ask the action to be smooth: moving a point around as the group element varies should be a smooth motion, not a jerky one. This is how continuous symmetry actually touches geometry.

Formally, a smooth (left) action of a Lie group G on a smooth manifold M is a smooth map phi: G x M -> M, written g . p, satisfying e . p = p (the identity does nothing) and g . (h . p) = (g*h) . p (composing actions matches multiplying in the group). For each fixed g the map p |-> g . p is then a diffeomorphism of M, with inverse given by g^(-1). Two notions organize an action: the orbit of a point p is G . p = {g . p : g in G}, the set of all places p can be sent; the stabilizer (isotropy subgroup) G_p = {g in G : g . p = p} is the set of symmetries fixing p. The orbit-stabilizer correspondence realizes each orbit as the quotient G/G_p. An action is transitive if there is a single orbit (you can reach any point from any other), and free if every stabilizer is trivial.

Smooth actions are the engine behind homogeneous spaces, principal bundles, and symmetry reduction. A caution: even when G and M are nicely behaved, the orbit space M/G need not be a manifold — orbits of different dimensions, or non-closed orbits, create singularities. When the action is free and proper, however, M/G is a smooth manifold and M -> M/G is a principal G-bundle. 'Smooth action' alone guarantees neither.