Homogeneous manifolds

LetGbe a Lie group and letHbe a closed subgroup. Consider the setG/H of left cosets ofHinGequipped with the quotient topology with respect to the projectionπ :G→G/H. Note also that left multiplication inGinduces a mapλ:G×G/H →G/H, namely,λ(g, xH) = (gx)H, and that

(3.6.1) π◦L_g =λ_g◦π

for allg∈G, whereλg(p) =λ(g, p)forp∈G/H.

3.6.2 Lemma A closed Lie subgroupHof a Lie group Gmust have the induced topology.

Proof. We need to prove that the inclusion mapι:H →Gis an embed-ding. Sinceιcommutes with left translations, it suffices to find an open sub-setV ofHsuch that the restrictionι|V is an embedding intoG. By the proof of Theorem 3.3.5, there exists a distinguished chart(U, ϕ= (x₁, . . . , x_n))of Garound1such that H∩U consists of at most countably many plaques, each plaque being a slice of the form

x_k+1 =c_k+1, . . . , x_n=c_n

for somec_k+1, . . . , cn ∈ R, wherek = dimH. Denote byτ : Rⁿ = R^k× R^n−k → R^n−k the projection. LetU˜ be a compact neighborhood of1 con-tained inU. Now H∩U˜ is compact, so τ(H ∩U˜) is a non-empty closed countable subset ofR^n−k which by the Baire category theorem must have an isolated point. This point specifies a isolated plaqueV ofH inU along whichιis an open mapping and hence a homeomorphism onto its image,

as desired.

3.6. HOMOGENEOUS MANIFOLDS 79 3.6.3 Theorem IfGis a Lie group andH is a closed subgroup ofG, then there is a unique smooth structure on the topological quotientG/H such thatλ:G× G/H → G/H is smooth. Moreover, π : G → G/H is a surjective submersion anddimG/H= dimG−dimH.

Proof. For an open set V of G/H we have that π⁻¹(π(V)) = S

g∈GgV is a union of open sets and thus open. This shows thatπ is an open map and hence the projection of a countable basis of open sets of G yields a countable basis of open sets ofG/H. To prove thatG/His Hausdorff, we use closedness ofH. Indeed it implies that the equivalence relationR ⊂ G×G, defined by specifying thatg∼g^′if and only ifg⁻¹g^′∈H, is a closed subset ofG×G. Now ifgH 6=g^′HinG/Hthen(g, g^′)6∈ Rand there exist open neighborhoodsW ofgandW^′ofg^′inGsuch that(W×W^′)∩ R=∅. It follows that π(W)andπ(W^′)are disjoint neighborhoods of g andg^′ in G/H, respectively.

We first construct a local chart of G/H around p₀ = π(1) = 1H. Re-call from Proposition 3.2.4 and (3.3.9) that the exponential mapexp = exp^G gives a parametrization ofG around the identity element and restricts to the exponential map ofh. Denote the Lie algebras ofGandHbygandh, resp., and choose a complementary subspacemtohing. We can choose a product neighborhood of0in gof the form U₀×V₀, whereU₀ is a neigh-borhood of0inh,V₀is a neighborhood of0inmsuch that the map

f :U₀×V₀ →G, f(X, Y) = expXexpY

is a diffeomorphism from onto its image (apply the Inverse Function Theo-rem 1.3.8 tof). Owing to Lemma 3.6.2,Hhas the topology induced fromG, so we may choose a neighborhoodW of1inGsuch thatW ∩H= exp(U₀). We also shrink V₀ so that (expV₀)⁻¹expV₀ ⊂ W. Now we claim that π◦exp|V0 is injective. Indeed, ifπ(expY) =π(expY^′)for someY,Y^′ ∈V₀, then (expY)⁻¹expY^′ ∈ H∩W = exp(U₀), so expY^′ = expY expX for some X ∈ U₀. Since f is injective on U₀ ×V₀, this implies that Y^′ = Y andX = 0and proves the claim. NoteexpV₀expU₀ is open inG, so the imageπ(expV₀) = π(expV₀expU₀) is open inG/H. We have shown that π ◦exp defines a homeomorphism from V₀ onto the open neighborhood V = π(expV₀) of p in G/H, whose inverse can then be used to define a local chart(V, ψ)ofG/Haroundp₀.

Now the collection{(V^g, ψ^g)}g∈Gdefines an atlas ofG/H, whereV^g = gV andψ^g = ψ◦L_g−1, and we need to check its smoothness. Supposeg, g^′ ∈Gare such thatV^g∩V^g′ 6=∅, and thatp= (gexpY)H= (g^′expY^′)H is an element there, namely, ψ^g(p) = Y andψ^g′(p) = Y^′. Then expY^′ = (g^′)⁻¹gexpY h∈expV₀ for someh∈H, so there exists a neighborhoodV˜₀ ofY inV0 such that(g^′)⁻¹g(exp ˜V0)h⊂expV0, and thusψ^g′◦(ψ^g)⁻¹|V^˜0 can be written as the composite map

τ ◦log◦R_h◦L_(g′)⁻¹g◦exp,

wherelogdenotes the inverse map ofexp : U0×V0 → exp(U0 ×V0), and τ : g → m denotes the projection along h. Hence the change of charts ψ^g′ ◦(ψ^g)⁻¹is smooth.

The local representation ofπ in the above charts isτ, namely, there is a commutative diagram

gexpV₀expU₀ π

> V^g

U0×V0

f⁻¹◦L_g−1

∨

τ > V0

L_g◦π◦exp

∧

which shows thatπis a submersion. Similarly, the commutative diagram G×gexpV₀ > G

G×V^g id×π

∨

λ|G×V^g

> G/H

∨π

proves thatλis smooth. The uniqueness of the smooth structure follows

from Proposition 3.6.4 below.

LetMbe a smooth manifold and letGbe a Lie group. AnactionofGon Mis a smooth mapµ:G×M →Msuch thatµ(1, p) =pandµ(g, λ(g^′, p)) = µ(gg^′, p) for allp ∈ M andg,g^′ ∈ G. For brevity of notation, in caseµis fixed and clear from the context, we will simply writeµ(g, p) =gp.

An action ofGisM is calledtransitiveif for everyp,q ∈M there exists g ∈ Gsuch thatgp =q. In this case,M is calledhomogeneous underG, G-homogeneous, or simply ahomogeneous manifold. Examples of homogeneous manifolds are given by the quotientsG/H, whereHis closed Lie subgroup of G, according to Theorem 3.6.3. Conversely, the next proposition that every homogeneous manifold is of this form. For an action ofGonM and p∈M, theisotropy groupatpis the subgroupG_pofGconsisting of elements that fixp, namely,Gp ={g∈G|gp=p}. Plainly,Gp is a closed subgroup ofG, and so a Lie subgroup ofG, owing to Theorem 3.7.1 below.

3.6.4 Proposition Letµ :G×M → M be a transitive action of a Lie groupG on a smooth manifoldM. Fixp₀ ∈ M and letH =G_p0 be the isotropy group at p₀. Define a map

f :G/H →M, f(gH) =µ(g, p₀).

Thenf is well-defined and a diffeomorphism.

Proof. It is easy to see thatf is well-defined, bijective and smooth. We can writef◦π =ω, whereω :G→ M is the “orbit map”ω(g) = gp0. For

3.6. HOMOGENEOUS MANIFOLDS 81 X∈g, we have

dω₁(X) = d ds

s=0(expsX)p₀=d(exp(−sX))d dt

t=s(exptX)p₀, so X ∈ kerdω₁ if and only ifexptX ∈ H for all t ∈ R if and only ifX belongs to the Lie algebrahofH, due to (3.3.10). Sincedf_1H ◦dπ₁ = dω₁ andkerdπ₁ = h, this implies that f is an immersion at 1H, and thus an immersion everywhere by the equivariance property f ◦λ_g = µ_g ◦f for allg∈G.

This already implies thatdimG/H≤dimM and that(G/H, f)is a sub-manifold ofM, but the strict inequality cannot hold asfis bijective and the image of a smooth map from a smooth manifold into a strictly higher di-mensional smooth manifold has measure zero (see Problem 24). It follows thatf is a local diffeomorphism and hence a diffeomorphism.

3.6.5 Examples (a) Let{e₁, . . . , e_n}be the canonical basis ofRⁿ and view elements ofRⁿas column-vectors (n×1matrices). ThenGL(n,R)acts on Rⁿby left-multiplication:

(3.6.6) GL(n,R)×Rⁿ →Rⁿ

The basis {e_i} is orthonormal with respect to the standard scalar product inRⁿ. The orthogonal groupO(n)precisely consists of those elements of GL(n,R)whose action onRⁿpreserves the lengths of vectors. In particu-lar, the action (3.6.6) restricts to an action

(3.6.7) O(n)×Sⁿ⁻¹→Sⁿ⁻¹

which is smooth, sinceSⁿ⁻¹ is an embedded submanifold ofRⁿ. The ac-tion (3.6.7) is transitive due to the facts that any unit vector can be com-pleted to an orthonormal basis ofRⁿ, and any two orthonormal bases of Rⁿ differ by an orthogonal transformation. The isotropy group of (3.6.7) ate₁consists of transformations that leave the orthogonal complemente^⊥₁ invariant, and indeed any orthogonal transformation ofe^⊥₁ ∼=Rⁿ⁻¹can oc-cur. It follows that the isotropy group is isomorphic toO(n−1)and hence

Sⁿ⁻¹ =O(n)/O(n−1)

presents the unit sphere as a homogeneous space, where a the diffeomor-phism is given bygO(n−1)7→g(e₁). If we use only orientation-preserving transformations onRⁿ, also the elements of the isotropy group will act by orientation-preserving transformations and hence

Sⁿ⁻¹=SO(n)/SO(n−1).

(b) The group SO(n) also acts transitively on the set of lines through the origin inRⁿ. Besides the orthogonal transformations ofe^⊥₁, the isotropy

group at the lineRe1now also contains transformations that mape1to−e1. It follows that

RPⁿ=SO(n)/O(n−1)

whereO(n−1)is identified with the subgroup ofSO(n)consisting of

ma-trices of the form

detA 0

0 A

whereA∈O(n−1).

(c) Let{e₁, . . . , e_n}be the canonical basis ofCⁿ. It is a unitary basis with respect to the standard Hermitian inner product inCⁿ. Similarly to (a), one shows thatU(n)andSU(n)act transitively on the set of unit vectors ofCⁿ, namely, the sphereS²ⁿ⁻¹. More interesting is to consider the setCPⁿ⁻¹of one-dimensional complex subspaces ofCⁿ. This set is homogeneous under SU(n) and the isotropy group at the line Ce1 consists of matrices of the

form

(detA)⁻¹ 0

0 A

whereA∈U(n−1). It follows from Theorem 3.6.3 thatCPⁿ⁻¹is a smooth manifold and

CPⁿ⁻¹ =SU(n)/U(n−1)

as a homogeneous manifold, calledcomplex projective space.

(d) Let{e1, . . . , en}be the canonical basis ofRⁿ, and letVk(Rⁿ)be the set of orthonormalk-frames inRⁿ, that is, orderedk-tuples of orthonormal vectors inRⁿ. There is an action

O(n)×V_k(Rⁿ)→V_k(Rⁿ), g·(v₁, . . . , v_k) = (gv₁, . . . , gv_k)

which is clearly transitive. The isotropy group at (e₁, . . . , e_k) is the sub-group ofO(n)consisting of matrices of the form

(3.6.8)

I 0 0 A

whereA∈O(n−k). The resulting homogeneous space V_k(Rⁿ) =O(n)/O(n−k)

is called theStiefel manifoldofk-frames inRⁿ. Note that the restricted action ofSO(n)onV_k(Rⁿ)is also transitive and

V_k(Rⁿ) =SO(n)/SO(n−k).