Chapter 18
Manifolds Arising from Group Actions
18.1 Proper Maps
We saw in Chapter 3 that many manifolds arise from a group action.
The scenario is that we have a smooth action ': G M M of a Lie group G acting on a manifold M (recall⇥ that! an action ' is smooth if it is a smooth map).
If G acts transitively on M,thenforanypointx M, 2 if Gx is the stabilizer of x,thenweknowfromProposi- tion ?? that G/Gx is di↵eomorphic to M and that the projection ⇡ : G G/G is a submersion. ! x
823 824 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS If the action is not transitive, then we consider the orbit space M/G of orbits G x. · However, in general, M/G is not even Hausdor↵.
It is thus desirable to look for su cient conditions that ensure that M/G is Hausdor↵.
Asu cientconditioncanbegivenusingthenotionofa proper map.
Before we go any further, let us observe that the case where our action is transitive is subsumed by the more general situation of an orbit space.
Indeed, if our action is transitive, for any x M,weknow 2 that the stabilizer H = Gx of x is a closed subgroup of G. 18.1. PROPER MAPS 825 Then, we can consider the right action G H G of H on G given by ⇥ !
g h = gh, g G, h H. · 2 2 The orbits of this (right) action are precisely the left cosets gH of H.
Therefore, the set of left cosets G/H (the homogeneous space induced by the action : G M M)istheset of orbits of the right action G· H⇥ G!. ⇥ ! Observe that we have a transitive left action of G on the space G/H of left cosets, given by
g g H = g g H. 1 · 2 1 2 The stabilizer of 1H is obviously H itself. 826 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Thus, we recover the original transitive left action of G on M = G/H.
Now, it turns out that an action of the form G H G, where H is a closed subgroup of a Lie group⇥ G!,isa special case of a free and proper action M G G,in which case the orbit space M/G is a manifold,⇥ ! and the projection ⇡ : G M/G is a submersion. ! Let us now define proper maps.
Definition 18.1. If X and Y are two Hausdor↵topo- logical spaces,1 acontinuousmap': X Y is proper i↵for every topological space Z,themap! ' id: X Z Y Z is a closed map (recall that f is⇥ a closed⇥ map! i↵the⇥ image of any closed set by f is a closed set).
If we let Z be a one-point space, we see that a proper map is closed.
1It is not necessary to assume that X and Y are Hausdor↵but, if X and/or Y are not Hausdor↵, we have to replace “compact” by “quasi-compact.” We have no need for this extra generality. 18.1. PROPER MAPS 827 If F is any closed subset of X,thentherestrictionof' to F is proper (see Bourbaki, General Topology [9], Chapter 1, Section 10).
The following result can be shown (see Bourbaki, General Topology [9], Chapter 1, Section 10):
Proposition 18.1. A continuous map ': X Y is 1 ! proper i↵ ' is closed and if ' (y) is compact for every y Y . 2
1 If ' is proper, it is easy to show that ' (K)iscompact in X whenever K is compact in Y .
Moreover, if Y is also locally compact, then we have the following result (see Bourbaki, General Topology [9], Chapter 1, Section 10). 828 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Proposition 18.2. If Y is locally compact, a map 1 ': X Y is a proper map i↵ ' (K) is compact in X whenever! K is compact in Y
In particular, this is true if Y is a manifold since manifolds are locally compact.
This explains why Lee [35] (Chapter 9) takes the property stated in Proposition 18.2 as the definition of a proper map (because he only deals with manifolds).
Finally, we can define proper actions. 18.2. PROPER AND FREE ACTIONS 829 18.2 Proper and Free Actions
Definition 18.2. Given a Hausdor↵topological group G and a topological space M,aleftaction : G M M is proper if it is continuous and if the map· ⇥ !
✓ : G M M M, (g, x) (g x, x) ⇥ ! ⇥ 7! · is proper.
If H is a closed subgroup of G and if : G M M is aproperaction,thentherestrictionofthisactionto· ⇥ ! H is also proper.
If we let M = G,thenG acts on itself by left translation, and the map ✓ : G G G G given by ✓(g, x)= (gx, x)isahomeomorphism,soitisproper.⇥ ! ⇥ 830 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS It follows that the action : H G G of a closed subgroup H of G on G (given· by⇥ (h, g)! hg)isproper. 7! The same is true for the right action of H on G.
As desired, proper actions yield Hausdor↵orbit spaces.
Proposition 18.3. If the action : G M M is proper (where G is Hausdor↵), then· the⇥ orbit! space M/G is Hausdor↵. Furthermore, M is also Haus- dor↵.
We also have the following properties (see Bourbaki, Gen- eral Topology [9], Chapter 3, Section 4). 18.2. PROPER AND FREE ACTIONS 831 Proposition 18.4. Let : G M M be a proper action, with G Hausdor↵·. For⇥ any x! M, let G x be 2 · the orbit of x and let Gx be the stabilizer of x. Then: (a) The map g g x is a proper map from G to M. 7! · (b) Gx is compact.
(c) The canonical map from G/Gx to G x is a home- omorphism. · (d) The orbit G x is closed in M. ·
If G is locally compact, we have the following character- ization of being proper (see Bourbaki, General Topology [9], Chapter 3, Section 4).
Proposition 18.5. If G and M are Hausdor↵and G is locally compact, then the action : G M M is proper i↵for all x, y M, there exist· some⇥ open! sets, V and V in M, with2x V and y V , so that the x y 2 x 2 y closure K of the set K = g G Vx g Vy = , is compact in G. { 2 | · \ 6 ;} 832 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS In particular, if G has the discrete topology, the above condition holds i↵the sets g G Vx g Vy = are finite. { 2 | · \ 6 ;}
Also, if G is compact, then K is automatically compact, so every compact group acts properly.
If M is locally compact, we have the following character- ization of being proper (see Bourbaki, General Topology [9], Chapter 3, Section 4).
Proposition 18.6. Let : G M M be a con- tinuous action, with G and· M⇥ Hausdor! ↵. For any compact subset K of M we have: (a) The set G = g G g K K = is closed. K { 2 | · \ 6 ;} (b) If M is locally compact, then the action is proper i↵ GK is compact for every compact subset K of M. 18.2. PROPER AND FREE ACTIONS 833 In the special case where G is discrete (and M is locally compact), condition (b) says that the action is proper i↵ GK is finite.
Remark: If G is a Hausdor↵topological group and if H is a subgroup of G,thenitcanbeshownthattheaction of G on G/H ((g1,g2H) g1g2H)isproperi↵H is compact in G. 7!
Definition 18.3. An action : G M M is free if for all g G and all x M,if· g =1then⇥ !g x = x. 2 2 6 · 6 An equivalent way to state that an action : G M M · ⇥ ! is free is as follows. For every g G,letLg : M M be the di↵eomorphism of M given2 by !
L (x)=g x, x M. g · 2 Then, the action : G M M is free i↵for all g G, if g =1thenL has· no⇥ fixed! point. 2 6 g 834 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Another equivalent statement is that for every x M, the stabilizer G of x is reduced to the trivial group2 1 . x { } If H is a subgroup of G,obviouslyH acts freely on G (by multiplication on the left or on the right).
Before stating the main results of this section, observe that in the definition of a fibre bundle (Definition ??), the local trivialization maps are of the form
1 ' : ⇡ (U ) U F, ↵ ↵ ! ↵ ⇥ where the fibre F appears on the right.
In particular, for a principal fibre bundle ⇠,thefibreF is equal to the structure group G,andthisisthereason why G acts on the right on the total space E of ⇠ (see Proposition ??). 18.2. PROPER AND FREE ACTIONS 835 To be more precise, we could call such bundles right bun- dles.
We can also define left bundles as bundles whose local trivialization maps are of the form
1 ' : ⇡ (U ) F U . ↵ ↵ ! ⇥ ↵ Then, if ⇠ is a left principal bundle, the group G acts on E on the left.
Duistermaat and Kolk [19] address this issue at the end of their Appendix B, and prove the theorem stated below (Chapter 1, Section 11).
Beware that in Duistermaat and Kolk [19], this theorem is stated for left bundles,whichisadeparturefromthe prevalent custom of dealing with right bundles. 836 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS However, the weaker version that does not mention princi- pal bundles is usually stated for left actions; for instance, in Lee [35] (Chapter 9, Theorem 9.16).
We formulate both versions at the same time.
Theorem 18.7. Let M be a smooth manifold, G be a Lie group, and let : G M M be a left (resp. right) smooth action which· ⇥ is proper! and free. Then, M/G is a principal left G-bundle (resp. right G- bundle) of dimension dim M dim G. Moreover, the canonical projection ⇡ : G M/G is a submersion ! (which means that d⇡g is surjective for all g G), and there is a unique manifold structure on M/G2 with this property. 18.2. PROPER AND FREE ACTIONS 837 Theorem 18.7 has some interesting corollaries.
Because a closed subgroup H of a Lie group G is a Lie group, and because the action of a closed subgroup is free and proper, we get the following result (proofs can be found in Br¨ocker and tom Dieck [10] (Chapter I, Section 4) and in Duistermaat and Kolk [19] (Chapter 1, Section 11)).
Theorem 18.8. If G is a Lie group and H is a closed subgroup of G, then the right action of H on G de- fines a principal (right) H-bundle ⇠ =(G,⇡, G/H, H), where ⇡ : G G/H is the canonical projection. More- ! over, ⇡ is a submersion (which means that d⇡g is sur- jective for all g G), and there is a unique manifold structure on G/H2 with this property. 838 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS If (N,h)isaRiemannianmanifoldandifG is a Lie group acting by isometries on N,whichmeansthatforevery g G,thedi↵eomorphismL : N N is an isometry 2 g ! (d(Lg)p : TpN TpN is an isometry for all p M), then ⇡ : N !N/G can be made into a Riemannian2 submersion. !
Theorem 18.9. Let (N,h) be a Riemannian manifold and let : G N N be a smooth, free and proper proper action,· ⇥ with!G a Lie group acting by isometries of N. Then, there is a unique Riemannian metric g on M = N/G such that ⇡ : N M is a Riemannian submersion. !
As an example, if N = S2n+1,thenthegroupG = S1 = SU(1) acts by isometries on S2n+1,andweobtainasub- n mersion ⇡ : S2n+1 CP . ! 18.2. PROPER AND FREE ACTIONS 839 If we pick the canonical metric on S2n+1,byTheorem n 18.9, we obtain a Riemannian metric on CP known as the Fubini–Study metric.
Using Proposition 16.7, it is possible to describe the n geodesics of CP ;seeGallot,Hulin,Lafontaine[23](Chap- ter 2).
Another situation where a group action yields a Rieman- nian submersion is the case where a transitive action is reductive, considered in the next section.
We now consider the case of a smooth action : G M M,whereG is a discrete group (and M is amanifold).Inthiscase,wewillseethat· ⇥ ! ⇡ : M M/G is a Riemannian covering map. !
Assume G is a discrete group. By Proposition 18.5, the action : G M M is proper i↵for all x, y M, there exist· some⇥ open! sets, V and V in M,withx2 V x y 2 x and y Vy,sothatthesetK = g G Vx g Vy = is finite.2 { 2 | · \ 6 ;} 840 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS By Proposition 18.6, the action : G M M is proper · ⇥ ! i↵ GK = g G g K K = is finite for every compact subset{ 2 K of| M· . \ 6 ;}
It is shown in Lee [35] (Chapter 9) that the above condi- tions are equivalent to the conditions below.
Proposition 18.10. If : G M M is a smooth action of a discrete group· G ⇥on a! manifold M, then this action is proper i↵ (i) For every x M, there is some open subset V with x V such2 that gV V = for only finitely many g2 G. \ 6 ; 2 (ii) For all x, y M, if y/G x (y is not in the orbit of x), then2 there exist2 some· open sets V,W with x V and y W such that gV W =0for all g 2 G. 2 \ 2 18.2. PROPER AND FREE ACTIONS 841 The following proposition gives necessary and su cient conditions for a discrete group to act freely and properly often found in the literature (for instance, O’Neill [44], Berger and Gostiaux [6], and do Carmo [17], but beware that in this last reference Hausdor↵separation is not re- quired!).
Proposition 18.11. If X is a locally compact space and G is a discrete group, then a smooth action of G on X is free and proper i↵the following conditions hold: (i) For every x X, there is some open subset V with x V such2 that gV V = for all g G such that2 g =1. \ ; 2 6 (ii) For all x, y X, if y/G x (y is not in the orbit of x), then2 there exist2 some· open sets V,W with x V and y W such that gV W =0for all g 2 G. 2 \ 2 842 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Remark: The action of a discrete group satisfying the properties of Proposition 18.11 is often called “properly discontinuous.” However, as pointed out by Lee ([35], just before Proposition 9.18), this term is self-contradictory since such actions are smooth, and thus continuous!
Then, we have the following useful result.
Theorem 18.12. Let N be a smooth manifold and let G be discrete group acting smoothly, freely and properly on N. Then, there is a unique structure of smooth manifold on N/G such that the projection map ⇡ : N N/G is a covering map. ! 18.2. PROPER AND FREE ACTIONS 843 Real projective spaces are illustrations of Theorem 18.12.
Indeed, if N is the unit n-sphere Sn Rn+1 and G = I, I ,where I is the antipodal map,✓ then the con- ditions{ } of Proposition 18.11 are easily checked (since Sn is compact), and consequently the quotient
n n RP = S /G is a smooth manifold and the projection map ⇡ : Sn n ! RP is a covering map.
1 n The fiber ⇡ ([x]) of every point [x] RP consists of two antipodal points: x, x Sn. 2 2 844 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS The next step is to see how a Riemannian metric on N induces a Riemannian metric on the quotient manifold N/G.ThefollowingtheoremistheRiemannianversion of Theorem 18.12.
Theorem 18.13. Let (N,h) be a Riemannian mani- fold and let G be discrete group acting smoothly, freely and properly on N, and such that the map x x is an isometry for all G. Then there is a7! unique· structure of Riemannian2 manifold on M = N/G such that the projection map ⇡ : N M is a Riemannian covering map. !
Theorem 18.13 implies that every Riemannian metric g on the sphere Sn induces a Riemannian metric g on the n projective space RP ,insuchawaythattheprojection n ⇡ : Sn RP is a Riemannian covering. ! b In particular, if U is an open hemisphere obtained by n 1 removing its boundary S from a closed hemisphere, n then ⇡ is an isometry between U and its image RP n 1 n n 1 ⇡(S ) RP RP . ⇡ 18.2. PROPER AND FREE ACTIONS 845 In summary, given a Riemannian manifold N and a group G acting on N,Theorem18.9givesusamethodforob- taining a Riemannian manifold N/G such that ⇡ : N N/G is a Riemannian submersion ( : G !N N is a free and proper action and G acts by· isometries).⇥ !
Theorem 18.13 gives us a method for obtaining a Rie- mannian manifold N/G such that ⇡ : N N/G is a Riemannian covering ( : G N N is a free! and proper action of a discrete group· G⇥acting! by isometries).
In the next section, we show that Riemannian submer- sions arise from a reductive homogeneous space. 846 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS 18.3 Reductive Homogeneous Spaces
If : G M M is a smooth action of a Lie group G on amanifold· ⇥ M!,thenacertainclassofRiemannianmetrics on M is particularly interesting.
Recall that for every g G, Lg : M M is the di↵eo- morphism of M given by2 !
L (p)=g p, for all p M. g · 2
Definition 18.4. Given a smooth action : G M · ⇥ ! M,ametric , on M is G-invariant if Lg is an isometry for allh g iG;thatis,forallp M,wehave 2 2
d(L ) (u),d(L ) (v) = u, v for all u, v T M. h g p g p ip h ip 2 p
If the action is transitive, then for any fixed p0 M and for every p M,thereissomeg G such that p2= g p , 2 2 · 0 so it is su cient to require that d(Lg)p0 be an isometry for every g G. 2 18.3. REDUCTIVE HOMOGENEOUS SPACES 847 In this section, we consider transitive actions such that for any given p M,ifH = G is the stabilizer of p , 0 2 p0 0 then the Lie algebra g of G factors as a direct sum
g = h m, for some subspace m of g,whereh is the Lie algebra of H,andwithAd(m) m for all h H. h ✓ 2 This implies that there is an isomorphism between T M T (G/H)and (where o denotes the point in p0 ⇠= o m G/H corresponding to the coset H).
It is also the case that if H is “nice” (for example, com- pact), then M = G/H will carry G-invariant metrics, and that under such metrics, the projection ⇡ : G G/H is a Riemannian submersion. ! 848 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS But first, we need to explain how an action : G M · ⇥ ! M associates a vector field X⇤ on M to every vector X g in the Lie algebra of G. 2 Definition 18.5. Given a smooth action ': G M ⇥ ! M of a Lie group on a manifold M,foreveryX g,we 2 define the vector field X⇤ (or XM )onM called an action field or infinitesimal generator of the action correspond- ing to X,by
d X⇤(p)= (exp(tX) p) ,p M. dt · 2 t=0 For a fixed X g,themapt exp(tX)isacurve through 1 in G,sothemap2 t 7!exp(tX) p is a curve 7! · through p in M,andX⇤(p)isthetangentvectortothis curve at p.
For example, in the case of the adjoint action Ad: G g g,foreveryX g,wehave ⇥ ! 2 X⇤(Y )=[X, Y ], so X⇤ =ad(X). 18.3. REDUCTIVE HOMOGENEOUS SPACES 849
For any p0 M,thereisadi↵eomorphism G/G G2 p onto the orbit G p of p viewed as a p0 ! · 0 · 0 0 manifold, and it is not hard to show that for any p G p0, we have an isomorphism 2 ·
T (G p )= X⇤(p) X g ; p · 0 { | 2 } see Marsden and Ratiu [36] (Chapter 9, Section 9.3).
It can also be shown that the Lie algebra gp of the stabi- lizer Gp of p is given by
g = X g X⇤(p)=0 . p { 2 | } The following proposition is shown in Marsden and Ratiu [36] (Chapter 9, Proposition 9.3.6 and lemma 9.3.7). 850 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Proposition 18.14. Given a smooth action ': G M M of a Lie group on a manifold M, the following⇥ properties! hold: (1) For every X g, we have 2 (AdgX)⇤ = L⇤ 1X⇤ =(Lg) X⇤, for every g G; g ⇤ 2 Here, L⇤ 1 is the pullback associated with Lg 1, and g (Lg) is the push-forward associated with Lg. ⇤ (2) The map X X⇤ from g to X(M) is a Lie algebra anti-homomorphism,7! which means that
[X⇤,Y⇤]= [X, Y ]⇤ for all X, Y g. 2
If the metric on M is G-invariant (that is, every Lg is an isometry of M), then the vector field X⇤ is a Killing vector field on M for every X g. 2 18.3. REDUCTIVE HOMOGENEOUS SPACES 851 Given a pair (G, H)whereG is a Lie group and H is aclosedsubgroupofG,wewillneedthenotionofthe isotropy representation of G/H.RecallthatG acts on the left on G/H via
g (g H)=g g H, g ,g G. 1 · 2 1 2 1 2 2
For any g1 G,denoteby⌧g1 the di↵eomorphism ⌧ : G/H 2 G/H,givenby g1 ! ⌧ (g H)=g (g H)=g g H. g1 2 1 · 2 1 2 Denote the point in G/H corresponding to the coset 1H = H by o.Then,wehaveahomomorphism