Orbits, stabilizers, and homogeneous spaces
When a Lie group moves the points of a space, each point sweeps out anorbit and is held fixed by a stabilizer. Together they explain why a sphere looks the same everywhere.
Pick a point on the sphere. Its orbit spreads over the whole sphere in a color wave, and the glowing ring shows the rotations that leave the point fixed.
A Lie group action on a manifold is a smooth map that assigns to each group element a diffeomorphism of the manifold, compatible with the group operation. The orbit of a point is the set of all points reachable from it via the group action. The stabilizer (or isotropy group) is the subgroup that fixes the point.
Pick a point on the sphere. The orbit under SO(3) fills the entire sphere in a color wave — the action is transitive, meaning any point can be rotated to any other. The stabilizer of a point is the set of rotations that fix it: rotations around the axis through that point, forming an SO(2) subgroup. The glowing ring with orbiting particles visualizes this stabilizer. Hence S² = SO(3)/SO(2).
Key insight: The sphere S² is a homogeneous space — it looks the same from every point. This is because SO(3) acts transitively: the orbit of any point is the entire sphere.
Step through the preset actions and watch the bar split dim(G) into an orbit part and a stabilizer part.
The orbit-stabilizer theorem says dim(Orbit) + dim(Stabilizer) = dim(Group). This gives rise to homogeneous spaces: the sphere S² = SO(3)/SO(2) because SO(3) acts transitively on the sphere with SO(2) stabilizer at each point. Grassmannians, flag manifolds, and many other important spaces arise as quotients of Lie groups.
dim(Orbit) + dim(Stabilizer) = dim(Group). This fundamental equation splits the dimension of the group into two complementary parts. Watch the bar animation divide dim(G) into colored segments representing the orbit and stabilizer dimensions. Explore multiple preset actions to see how the split varies.
Select a space, such as a sphere, a projective space, or a Grassmannian, to see which group quotient builds it and watch the group action that generates it.
Many important spaces in geometry arise as quotients of Lie groups: spheres Sⁿ = SO(n+1)/SO(n), real projective spaces ℝPⁿ = O(n+1)/(O(1) × O(n)), Grassmannians, and flag manifolds. Select a homogeneous space to see its structure and the animated group action that generates it.
Click any point to see its orbit under SO(2). The heatmap shows orbit dimension: zero at the fixed origin, one everywhere else.
When a group acts on a space, different points can have different stabilizers — this is symmetry breaking. For SO(2) acting on ℝ², the origin has the full group as its stabilizer (it's a fixed point), while every other point has trivial stabilizer (its orbit is a full circle). The heatmap shows orbit dimension across the space, with animated particle flow along the circular orbits.
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