Homogeneous spaces and stabilizers

For a closed Lie subgroup H of G, the space G/H of left cosets gH, the orbits of right multiplication by H, has a canonical smooth manifold structure with the projection G→G/H a surjective submersion. It is a principal H-bundle under right multiplication. Its dimension is dim G minus dim H, and the tangent space at the identity coset is naturally the vector quotient g/h. This vector quotient exists without requiring h to be an ideal; it is a tangent space, not automatically a Lie algebra.

A quotient group structure on G/H induced by multiplication requires H to be normal. Most stabilizers in geometric actions are not normal, so their quotients are homogeneous spaces rather than groups. Closedness serves a different purpose: it ensures the appropriate Hausdorff quotient topology and embedded subgroup structure. An irrational dense subgroup of a torus fails this requirement and does not produce an ordinary Hausdorff manifold quotient through the same construction.

For a smooth left action on a Hausdorff manifold M, the stabilizer G-p is the inverse image of the point p under the continuous orbit map, so it is closed. The orbit map sends gH to g acting on p, with H=G-p. It is a smooth bijection onto the orbit and an injective immersion into M. The orbit carries a natural immersed manifold structure from the quotient, which need not coincide with its ambient subset topology.

If the action is transitive on M, the quotient map identifies G/H diffeomorphically with M. More generally, a proper smooth action has closed embedded orbits and compact stabilizers. Proper means that the map taking (g,x) to (g acting on x,x) is proper, with compact inverse images of compact sets. It is a sufficient guarantee for these orbit properties, not a necessary condition for one particular orbit to be embedded. Compact group actions on Hausdorff manifolds provide an important proper example.

The derivative of the orbit map sends a generator to its instantaneous action at p. Its kernel is the stabilizer algebra, so the quotient dimension counts genuinely moving directions. A discrete stabilizer contributes dimension zero even if it has many elements. Conversely, a continuous stabilizer can remove several tangent directions while leaving the orbit highly symmetric. Counting the number of matrix entries or group parameters without subtracting the stabilizer dimension overstates the configuration's independent motion.

Choosing a different orbit point conjugates the stabilizer rather than making it identical as a subset. Quotients by conjugate subgroups are naturally isomorphic as homogeneous spaces. In applications a convenient base point makes the stabilizer equations transparent, and an explicit rotation or basis change transports the result elsewhere. Keep track of whether the application asks for a group, an orbit manifold, a principal bundle, or a numerical coordinate chart: they are related constructions with different equivalence relations and validity ranges.