Symmetry Algebras, Orbits, and Fixed Sets

A continuous symmetry group partitions a manifold into orbits. At a point $p$, the orbit dimension counts directions actually generated there, while the stabilizer records transformations that leave $p$ fixed. Their relation, $\dim(G\!\cdot\!p)=\dim G-\dim G_p$, explains why orbit dimension can drop at axes or other fixed sets and why a quotient may fail to be a smooth manifold.

Required background. Tensor Fields and Natural Operations supplies pointwise tensor operations; Smooth Maps, Pushforwards, and Pullbacks supplies smooth maps, differentials, and rank, while Vector Fields, Flows, and Lie Brackets supplies vector fields, local flows, and brackets. Affine Connections and Covariant Derivatives supplies the general covariant-derivative framework; the orbit and stabilizer results below are connection-free.

Helpful background. Calculus and Linear Algebra for Relativity supplies kernels and rank; Affine and Variational Geodesics supplies the affine equation used in later motion examples, while symmetry reduction itself is developed here.

Actions and infinitesimal generators

A smooth left action of a Lie group $G$ on $M$ is a smooth map $(g,p)\mapsto g\cdot p$ satisfying $e\cdot p=p$ and $g\cdot(h\cdot p)=(gh)\cdot p$. For $\xi$ in the Lie algebra $\mathfrak g$, choose the convention

$$\xi_M(p)=\left.\frac{d}{dt}\right|_{0}\exp(-t\xi)\cdot p.$$

With the minus sign, $\xi\mapsto\xi_M$ is a Lie-algebra homomorphism:

$$[\xi_M,\eta_M]=[\xi,\eta]_M.$$

Changing the convention changes both signs together and no geometry. For an isometric action, each fundamental field is Killing. Conversely, complete Killing fields that close to a suitable finite-dimensional algebra can integrate to an isometry-group action, subject to global group and completeness issues.

The construction of actions, fundamental fields, and their brackets is presented in Meinrenken 2003, §§1.1–1.2, pp. 4–7.

Orbits, stabilizers, and fixed points

The orbit and stabilizer at $p$ are

$$G\cdot p=\{g\cdot p:g\in G\}, \qquad G_p=\{g\in G:g\cdot p=p\}.$$

The tangent space to the orbit is the image of the evaluation map $\mathfrak g\to T_pM$, $\xi\mapsto\xi_M(p)$. Its kernel is the Lie algebra of $G_p$, so

$$\dim(G\cdot p)=\dim G-\dim G_p.$$

A point fixed by the whole group has $G_p=G$ and a zero-dimensional orbit. For a proper smooth action, the connected pieces with a fixed conjugacy class of stabilizers form smooth orbit-type strata, and their images stratify the quotient. Orbit dimension is constant on each such piece and cannot increase in a limiting orbit. Without properness, this well-behaved stratification is not automatic. The precise result is Meinrenken 2003, §1.5, Theorem 1.30, pp. 13–14, PDF.

At a fixed point, the first nontrivial information is the derivative of the action on $T_pM$. For an isometric action this is a linear orthogonal representation. A rotation generator may vanish at $p$ while its antisymmetric derivative remains nonzero, encoding how nearby points move. Thus “the Killing field vanishes” does not mean that the symmetry is locally trivial.

Rotation of the two-sphere

Let $SO(2)$ rotate the round sphere about its vertical axis:

$$(\theta,\phi)\longmapsto(\theta,\phi+\alpha).$$

At $0\lt\theta\lt\pi$, the stabilizer is trivial and the orbit is a latitude circle, hence one-dimensional. At the north and south poles, every rotation fixes the point, so $G_p=SO(2)$ and the orbit dimension drops to zero. The field that generates increasing $\phi$ is $X=\partial_\phi$; with the minus-sign definition of $\xi_M$ above it corresponds to the negative of the usual positive-angle Lie-algebra generator. Its norm $\sin\theta$ tends to zero at both poles. In regular Cartesian coordinates on the unit sphere, $X=-y\partial_x+x\partial_y$, so the vanishing at the poles is explicit.

The quotient is naturally labeled by $z=\cos\theta\in[-1,1]$. It is an interval with special endpoint strata, not a smooth one-dimensional manifold without boundary obtained by forgetting the fixed points. The action is proper because $SO(2)$ is compact, but it is not free. Properness alone ensures good separation properties; freeness as well as properness is the standard hypothesis under which $M/G$ is a smooth manifold and $M\to M/G$ is a principal bundle. These distinctions and the quotient theorem appear in Meinrenken 2003, §§1.3–1.4, pp. 7–11.

Freeness means every stabilizer is trivial. Properness means the map $G\times M\to M\times M$, $(g,p)\mapsto(g\cdot p,p)$, is proper. The first controls local orbit dimension; the second prevents non-Hausdorff pathologies in the quotient. Neither condition follows from the other for an arbitrary group action. Compact groups act properly, which is why the sphere example has a well-behaved stratified quotient despite its fixed points.

What orbit data do and do not say

Lorentzian signature introduces another important distinction: zero norm need not mean a zero generator. In two-dimensional Minkowski space the boost field $B=x\partial_t+t\partial_x$ has

$$g(B,B)=t^2-x^2, \qquad B(x^2-t^2)=0.$$

Its non-null orbits are branches of hyperbolas with $x^2-t^2$ constant. On $t=\pm x$ away from the origin, $B$ is nonzero and null, so those points still lie on one-dimensional orbits. Only at $t=x=0$ does $B$ vanish and the orbit reduce to a fixed point. Hence a null symmetry direction does not by itself indicate a fixed set or an axis. This is a direct check on the rank definition of orbit dimension, independent of the sign of a norm.

The Lie algebra records infinitesimal composition laws, while orbit and stabilizer data record how that algebra is realized at each point. Two actions of the same abstract group can have different fixed sets and quotients. Likewise, knowing the dimension of an isometry group does not classify the metric: curvature and global topology provide additional information.

Near a fixed set, smoothness must be checked using regular geometric variables, not only symmetry-adapted coordinates. This chapter uses axes as a model for orbit degeneration but does not assign horizon types, physical charges, or a classification of particular spacetimes.

One should also distinguish the abstract Lie algebra from a chosen basis of generators. Structure constants change under a basis change, while the bracket operation does not. Orbit dimension is similarly basis-independent: it is the rank of the entire evaluation map, not the number of coordinate components that happen to be nonzero.

Connected group orbits capture continuous motion only. A disconnected component may identify additional points without adding a Killing generator, so the full quotient by an isometry group can be smaller than the quotient inferred from its Lie algebra alone.

Exercises

For the $SO(2)$ action on $S^2$, determine every orbit type and show explicitly why the quotient map $q:S^2\to[-1,1]$, $q(x,y,z)=z$, separates the orbits.

Solution

If $-1<z<1$, the intersection of $S^2$ with the plane of fixed $z$ is a circle, and rotations act transitively on it with trivial stabilizer. For $z=1$ or $z=-1$, the intersection is one pole and its stabilizer is all of $SO(2)$. Two points have the same $z$ exactly when their $(x,y)$ coordinates have the same radius $\sqrt{1-z^2}$, so one rotation carries one to the other. Thus the fibers of $q$ are precisely the orbits. The endpoint fibers exhibit the two fixed-point strata.

References

• Eckhard Meinrenken, Group Actions on Manifolds, University of Toronto lecture notes (2003), PDF.