Chapter 18 Manifolds Arising from Group Actionscis610/cis610-15-sl18.pdf · 2015. 4. 30. · F is...

Chapter 18

Manifolds Arising from GroupActions

18.1 Proper Maps

We saw in Chapter 3 that many manifolds arise from agroup action.

The scenario is that we have a smooth action' : G ⇥ M ! M of a Lie group G acting on a manifoldM (recall that an action ' is smooth if it is a smoothmap).

If G acts transitively on M , then for any point x 2 M ,if Gx is the stabilizer of x, then we know from Proposi-tion ?? that G/Gx is di↵eomorphic to M and that theprojection ⇡ : G ! G/Gx is a submersion.

823

824 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS

If the action is not transitive, then we consider the orbitspace 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 thatensure that M/G is Hausdor↵.

A su�cient condition can be given using the notion of aproper map.

Before we go any further, let us observe that the casewhere our action is transitive is subsumed by the moregeneral situation of an orbit space.

Indeed, if our action is transitive, for any x 2 M , we knowthat the stabilizer H = Gx of x is a closed subgroup ofG.

18.1. PROPER MAPS 825

Then, we can consider the right action G ⇥ H ! G ofH on G given by

g · h = gh, g 2 G, h 2 H.

The orbits of this (right) action are precisely the leftcosets gH of H .

Therefore, the set of left cosets G/H (the homogeneousspace induced by the action · : G ⇥ M ! M) is the setof orbits of the right action G ⇥ H ! G.

Observe that we have a transitive left action of G on thespace G/H of left cosets, given by

g1 · g2H = g1g2H.

The stabilizer of 1H is obviously H itself.


Thus, we recover the original transitive left action of Gon 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, is aspecial case of a free and proper action M ⇥ G ! G, inwhich case the orbit space M/G is a manifold, and theprojection ⇡ : 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 a continuous map ' : X ! Y is properi↵ for every topological space Z, the map' ⇥ id : X ⇥ Z ! Y ⇥ Z is a closed map (recall that fis a closed map i↵ the image of any closed set by f is aclosed set).

If we let Z be a one-point space, we see that a propermap is closed.

1It is not necessary to assume that X and Y are Hausdor↵ but, if X and/or Y are not Hausdor↵, wehave 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 , then the restriction of ' toF is proper (see Bourbaki, General Topology [9], Chapter1, Section 10).

The following result can be shown (see Bourbaki, GeneralTopology [9], Chapter 1, Section 10):

Proposition 18.1. A continuous map ' : X ! Y isproper i↵ ' is closed and if '�1(y) is compact forevery y 2 Y .

If ' is proper, it is easy to show that '�1(K) is compactin X whenever K is compact in Y .

Moreover, if Y is also locally compact, then we havethe following result (see Bourbaki, General Topology [9],Chapter 1, Section 10).


Proposition 18.2. If Y is locally compact, a map' : X ! Y is a proper map i↵ '�1(K) is compact inX whenever K is compact in Y

In particular, this is true if Y is a manifold since manifoldsare locally compact.

This explains why Lee [35] (Chapter 9) takes the propertystated in Proposition 18.2 as the definition of a propermap (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 groupG and a topological space M , a left action· : G ⇥ M ! M is proper if it is continuous and if themap

✓ : G ⇥ M �! M ⇥ M, (g, x) 7! (g · x, x)

is proper.

If H is a closed subgroup of G and if · : G ⇥ M ! M isa proper action, then the restriction of this action to His also proper.

If we let M = G, then G acts on itself by left translation,and the map ✓ : G ⇥ G ! G ⇥ G given by ✓(g, x) =(gx, x) is a homeomorphism, so it is proper.


It follows that the action · : H ⇥ G ! G of a closedsubgroup H of G on G (given by (h, g) 7! hg) is proper.

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 isproper (where G is Hausdor↵), then the orbit spaceM/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).


Proposition 18.4. Let · : G ⇥ M ! M be a properaction, with G Hausdor↵. For any x 2 M , let G ·x bethe orbit of x and let Gx be the stabilizer of x. Then:

(a) The map g 7! g · x is a proper map from G to M .

(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 Gis locally compact, then the action · : G ⇥ M ! M isproper i↵ for all x, y 2 M , there exist some open sets,Vx and Vy in M , with x 2 Vx and y 2 Vy, so that theclosure K of the set K = {g 2 G | Vx · g \ Vy 6= ;}, iscompact in G.


In particular, if G has the discrete topology, the abovecondition holds i↵ the sets {g 2 G | Vx · g \ Vy 6= ;} arefinite.

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 anycompact subset K of M we have:

(a) The set GK = {g 2 G | g · K \ K 6= ;} is closed.

(b) If M is locally compact, then the action is properi↵ GK is compact for every compact subset K ofM .


In the special case where G is discrete (and M is locallycompact), condition (b) says that the action is proper i↵GK is finite.

Remark: If G is a Hausdor↵ topological group and if His a subgroup of G, then it can be shown that the actionof G on G/H ((g1, g2H) 7! g1g2H) is proper i↵ H iscompact in G.

Definition 18.3. An action · : G ⇥ M ! M is free iffor all g 2 G and all x 2 M , if g 6= 1 then g · x 6= x.

An equivalent way to state that an action · : G⇥M ! Mis free is as follows. For every g 2 G, let Lg : M ! Mbe the di↵eomorphism of M given by

Lg(x) = g · x, x 2 M.

Then, the action · : G⇥M ! M is free i↵ for all g 2 G,if g 6= 1 then Lg has no fixed point.


Another equivalent statement is that for every x 2 M ,the stabilizer Gx of x is reduced to the trivial group {1}.

If H is a subgroup of G, obviously H acts freely on G(by multiplication on the left or on the right).

Before stating the main results of this section, observethat 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 ⇠, the fibre Fis equal to the structure group G, and this is the reasonwhy G acts on the right on the total space E of ⇠ (seeProposition ??).


To be more precise, we could call such bundles right bun-dles .

We can also define left bundles as bundles whose localtrivialization maps are of the form

'↵ : ⇡�1(U↵) ! F ⇥ U↵.

Then, if ⇠ is a left principal bundle, the group G acts onE on the left .

Duistermaat and Kolk [19] address this issue at the endof their Appendix B, and prove the theorem stated below(Chapter 1, Section 11).

Beware that in Duistermaat and Kolk [19], this theoremis stated for left bundles , which is a departure from theprevalent custom of dealing with right bundles.


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 bea 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 dimM � dimG. Moreover, thecanonical projection ⇡ : G ! M/G is a submersion(which means that d⇡g is surjective for all g 2 G),and there is a unique manifold structure on M/G withthis property.


Theorem 18.7 has some interesting corollaries.

Because a closed subgroup H of a Lie group G is a Liegroup, and because the action of a closed subgroup is freeand proper, we get the following result (proofs can befound in Brocker and tom Dieck [10] (Chapter I, Section4) and in Duistermaat and Kolk [19] (Chapter 1, Section11)).

Theorem 18.8. If G is a Lie group and H is a closedsubgroup 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 2 G), and there is a unique manifoldstructure on G/H with this property.


If (N, h) is a Riemannian manifold and if G is a Lie groupacting by isometries on N , which means that for everyg 2 G, the di↵eomorphism Lg : N ! N is an isometry(d(Lg)p : TpN ! TpN is an isometry for all p 2 M),then ⇡ : N ! N/G can be made into a Riemanniansubmersion.

Theorem 18.9. Let (N, h) be a Riemannian manifoldand let · : G ⇥ N ! N be a smooth, free and properproper action, with G a Lie group acting by isometriesof N . Then, there is a unique Riemannian metric gon M = N/G such that ⇡ : N ! M is a Riemanniansubmersion.

As an example, if N = S2n+1, then the group G = S1 =SU(1) acts by isometries on S2n+1, and we obtain a sub-mersion ⇡ : S2n+1 ! CPn.


If we pick the canonical metric on S2n+1, by Theorem18.9, we obtain a Riemannian metric on CPn known asthe Fubini–Study metric.

Using Proposition 16.7, it is possible to describe thegeodesics ofCPn; see Gallot, Hulin, Lafontaine [23] (Chap-ter 2).

Another situation where a group action yields a Rieman-nian submersion is the case where a transitive action isreductive, considered in the next section.

We now consider the case of a smooth action· : G ⇥ M ! M , where G is a discrete group (and M isa manifold). In this case, we will see that ⇡ : M ! M/Gis a Riemannian covering map.

Assume G is a discrete group. By Proposition 18.5, theaction · : G ⇥ M ! M is proper i↵ for all x, y 2 M ,there exist some open sets, Vx and Vy in M , with x 2 Vx

and y 2 Vy, so that the set K = {g 2 G | Vx ·g\Vy 6= ;}is finite.


By Proposition 18.6, the action · : G⇥M ! M is properi↵ GK = {g 2 G | g · K \ K 6= ;} is finite for everycompact subset K of M .

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 smoothaction of a discrete group G on a manifold M , thenthis action is proper i↵

(i) For every x 2 M , there is some open subset Vwith x 2 V such that gV \ V 6= ; for only finitelymany g 2 G.

(ii) For all x, y 2 M , if y /2 G ·x (y is not in the orbitof x), then there exist some open sets V, W withx 2 V and y 2 W such that gV \ W = 0 for allg 2 G.


The following proposition gives necessary and su�cientconditions for a discrete group to act freely and properlyoften found in the literature (for instance, O’Neill [44],Berger and Gostiaux [6], and do Carmo [17], but bewarethat in this last reference Hausdor↵ separation is not re-quired!).

Proposition 18.11. If X is a locally compact spaceand G is a discrete group, then a smooth action of Gon X is free and proper i↵ the following conditionshold:

(i) For every x 2 X, there is some open subset V withx 2 V such that gV \ V = ; for all g 2 G suchthat g 6= 1.

(ii) For all x, y 2 X, if y /2 G · x (y is not in the orbitof x), then there exist some open sets V, W withx 2 V and y 2 W such that gV \ W = 0 for allg 2 G.


Remark: The action of a discrete group satisfying theproperties of Proposition 18.11 is often called “properlydiscontinuous.” However, as pointed out by Lee ([35], justbefore Proposition 9.18), this term is self-contradictorysince such actions are smooth, and thus continuous!

Then, we have the following useful result.

Theorem 18.12. Let N be a smooth manifold andlet G be discrete group acting smoothly, freely andproperly on N . Then, there is a unique structure ofsmooth manifold on N/G such that the projection map⇡ : N ! N/G is a covering map.


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

RPn = Sn/G

is a smooth manifold and the projection map ⇡ : Sn !RPn is a covering map.

The fiber ⇡�1([x]) of every point [x] 2 RPn consists oftwo antipodal points: x, �x 2 Sn.


The next step is to see how a Riemannian metric on Ninduces a Riemannian metric on the quotient manifoldN/G. The following theorem is the Riemannian versionof Theorem 18.12.

Theorem 18.13. Let (N, h) be a Riemannian mani-fold and let G be discrete group acting smoothly, freelyand properly on N , and such that the map x 7! � · xis an isometry for all � 2 G. Then there is a uniquestructure of Riemannian manifold on M = N/G suchthat the projection map ⇡ : N ! M is a Riemanniancovering map.

Theorem 18.13 implies that every Riemannian metric gon the sphere Sn induces a Riemannian metric bg on theprojective space RPn, in such a way that the projection⇡ : Sn ! RPn is a Riemannian covering.

In particular, if U is an open hemisphere obtained byremoving its boundary Sn�1 from a closed hemisphere,then ⇡ is an isometry between U and its image RPn �⇡(Sn�1) ⇡ RPn � RPn�1.


In summary, given a Riemannian manifold N and a groupG acting on N , Theorem 18.9 gives us a method for ob-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 actsby isometries).

Theorem 18.13 gives us a method for obtaining a Rie-mannian manifold N/G such that ⇡ : N ! N/G is aRiemannian covering (· : G⇥N ! N is a free and properaction of a discrete group G acting by isometries).

In the next section, we show that Riemannian submer-sions arise from a reductive homogeneous space.


18.3 Reductive Homogeneous Spaces

If · : G⇥M ! M is a smooth action of a Lie group G ona manifold M , then a certain class of Riemannian metricson M is particularly interesting.

Recall that for every g 2 G, Lg : M ! M is the di↵eo-morphism of M given by

Lg(p) = g · p, for all p 2 M.

Definition 18.4. Given a smooth action · : G ⇥ M !M , a metric h�, �i on M is G-invariant if Lg is anisometry for all g 2 G; that is, for all p 2 M , we have

hd(Lg)p(u), d(Lg)p(v)ip = hu, vip for all u, v 2 TpM.

If the action is transitive, then for any fixed p0 2 M andfor every p 2 M , there is some g 2 G such that p = g ·p0,so it is su�cient to require that d(Lg)p0 be an isometryfor every g 2 G.

18.3. REDUCTIVE HOMOGENEOUS SPACES 847

In this section, we consider transitive actions such thatfor any given p0 2 M , if H = Gp0 is the stabilizer of p0,then the Lie algebra g of G factors as a direct sum

g = h � m,

for some subspace m of g, where h is the Lie algebra ofH , and with Adh(m) ✓ m for all h 2 H .

This implies that there is an isomorphism betweenTp0M

⇠= To(G/H) and m (where o denotes the point inG/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.


But first, we need to explain how an action · : G ⇥ M !M associates a vector field X⇤ on M to every vectorX 2 g in the Lie algebra of G.

Definition 18.5. Given a smooth action ' : G ⇥ M !M of a Lie group on a manifold M , for every X 2 g, wedefine the vector field X⇤ (or XM) on M called an actionfield or infinitesimal generator of the action correspond-ing to X , by

X⇤(p) =d

dt(exp(tX) · p)

��t=0

, p 2 M.

For a fixed X 2 g, the map t 7! exp(tX) is a curvethrough 1 in G, so the map t 7! exp(tX) · p is a curvethrough p in M , and X⇤(p) is the tangent vector to thiscurve at p.

For example, in the case of the adjoint actionAd: G ⇥ g ! g, for every X 2 g, we have

X⇤(Y ) = [X, Y ],

so X⇤ = ad(X).


For any p0 2 M , there is a di↵eomorphismG/Gp0 ! G · p0 onto the orbit G · p0 of p0 viewed as amanifold, and it is not hard to show that for any p 2 G·p0,we have an isomorphism

Tp(G · p0) = {X⇤(p) | X 2 g};

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

gp = {X 2 g | X⇤(p) = 0}.

The following proposition is shown in Marsden and Ratiu[36] (Chapter 9, Proposition 9.3.6 and lemma 9.3.7).


Proposition 18.14. Given a smooth action' : G⇥M ! M of a Lie group on a manifold M , thefollowing properties hold:

(1) For every X 2 g, we have

(AdgX)⇤ = L⇤g�1X

⇤ = (Lg)⇤X⇤, for every g 2 G;

Here, L⇤g�1 is the pullback associated with Lg�1, and

(Lg)⇤ is the push-forward associated with Lg.

(2) The map X 7! X⇤ from g to X(M) is a Lie algebraanti-homomorphism, which means that

[X⇤, Y ⇤] = �[X, Y ]⇤ for all X, Y 2 g.

If the metric on M is G-invariant (that is, every Lg isan isometry of M), then the vector field X⇤ is a Killingvector field on M for every X 2 g.


Given a pair (G, H) where G is a Lie group and H isa closed subgroup of G, we will need the notion of theisotropy representation of G/H . Recall that G acts onthe left on G/H via

g1 · (g2H) = g1g2H, g1, g2 2 G.

For any g1 2 G, denote by ⌧g1 the di↵eomorphism⌧g1 : G/H ! G/H , given by

⌧g1(g2H) = g1 · (g2H) = g1g2H.

Denote the point in G/H corresponding to the coset1H = H by o. Then, we have a homomorphism

�G/H : H ! GL(To(G/H)),

given by

�G/H(h) = (d⌧h)o, for all h 2 H.


The homomorphism �G/H is called the isotropy repre-sentation of the homogeneous space G/H .

Actually, we have an isomorphism

To(G/H) ⇠= g/h

induced by d⇡1 : g ! To(G/H), where ⇡ : G ! G/H isthe canonical projection.

Proposition 18.15. Let (G, H) be a pair where G isa Lie group and H is a closed subgroup of G. Therepresentations �G/H : H ! GL(To(G/H)) andAdG/H : H ! GL(g/h) are equivalent; this means thatfor every h 2 H, we have the commutative diagram

g/hAd

G/Hh //

d⇡1

✏✏

g/h

d⇡1

✏✏

To(G/H)(d⌧h)o

// To(G/H),

where ⇡ : G ! G/H is the canonical projection.


The significance of Proposition 18.15 is that it can beused to show that G/H has some G-invariant metric i↵the closure of AdG/H(H) is compact in GL(g/h).

The proof is very similar to the proof of Theorem 17.5.

Furthermore, this metric is unique up to a scalar if theisotropy representation is irreducible.

We now consider the case where there is a subspace m ofg such that g = h � m, so that g/h is isomorphic to m.

Definition 18.6. Let (G, H) be a pair where G is aLie group and H is a closed subgroup of G. We say thatthe homogeneous space G/H is reductive if there is somesubspace m of g such that

g = h � m,

and

Adh(m) ✓ m for all h 2 H.


Observe that unlike h, which is a Lie subalgebra of g,the subspace m is not necessarily closed under the Liebracket, so in general it is not a Lie algebra.

Also, since m is finite-dimensional and since Adh is anisomorphism, we actually have Adh(m) = m.

Since g/h is isomorphic to m, by previous considerations,the map d⇡1 : g ! To(G/H) restricts to an isomorphismbetween m and To(G/H) (where o denotes the point inG/H corresponding to the coset H).

For any X 2 g, we can express d⇡1(X) in terms of thevector field X⇤ introduced in Definition 18.5.


Indeed to compute d⇡1(X), we can use the curve t 7!exp(tX), and we have

d⇡1(X) =d

dt(⇡(exp(tX)))

��t=0

=d

dt(exp(tX)H)

��t=0

= X⇤o .

For every X 2 h, since the curve t 7! exp(tX)H in G/Hhas the constant value o, we see that

Ker d⇡1 = h,

and thus, the restriction of d⇡1 to m is an isomorphismonto T0(G/H), given by X 7! X⇤

o .

Also, for every X 2 g, since g = h � m, we can writeX = X

h

+Xm

, for some unique Xh

2 h and some uniqueX

m

2 m, and

d⇡1(X) = d⇡1(Xm

) = X⇤o .


We use the isomorphism d⇡1 to transfer any inner producth�, �i

m

on m to an inner product h�, �i on To(G/H),and vice-versa, by stating that

hX, Y im

= hX⇤0 , Y

⇤0 i, for all X, Y 2 m;

that is, by declaring d⇡1 to be an isometry between m

and To(G/H).

If the metric on G/H is G-invariant, then the map p 7!exp(tX)·p = exp(tX)aH (with p = aH 2 G/H , a 2 G)is an isometry of G/H for every t 2 R, so X⇤ is a Killingvector field.

A key property of a reductive space G/H is that there isa criterion for the existence of a G-invariant metrics onG/H in terms of Ad(H)-invariant inner products on m.


Recall that Adg1(g2) = g1g2g�11 . Then, following O’Neill

[44] (see Proposition 22, Chapter 11), observe that

⌧h � ⇡ = ⇡ � Adh for all h 2 H,

since h 2 H implies that h�1H = H , so for all g 2 G,

(⌧h � ⇡)(g) = hgH = hgh�1H = (⇡ � Adh)(g).

By taking derivatives at 1, we get

(d⌧h)o � d⇡1 = d⇡1 � Adh.


Proposition 18.16. Let (G, H) be a pair of Lie groupsdefining a reductive homogeneous space M = G/H,with reductive decomposition g = h � m. The follow-ing properties hold:

(1) The isotropy representation�G/H : H ! GL(To(G/H)) is equivalent to therepresentation AdG : H ! GL(m) (where Adh isrestricted to m for every h 2 H): this means thatfor every h 2 H, we have the commutative dia-gram

m

AdGh //

d⇡1

✏✏

m

d⇡1

✏✏

To(G/H)(d⌧h)o

// To(G/H),

where ⇡ : G ! G/H is the canonical projection.


(2) By making d⇡1 an isometry between m and To(G/H)(as explained above), there is a one-to-one corre-spondence between G-invariant metrics on G/Hand Ad(H)-invariant inner products on m (innerproducts h�, �i

m

such that

hu, vim

= hAdh(u),Adh(v)im,

for all h 2 H and all u, v 2 m).

(3) The homogeneous space G/H has some G-invariantmetric i↵ the closure of AdG(H) is compact inGL(m). Furthermore, if the representationAdG : H ! GL(m) is irreducible, then such a met-ric is unique up to a scalar. In particular, if H iscompact, then a G-invariant metric on G/H al-ways exists.


A proof of the following Proposition is given in O’Neill[44] (Chapter 11, Lemma 24).

Proposition 18.17. Let (G, H) be a pair of Lie groupsdefining a reductive homogeneous space M = G/H,with reductive decomposition g = h�m. If m has someAd(H)-invariant inner product h�, �i

m

, then for anyleft invariant metric on G extending h�, �i

m

suchthat h and m are orthogonal, the map ⇡ : G ! G/His a Riemannian submersion.

By Proposition 16.7, a Riemannian submersion carrieshorizontal geodesics to geodesics. For naturally reductivehomogeneous manifolds (see next), the geodesics can bedetermined.


When M = G/H is a reductive homogeneous space thathas a G-invariant metric, it is possible to give an expres-sion for (rX⇤Y ⇤)o (where X⇤ and Y ⇤ are the vector fieldscorresponding to X, Y 2 m).

If X⇤, Y ⇤, Z⇤ are the Killing vector fields associated withX, Y, Z 2 m, then it can be shown that

2hrX⇤Y ⇤, Z⇤i = �h[X, Y ]⇤, Z⇤i � h[X, Z]⇤, Y ⇤i� h[Y, Z]⇤, X⇤i.

The problem is that the vector field rX⇤Y ⇤ is not neces-sarily of the form W ⇤ for some W 2 g. However, we canfind its value at o.

By evaluating at o and using the fact that X⇤o = (X⇤

m

)ofor any X 2 g, we obtain

2h(rX⇤Y ⇤)o, Z⇤o i + h([X, Y ]⇤

m

)o, Z⇤o i

= h([Z, X ]⇤m

)o, Y⇤o i + h([Z, Y ]⇤

m

)o, X⇤o i.


Consequently,

(rX⇤Y ⇤)o = �1

2([X, Y ]⇤

m

)o + U(X, Y )⇤o,

where [X, Y ]m

is the component of [X, Y ] on m andU(X, Y ) is determined by

2hU(X, Y ), Zi = h[Z, X ]m

, Y i + hX, [Z, Y ]m

i,

for all Z 2 m.

Here, we are using the isomorphism X 7! X⇤0 between m

and To(G/H) and the fact that the inner product on m

is chosen so that m and To(G/H) are isometric


Since the term U(X, Y ) clearly complicates matters, it isnatural to make the following definition, which is equiva-lent to requiring that U(X, Y ) = 0 for all X, Y 2 m.

Definition 18.7. A homogeneous space G/H is natu-rally reductive if it is reductive, if it has a G-invariantmetric, and if

h[X, Z]m

, Y i = hX, [Z, Y ]m

i, for all X, Y, Z 2 m.

If G/H is a naturally reductive homogeneous space, then

(rXY )⇤o = �1

2([X, Y ]⇤

m

)o.


Since G/H has a G-invariant metric, X⇤, Y ⇤ are Killingvector fields on G/H .

We can now find the geodesics on a naturally reductivehomogenous space.

Indeed, if M = (G, H) is a reductive homogeneous spaceand M has a G-invariant metric, then there is an Ad(H)-invariant inner product h�, �i

m

on m.

Pick any inner product h�, �ih

on h, and define an innerproduct on g = h�m by setting h andm to be orthogonal.

Then, we get a left-invariant metric on G for which theelements of h are vertical vectors and the elements of mare horizontal vectors.


Observe that in this situation, the condition for beingnaturally reductive extends to left-invariant vector fieldson G induced by vectors in m.

Since (dLg)1 : g ! TgG is a linear isomorphism for allg 2 G, the direct sum decomposition g = h�m yields adirect sum decomposition TgG = (dLg)1(h)� (dLg)1(m).

Given a left-invariant vector field XL induced by a vectorX 2 g, if X = X

h

+ Xm

is the decomposition of X ontoh � m, we obtain a decomposition

XL = XLh

+ XLm

,

into a left-invariant vector field XLh

2 h

L and a left-invariant vector field XL

m

2 m

L, with

XLh

(g) = (dLg)1(Xh

), XLm

= (dLg)1(Xm

).


Since the (dLg)1 are isometries, if h andm are orthogonal,so are (dLg)1(h) and (dLg)1(m), and so XL

h

and XLm

areorthogonal vector fields.

Then, it is easy to show that the following condition holdsfor left-invariant vector fields:

h[XL, ZL]m

, Y Li = hXL, [ZL, Y L]m

i,for all XL, Y L, ZL 2 m

L.

Recall that the left action of G on G/H is given by g1 ·g2H = g1g2H , and that o denotes the coset 1H .

Proposition 18.18. If M = G/H is a naturally re-ductive homogeneous space, for every G-invariant met-ric on G/H, for every X 2 m, the geodesic �d⇡1(X)

through o is given by

�d⇡1(X)(t) = ⇡ � exp(tX) = exp(tX) · o, for all t 2 R.


Proposition 18.18 shows that the geodesics in G/H aregiven by the obits of the one-parameter groups (t 7!exp tX) generated by the members of m.

We can also obtain a formula for the geodesic throughevery point p = gH 2 G/H .

We find that the geodesic through p = gH with initialvelocity (AdgX)⇤o = (Lg)⇤X⇤

o is given by

t 7! g exp(tX) · o.

An important corollary of Proposition 18.18 is that nat-urally reductive homogeneous spaces are complete.

Indeed, the one-parameter group t 7! exp(tX) is definedfor all t 2 R.


One can also figure out a formula for the sectional curva-ture (see (O’Neill [44], Chapter 11, Proposition 26).

Under the identification of m and To(G/H) given by therestriction of d⇡1 to m, we have

hR(X, Y )X, Y i = 1

4h[X, Y ]

m

, [X, Y ]m

i+ h[[X, Y ]

h

, X ]m

, Y i, for all X, Y 2 m.

Conditions on a homogeneous space that ensure that sucha space is naturally reductive are obviously of interest.Here is such a condition.


Proposition 18.19. Let M = G/H be a homoge-neous space with G a connected Lie group, assumethat g admits an Ad(G)-invariant inner product h�, �i,and let m = h

? be the orthogonal complement of h

with respect to h�, �i. Then, the following propertieshold:

(1) The space G/H is reductive with respect to the de-composition g = h � m.

(2) Under the G-invariant metric induced by h�, �i,the homogeneous space G/H is naturally reductive.

(3) The sectional curvature is determined by

hR(X, Y )X, Y i = 1

4h[X, Y ]

m

, [X, Y ]m

i+ h[X, Y ]

h

, [X, Y ]h

i.


Recall a Lie group G is said to be semisimple if its Liealgebra g is semisimple.

From Theorem 17.22, a Lie algebra g is semisimple i↵its Killing form B is nondegenerate, and from Theorem17.24, a connected Lie group G is compact and semisim-ple i↵ its Killing form B is negative definite.

By Proposition 17.21, the Killing form is Ad(G)-invariant.

Thus, for any connected compact semisimple Lie groupG, for any constant c > 0, the bilinear form �cB is anAd(G)-invariant inner product on g.

Then, as a corollary of Proposition 18.19, we obtain thefollowing result.


Proposition 18.20. Let M = G/H be a homoge-neous space such that G is a connected compactsemisimple group. Then, under any inner producth�, �i on g given by �cB, where B is the Killingform of g and c > 0 is any positive real, the spaceG/H is naturally reductive with respect to the decom-position g = h � m, where m = h

? be the orthogonalcomplement of h with respect to h�, �i. The sectionalcurvature is non-negative.

A homogeneous space as in Proposition 18.20 is called anormal homogeneous space .

Since SO(n) is semisimple and compact for n � 3, theStiefel manifolds S(k, n) and the Grassmannian mani-folds G(k, n) are examples of homogeneous spaces satisy-ing the assumptions of Proposition 18.20 (with an innerproduct induced by a scalar factor of the Killing form onSO(n)).


Therefore, Stiefel manifolds S(k, n) and Grassmannianmanifolds G(k, n) are naturally reductive homogeneousspaces for n � 3 (under the reduction g = h�m inducedby the Killing form).

If n = 2, then SO(2) is an abelian group, and thus notsemisimple. However, in this case, G(1, 2) = RP(1) ⇠=SO(2)/S(O(1) ⇥ O(1)) ⇠= SO(2)/O(1), and S(1, 2) =S1 ⇠= SO(2)/SO(1) ⇠= SO(2).

These are special cases of symmetric cases discussed inSection 18.5.

In the first case, H = S(O(1)⇥O(1)), and in the secondcase, H = SO(1). In both cases,

h = (0),

and we can pickm = so(2),

which is trivially Ad(H)-invariant.


In Section 18.5, we show that the inner product on so(2)given by

hX, Y i = tr(X>Y )

is Ad(H)-invariant, and with the induced metric, RP(1)and S1 ⇠= SO(2) are naturally reductive manifolds.

For n � 3, we have S(1, n) = Sn�1 and S(n � 1, n) =SO(n), which are symmetric spaces.

On the other hand, S(k, n) it is not a symmetric space if2 k n � 2. A justification is given in Section 18.6.

Since the Grassmannian manifolds G(k, n) have morestructure (they are symmetric spaces), let us first con-sider the Stiefel manifolds S(k, n) in more detail.


Readers may find material from Absil, Mahony and Sepul-chre [1], especially Chapters 1 and 2, a good complementto our presentation, which uses more advanced concepts(reductive homogeneous spaces).

We may assume that n � 3. Then, for 1 k n � 1,we have S(k, n) ⇠= G/H , with G = SO(n) andH ⇠= SO(n � k).

Then,

H =

⇢✓I 00 R

◆ �� R 2 SO(n � k)

�,

h =

⇢✓0 00 S

◆ �� S 2 so(n � k)

�,

and the points of G/H ⇠= S(k, n) are the cosets QH ,with Q 2 SO(n); that is, the equivalence classes [Q],with the equivalence relation on SO(n) given by

Q1 ⌘ Q2 i↵ Q2 = Q1eR, for some eR 2 H.


If we write Q = [Y Y?], where Y consists of the first kcolumns of Q and Y? consists of the last n � k columnsof Q, it is clear that [Q] is uniquely determined by Y .

In fact, if Pn,k denotes the projection matrix consistingof the first k columns of the identity matrix In,

Pn,k =

✓Ik

0n�k,k

◆,

for any Q = [Y Y?], the unique representative Y of theequivalence class [Q] is given by

Y = QPn,k.

Observe that Y? is characterized by the fact that Q =[Y Y?] is orthogonal, namely, Y Y > + Y?Y >

? = I .


We also know that the Killing form on so(n) is given byB(X, Y ) = (n � 2)tr(XY ). Now, observe that if we let

m =

⇢✓T �A>

A 0

◆ �� T 2 so(k), A 2 Mn�k,k

�,

then

tr

✓✓0 00 S

◆✓T �A>

A 0

◆◆= tr

✓0 0

SA 0

◆= 0.

Furthermore, it is clear that dim(m) = dim(g)� dim(h),so m is the orthogonal complement of h with respect tothe Killing form.


If X, Y 2 m, with

X =

✓S �A>

A 0

◆, Y =

✓T �B>

B 0

◆,

observe that

tr

✓✓S �A>

A 0

◆✓T �B>

B 0

◆◆= tr(ST ) � 2tr(A>B),

and since S> = �S, we have

tr(ST ) � 2tr(A>B)) = �tr(S>T ) � 2tr(A>B),

so we define an inner product on m by

hX, Y i = �1

2tr(XY ) =

1

2tr(S>T ) + tr(A>B).

We give h the same inner product.


Observe that there is a bijection between the space m ofn ⇥ n matrices of the form

X =

✓S �A>

A 0

◆

and the set of n ⇥ k matrices of the form

� =

✓SA

◆,

but the inner product given by

h�1,�2i = tr(�>1 �2)

yields

h�1,�2i = tr(S>T ) + tr(A>B),

without the factor 1/2 in front of S>T . These metricsare di↵erent.


The vector space m is the tangent space ToS(k, n) toS(k, n) at o = [I ], the coset of the point correspondingto I .

For any other point [Q] 2 G/H ⇠= S(k, n), the tangentspace T[Q]S(k, n) is given by

T[Q]S(k, n) =

⇢Q

✓S �A>

A 0

◆ �� S 2 so(k), A 2 Mn�k,k

�.

Skip upon first reading ~

Using the decomposition Q = [Y Y?], where Y consistsof the first k columns of Q and Y? consists of the lastn � k columns of Q, we have

[Y Y?]

✓S �A>

A 0

◆= [Y S + Y?A � Y A>].


If we write X = Y S + Y?A, since Y and Y? are partsof an orthogonal matrix, we have Y >

? Y = 0, Y >Y = I ,and Y >

? Y? = I , so we can recover A from X and Y? andS from X and Y , by

Y >? X = Y >

? (Y S + Y?A) = A,

and

Y >X = Y >(Y S + Y?A) = S.

Since A = Y >? X , we also have A>A = X>Y?Y >

? X =X>(I � Y Y >)X .

Therefore, given Q = [Y Y?], the matrices

[Y Y?]

✓S �A>

A 0

◆

are in one-to-one correspondence with the n⇥k matricesof the form Y S + Y?A.


Since Y describes an element of S(k, n), we can say thatthe tangent vectors to S(k, n) at Y are of the form

X = Y S + Y?A, S 2 so(k), A 2 Mn�k,k.

Since [Y Y?] is an orthogonal matrix, we get Y >X = S,which shows that Y >X is skew-symmetric.

Conversely, since the columns of [Y Y?] form an orthonor-mal basis of Rn, every n ⇥ k matrix X can be writtenas

X =�Y Y?

�✓SA

◆= Y S + Y?A,

where S 2 Mk,k and A 2 Mn�k,k(R), and if Y >X isskew-symmetric, then S = Y >X is also skew-symmetric.


Therefore, the tangent vectors to S(k, n) at Y are thevectors X 2 Mn,k(R) such that Y >X is skew-symmetric.

This is the description given in Edelman, Arias and Smith[20].

Another useful observation is that if X = Y S + Y?A isa tangent vector to S(k, n) at Y , then the square normhX, Xi (in the canonical metric) is given by

hX, Xi = tr⇣X>⇣I � 1

2Y Y >

⌘X⌘.

By polarization, we find that the canonical metric is givenby

hX1, X2i = tr⇣X>

1

⇣I � 1

2Y Y >

⌘X2

⌘.


In that paper, it is also observed that because Y? has rankn�k (since Y >

? Y? = I), for every (n�k)⇥k matrix A,there is some n⇥k matrix C such that A = Y >

? C (everycolumn of A must be a linear combination of the n � kcolumns of Y?, which are linearly independent).

Thus, we have

Y S + Y?A = Y S + Y?Y >? C = Y S + (I � Y Y >)C.

End ~

By Proposition 18.18, the geodesic through o with initialvelocity

X =

✓S �A>

A 0

◆

is given by

�(t) = exp

✓t

✓S �A>

A 0

◆◆Pn,k.


This is not a very explicit formula. It is possible to dobetter, see Edelman, Arias and Smith [20] for details.

Let us consider the case where k = n � 1, which issimpler.

If k = n � 1, then n � k = 1, so S(n � 1, n) = SO(n),H ⇠= SO(1) = {1}, h = (0) and m = so(n).

The inner product on so(n) is given by

hX, Y i = �1

2tr(XY ) =

1

2tr(X>Y ), X, Y 2 so(n).

Every matrix X 2 so(n) is a skew-symmetric matrix, andwe know that every such matrix can be written as X =P>DP , where P is orthogonal and where D is a blockdiagonal matrix whose blocks are either a 1-dimensionalblock consisting of a zero, of a 2 ⇥ 2 matrix of the form

Dj =

✓0 �✓j

✓j 0

◆,

with ✓j > 0.


Then, eX = P>eDP = P>⌃P , where ⌃ is a block diago-nal matrix whose blocks are either a 1-dimensional blockconsisting of a 1, of a 2 ⇥ 2 matrix of the form

Dj =

✓cos ✓j � sin ✓j

sin ✓j cos ✓j

◆.

We also know that every matrix R 2 SO(n) can bewritten as

R = eX,

for some matrix X 2 so(n) as above, with 0 < ✓j ⇡.

Then, we can give a formula for the distance d(I, Q) be-tween the identity matrix and any matrix Q 2 SO(n).


Since the geodesics from I through Q are of the fom

�(t) = etX with eX = Q,

and since the length L(�) of the geodesic from I to eX is

L(�) =

Z 1

0

h�0(t), �0(t)i12dt.

We find that

d(I, Q) = (✓21 + · · · + ✓2

m)12 ,

where ✓1, . . . , ✓m are the angles associated with the eigen-values e±i✓1, . . . , e±i✓m of Q distinct from 1, and with0 < ✓j ⇡.


If Q, R 2 SO(n), then

d(Q, R) = (✓21 + · · · + ✓2

m)12 ,

where ✓1, . . . , ✓m are the angles associated with the eigen-values e±i✓1, . . . , e±i✓m of Q�1R = Q>R distinct from 1,and with 0 < ✓j ⇡.

Remark: SinceX> = �X , the square distance d(I, Q)2

can also be expressed as

d(I, Q)2 = �1

2min

X|eX=Qtr(X2),

or even (with some abuse of notation, since log is multi-valued) as

d(I, Q)2 = �1

2min tr((logQ)2).


In the other special case where k = 1, we haveS(1, n) = Sn�1, H ⇠= SO(n � 1),

h =

⇢✓0 00 S

◆ �� S 2 so(n � 1)

�,

and

m =

⇢✓0 �u>

u 0

◆ �� u 2 Rn�1

�.

Therefore, there is a one-to-one correspondence betweenm and Rn�1.

Given any Q 2 SO(n), the equivalence class [Q] of Qis uniquely determined by the first column of Q, and weview it as a point on Sn�1.


If we let kuk =p

u>u, we leave it as an exercise to provethat for any

X =

✓0 �u>

u 0

◆,

we have

etX =

0

@cos(kuk t) � sin(kuk t) u>

kuk

sin(kuk t) ukuk I + (cos(kuk t) � 1) uu>

kuk2

1

A .

Consequently (under the identification of Sn�1 with thefirst column of matrices Q 2 SO(n)), the geodesic �through e1 (the column vector corresponding to the pointo 2 Sn�1) with initial tangent vector u is given by

�(t) =

cos(kuk t)

sin(kuk t) ukuk

!= cos(kuk t)e1 + sin(kuk t)

u

kuk,

where u 2 Rn�1 is viewed as the vector in Rn whose firstcomponent is 0.


Then, we have

�0(t) = kuk✓

� sin(kuk t)e1 + cos(kuk t)u

kuk

◆,

and we find the that the length L(�)(✓) of the geodesicfrom e1 to the point

p(✓) = �(✓) = cos(kuk ✓)e1 + sin(kuk ✓)u

kuk

is given by

L(�)(✓) =

Z ✓

0

h�0(t), �0(t)i12 dt = ✓ kuk .


Since

he1, p(✓)i = cos(✓ kuk),

we see that for a unit vector u and for any angle ✓ suchthat 0 ✓ ⇡, the length of the geodesic from e1 top(✓) can be expressed as

L(�)(✓) = ✓ = arccos(he1, pi);

that is, the angle between the unit vectors e1 and p. Thisis a generalization of the distance between two points ona circle.


Geodesics can also be determined in the general casewhere 2 k n � 2; we follow Edelman, Arias andSmith [20], with one change because some point in thatpaper requires some justification which is not provided.

Given a point [Y Y?] 2 S(k, n), and given and any tan-gent vector X = Y S + Y?A, we need to compute

�(t) = [Y Y?] exp

✓t

✓S �A>

A 0

◆◆Pn,k.

We can compute this exponential if we replace the matrixby a more “regular matrix,” and for this, we use a QR-decomposition of A. Let

A = U

✓R0

◆

be a QR-decomposition of A, with U an orthogonal (n�k) ⇥ (n � k) matrix and R an upper triangular k ⇥ kmatrix.


We can write U = [U1 U2], where U1 consists of the firstk columns on U and U2 of the last n � 2k columns of U(if 2k n).

We have

A = U1R,

and we can write

✓S �A>

A 0

◆=

✓I 00 U1

◆✓S �R>

R 0

◆✓I 00 U>

1

◆.

Then, we find

�(t) = [Y Y?U1] exp t

✓S �R>

R 0

◆✓Ik

0

◆.

This is essentially the formula given in Section 2.4.2 ofEdelman, Arias and Smith [20], except for the term Y?U1.


We can easily compute the length L(�)(s) of the geodesic� from o to p = esX · o, for any X 2 m.

Indeed, for any

X =

✓S �A>

A 0

◆2 m,

we know that the geodesic from o with initial velocity Xis �(t) = etX · o, so we have

L(�)(s) =

Z s

0

h(etX)0, (etX)0i12dt,

but we already did this computation and found that

(L(�)(s))2 = s2

✓1

2tr(X>X)

◆

= s2

✓1

2tr(S>S) + tr(A>A)

◆.


We can compute these traces using the eigenvalues of Sand the singular values of A.

If ±i✓1, . . . , ±i✓m are the nonzero eigenvalues of S and�1, . . . , �k are the singular values of A, then

L(�)(s) = s(✓21 + · · · + ✓2

m + �21 + · · · + �2

k)12 .


We conclude this section with a proposition that showsthat under certain conditions, G is determined by m andH .

A point p 2 M = G/H is called a pole if the exponentialmap at p is a di↵eomorphism. The following propositionis proved in O’Neill [44] (Chapter 11, Lemma 27).

Proposition 18.21. If M = G/H is a naturally re-ductive homogeneous space, then for any pole o 2 M ,there is a di↵eomorphism m ⇥ H ⇠= G given by themap (X, h) 7! (exp(X))h.

Next, we will see that there exists a large supply of nat-urally reductive homogeneous spaces: symmetric spaces.

18.4. A GLIMPSE AT SYMMETRIC SPACES 897

18.4 A Glimpse at Symmetric Spaces

There is an extensive theory of symmetric spaces and ourgoal is simply to show that the additional structure af-forded by an involutive automorphism of G yields spacesthat are naturally reductive.

The theory of symmetric spaces was entirely created byone person, Elie Cartan, who accomplished the tour deforce of giving a complete classification of these spacesusing the classification of semisimple Lie algebras that hehad obtained earlier.

One of the most complete exposition is given in Helgason[25]. O’Neill [44], Petersen [45], Sakai [49] and Jost [28]have nice and more concise presentations. Ziller [54] isalso an excellent introduction.


Given a homogeneous space G/K, the new ingredient isthat we have an automorphism � of G such that � 6= idand �2 = id.

Then, let G� be the set of fixed points of �, the subgroupof G given by

G� = {g 2 G | �(g) = g},

and let G�0 be the identity component of G� (the con-

nected component of G� containing 1).

If G�0 ✓ K ✓ G�, then we can consider the +1 and �1

eigenspaces of d�1 : g ! g, given by

k = {X 2 g | d�1(X) = X}m = {X 2 g | d�1(X) = �X}.


An involutive automorphism of G satisfyingG�

0 ✓ K ✓ G� is called a Cartan involution .

The map d�1 is often denoted by ✓.

For all X 2 m and all Y 2 k, we have

B(X, Y ) = 0,

where B is the Killing form of g.

Remarkably, k and m yield a reductive decomposition ofG/K.


Proposition 18.22.Given a homogeneous space G/Kwith a Cartan involution � (G�

0 ✓ K ✓ G�), if k andm are defined as above, then

(1) k is indeed the Lie algebra of K.

(2) We have a reductive decomposition

g = k � m,

and[k, k] ✓ k, [m,m] ✓ k.

(3) We have Ad(K)(m) ✓ m; in particular [k,m] ✓ m.

If we also assume that G is connected and that G�0 is

compact, then we obtain the following remarkable resultproved in O’Neill [44] (Chapter 11) and Ziller [54] (Chap-ter 6).


Theorem 18.23. Let G be a connected Lie group andlet � : G ! G be an automorphism such that �2 =id, � 6= id (an involutive automorphism), and G�

0 iscompact. For every compact subgroup K of G, ifG�

0 ✓ K ✓ G�, then G/K has G-invariant metrics,and for every such metric, G/K is naturally reductive.The reductive decomposition g = k�m is given by the+1 and �1 eigenspaces of d�1. Furthermore, for everyp 2 G/K, there is an isometry sp : G/K ! G/K suchthat sp(p) = p, d(sp)p = �id, and

sp � ⇡ = ⇡ � �,

as illustrated in the diagram below:

G � //

⇡✏✏

G⇡✏✏

G/K sp// G/K.


A triple (G, K, �) satisfying the assumptions of Theorem18.23 is often called a symmetric pair .

Such a triple defines a special kind of naturally homoge-neous space G/K known as a symmetric space.

IfM is a connected Riemannian manifold, for any p 2 M ,an isometry sp such that sp(p) = p and d(sp)p = �id isa called a global symmetry at p.

A connected Riemannian manifold M for which thereis a global symmetry for every point of M is called asymmetric space .

Theorem 18.23 implies that the naturally reductive homo-geneous spaceG/K defined by a symmetric pair (G, K, �)is a symmetric space.


It can be shown that a global symmetry sp reversesgeodesics at p and that s2

p = id, so sp is an involution.

It should be noted that although sp 2 Isom(M), theisometry sp does not necessarily lie in Isom(M)0.

The following facts are proved in O’Neill [44] (Chapters 9and 11), Ziller [54] (Chapter 6), and Sakai [49] (ChapterIV).

Every symmetric space is complete, and Isom(M) actstransitively on M .

In fact the identity component Isom(M)0 acts transitivelyon M .

As a consequence, every symmetric space is a homoge-neous space of the form Isom(M)0/K, where K is theisotropy group of any chosen point p 2 M (it turns outthat K is compact).


The symmetry sp gives rise to a Cartan involution � ofG = Isom(M)0 defined so that

�(g) = sp � g � sp g 2 G.

Then, we have

G�0 ✓ K ✓ G�.

Therefore, every symmetric space M is presented by asymmetric pair (Isom(M)0, K, �).

However, beware that in the presentation of the symmet-ric space M = G/K given by a symmetric pair (G, K, �),the group G is not necessarily equal to Isom(M)0.


Thus, we do not have a one-to-one correspondence be-tween symmetric spaces and homogeneous spaces with aCartan involution.

From our point of view, this does not matter since weare more interested in getting symmetric spaces from thedata (G, K, �).

By abuse of terminology (and notation), we refer to thehomogeneous space G/K defined by a symmetric pair(G, K, �) as the symmetric space (G, K, �).

Remark: The reader may have noticed that in this sec-tion on symmetric spaces, the closed subgroup of G isdenoted by K rather than H . This is in accordance withthe convention that G/K usually refers to a symmetricspace rather than just a homogeneous space.


Since the homogeneous space G/H defined by a sym-metric pair (G, K, �) is naturally reductive and has aG-invariant metric, by Proposition 18.18, its geodesicscoincide with the one-parameter groups (they are givenby the Lie group exponential).

The Levi-Civita connection on a symmetric space de-pends only on the Lie bracket on g. Indeed, we havethe following formula proved in Ziller [54] (Chapter 6).

Proposition 18.24. Given any symmetric space Mdefined by the triple (G, K, �), for any X 2 m andand vector field Y on M ⇠= G/K, we have

(rX⇤Y )o = [X⇤, Y ]o.

Another nice property of symmetric space is that the cur-vature formulae are quite simple.


If we use the isomorphism between m and T0(G/K) in-duced by the restriction of d⇡1 tom, then for allX, Y, Z 2m we have:

1. The curvature at o is given by

R(X, Y )Z = [[X, Y ]h

, Z],

or more precisely by

R(d⇡1(X), d⇡1(Y ))d⇡1(Z) = d⇡1([[X, Y ]h

, Z]).

In terms of the vector fields X⇤, Y ⇤, Z⇤, we have

R(X⇤, Y ⇤)Z⇤ = [[X, Y ], Z]⇤ = [[X⇤, Y ⇤], Z⇤].

2. The sectional curvature K(X⇤, Y ⇤) at o is determinedby

hR(X⇤, Y ⇤)X⇤, Y ⇤i = h[[X, Y ]h

, X ], Y i.


3. The Ricci curvature at o is given by

Ric(X⇤, X⇤) = �1

2B(X, X),

where B is the Killing form associated with g.

Proof of the above formulae can be found in O’Neill [44](Chapter 11), Ziller [54] (Chapter 6), Sakai [49] (ChapterIV) and Helgason [25] (Chapter IV, Section 4).

However, beware that Ziller, Sakai and Helgason use theopposite of the sign convention that we are using for thecurvature tensor (which is the convention used by O’Neill[44], Gallot, Hulin, Lafontaine [23], Milnor [37], and Ar-vanitoyeorgos [2]).


Recall that we define the Riemann tensor by

R(X, Y ) = r[X,Y ] + rY � rX � rX � rY ,

whereas Ziller, Sakai and Helgason use

R(X, Y ) = �r[X,Y ] � rY � rX + rX � rY .

With our convention, the sectional curvature K(x, y) isdetermined by hR(x, y)x, yi, and the Ricci curvatureRic(x, y) as the trace of the map v 7! R(x, v)y.


With the opposite sign convention, the sectional curva-tureK(x, y) is determined by hR(x, y)y, xi, and the Riccicurvature Ric(x, y) as the trace of the map v 7! R(v, x)y.

Therefore, the sectional curvature and the Ricci curvatureare identical under both conventions (as they should!).

In Ziller, Sakai and Helgason, the curvature formula is

R(X⇤, Y ⇤)Z⇤ = �[[X, Y ], Z]⇤.

We are now going to see that basically all of the familiarspaces are symmetric spaces.

18.5. EXAMPLES OF SYMMETRIC SPACES 911

18.5 Examples of Symmetric Spaces

Recall that the set of all k-dimensional spaces of Rn isa homogeneous space G(k, n), called a Grassmannian ,and that G(k, n) ⇠= SO(n)/S(O(k) ⇥ O(n � k)).

We can also consider the set of k-dimensional orientedsubspaces of Rn.

An oriented k-subspace is a k-dimensional subspace Wtogether with the choice of a basis (u1, . . . , uk) determin-ing the orientation of W .

Another basis (v1, . . . , vk) of W is positively oriented ifdet(f ) > 0, where f is the unique linear map f such thatf (ui) = vi, i = 1, . . . , k.

The set of of k-dimensional oriented subspaces of Rn isdenoted by G0(k, n).


The group SO(n) acts transitively on G0(k, n), and us-ing a reasoning similar to the one used in the case whereSO(n) acts on G(k, n), we find that the stabilizer of theoriented subspace (e1, . . . , ek) is the set of orthogonal ma-trices of the form

✓Q 00 R

◆,

where Q 2 SO(k) and R 2 SO(n � k), because thistime, Q has to preserve the orientation of the subspacespanned by (e1, . . . , ek).

Thus, the isotropy group is isomorphic to

SO(k) ⇥ SO(n � k).


It follows that

G0(k, n) ⇠= SO(n)/SO(k) ⇥ SO(n � k).

Let us see how both G0(k, n) and G(k, n) are presentedas symmetric spaces.

Again, readers may find material from Absil, Mahony andSepulchre [1], especially Chapters 1 and 2, a good com-plement to our presentation, which uses more advancedconcepts (symmetric spaces).


1. Grassmannians as Symmetric Spaces

Let G = SO(n) (with n � 2), let

Ik,n�k =

✓Ik 00 �In�k

◆,

where Ik is the k ⇥ k-identity matrix, and let � be givenby

�(P ) = Ik,n�kPIk,n�k, P 2 SO(n).

It is clear that � is an involutive automorphism of G.

The set F = G� of fixed points of � is given by

F = G� = S(O(k) ⇥ O(n � k)),

and

G�0 = SO(k) ⇥ SO(n � k).


Therefore, there are two choices for K:

1. K = SO(k) ⇥ SO(n � k), in which case we get theGrassmannian G0(k, n) of oriented k-subspaces.

2. K = S(O(k) ⇥ O(n � k)), in which case we get theGrassmannian G(k, n) of k-subspaces.

As in the case of Stiefel manifolds, given any Q 2 SO(n),the first k columns Y of Q constitute a representative ofthe equivalence class [Q], but these representatives arenot unique; there is a further equivalence relation givenby

Y1 ⌘ Y2 i↵ Y2 = Y1R for some R 2 O(k).

Nevertheless, it is useful to consider the first k columnsof Q, given by QPn,k, as representative of [Q] 2 G(k, n).


Because � is a linear map, its derivative d� is equal to�, and since so(n) consists of all skew-symmetric n ⇥ nmatrices, the +1-eigenspace is given by

k =

⇢✓S 00 T

◆ �� S 2 so(k), T 2 so(n � k)

�,

and the �1-eigenspace by

m =

⇢✓0 �A>

A 0

◆ �� A 2 Mn�k,k(R)�

.

Thus, m is isomorphic to Mn�k,k(R) ⇠= R(n�k)k.

It is also easy to show that the isotropy representation isgiven by

Ad((Q, R))A = QAR>,

where (Q, R) represents an element ofS(O(k)⇥O(n � k)), and A represents an element of m.


It can be shown that this representation is irreducible i↵(k, n) 6= (2, 4).

It can also be shown that if n � 3, then G0(k, n) is simplyconnected, ⇡1(G(k, n)) = Z2, and G0(k, n) is a doublecover of G(n, k).

An Ad(K)-invariant inner product on m is given by

⌧✓0 �A>

A 0

◆,

✓0 �B>

B 0

◆�

= �1

2tr

✓✓0 �A>

A 0

◆✓0 �B>

B 0

◆◆= tr(A>B).

We also give g the same inner product. Then, we imme-diately check that k and m are orthogonal.

In the special case where k = 1, we have G0(1, n) = Sn�1

and G(1, n) = RPn�1, and then the SO(n)-invariantmetric on Sn�1 (resp. RPn�1) is the canonical one.


Skip upon first reading ~

For any point [Q] 2 G(k, n) with Q 2 SO(n), if we writeQ = [Y Y?], where Y denotes the first k columns of Qand Y? denotes the last n � k columns of Q, the tangentvectors X 2 T[Q]G(k, n) are of the form

X = [Y Y?]

✓0 �A>

A 0

◆= [Y?A � Y A>],

A 2 Mn�k,k(R).

Consequently, there is a one-to-one correspondence be-tween matrices X as above and n ⇥ k matrices of theform X 0 = Y?A, for any matrix A 2 Mn�k,k(R).

As noted in Edelman, Arias and Smith [20], because thespaces spanned by Y and Y? form an orthogonal directsum in Rn, there is a one-to-one correspondence betweenn ⇥ k matrices of the form Y?A for any matrix A 2Mn�k,k(R), and matrices X 0 2 Mn,k(R) such that

Y >X 0 = 0.


This second description of tangent vectors to G(k, n) at[Y ] is sometimes more convenient.

The tangent vectors X 0 2 Mn,k(R) to the Stiefel manifoldS(k, n) at Y satisfy the weaker condition that Y >X 0 isskew-symmetric.

End ~

Given any X 2 m of the form

X =

✓0 �A>

A 0

◆,

the geodesic starting at o is given by

�(t) = exp(tX) · o.

Thus, we need to compute

exp(tX) = exp

✓0 �tA>

tA 0

◆.


This can be done using SVD.

Since G(k, n) and G(n � k, n) are isomorphic, withoutloss of generality, assume that 2k n. Then, let

A = U

✓⌃0

◆V >

be an SVD for A, with U a (n� k)⇥ (n� k) orthogonalmatrix, ⌃ a k ⇥ k matrix, and V a k ⇥ k orthogonalmatrix.

Since we assumed that k n � k, we can write

U = [U1 U2],

with U1 is a (n�k)⇥k matrix and U2 an (n�k)⇥(n�2k)matrix.


We find that

exp(tX) = exp t

✓0 �A>

A 0

◆

=

✓V 00 U

◆0

@cos t⌃ � sin t⌃ 0sin t⌃ cos t⌃ 00 0 I

1

A✓

V > 00 U>

◆.

Now, exp(tX)Pn,k is certainly a representative of theequivalence class of [exp(tX)], so as a n ⇥ k matrix, thegeodesic through o with initial velocity

X =

✓0 �A>

A 0

◆

(with A any (n� k)⇥ k matrix with n� k � k) is givenby

�(t) =

✓V 00 U1

◆✓cos t⌃sin t⌃

◆V >,

where A = U1⌃V >, a compact SVD of A.


Because symmetric spaces are geodesically complete, weget an interesting corollary. Indeed, every equivalenceclass [Q] 2 G(k, n) possesses some representative of theform eX for some X 2 m, so we conclude that for everyorthogonal matrix Q 2 SO(n), there exist some orthog-onal matrices V, eV 2 O(k) and U, eU 2 O(n � k), andsome diagonal matrix ⌃ with nonnegatives entries, so that

Q =

✓V 00 U

◆0

@cos⌃ � sin⌃ 0sin⌃ cos⌃ 00 0 I

1

A (eV )> 0

0 (eU)>

!.

This is an instance of the CS-decomposition; see Goluband Van Loan [24].


The matrices cos⌃ and sin⌃ are actually diagonal ma-trices of the form

cos⌃ = diag(cos ✓1, . . . , cos ✓k)

and sin⌃ = diag(sin ✓1, . . . , sin ✓k),

so we may assume that 0 ✓i ⇡/2, because if cos ✓i orsin ✓i is negative, we can change the sign of the ith rowof V (resp. the sign of the i-th row of U) and still obtainorthogonal matrices U 0 and V 0 that do the job.

Now, it is known that (✓1, . . . , ✓k) are the principal an-gles (or Jordan angles) between the subspaces spannedthe first k columns of In and the subspace spanned bythe columns of Y (see Golub and van Loan [24]).


Recall that given two k-dimensional subspaces U and Vdetermined by two n⇥k matrices Y1 and Y2 of rank k, theprincipal angles ✓1, . . . , ✓k between U and V are definedrecursively as follows: Let U1 = U , V1 = V , let

cos ✓1 = maxu2U ,v2V

kuk2=1,kvk2=1

hu, vi,

let u1 2 U and v1 2 V be any two unit vectors such thatcos ✓1 = hu1, v1i, and for i = 2, . . . , k, ifUi = Ui�1 \ {ui�1}? and Vi = Vi�1 \ {vi�1}?, let

cos ✓i = maxu2Ui,v2Vi

kuk2=1,kvk2=1

hu, vi,

and let ui 2 Ui and vi 2 Vi be any two unit vectors suchthat cos ✓i = hui, vii.


The vectors ui and vi are not unique, but it is shown inGolub and van Loan [24] that (cos ✓1, . . . , cos ✓k) are thesingular values of Y >

1 Y2 (with 0 ✓1 ✓2 . . . ✓k ⇡/2).

We can also determine the length L(�)(s) of the geodesic�(t) from o to p = esX , for any X 2 m, with

X =

✓0 �A>

A 0

◆.

The computation from Section 18.3 remains valid and weobtain

(L(�)(s))2 = s2

✓1

2tr(X>X)

◆= s2tr(A>A).

Then, if ✓1, . . . , ✓k are the singular values of A, we get

L(�)(s) = s(✓21 + · · · + ✓2

k)12 .


In view of the above discussion regarding principal angles,we conclude that if Y1 consists of the first k columns ofan orthogonal matrix Q1 and if Y2 consists of the firstk columns of an orthogonal matrix Q2 then the distancebetween the subspaces [Q1] and [Q2] is given by

d([Q1], [Q2]) = (✓21 + · · · + ✓2

k)12 ,

where (cos ✓1, . . . , cos ✓k) are the singular values of Y >1 Y2

(with 0 ✓i ⇡/2); the angles (✓1, . . . , ✓k) are theprincipal angles between the spaces [Q1] and [Q2].

In Golub and van Loan, a di↵erent distance between sub-spaces is defined, namely

dp2([Q1], [Q2]) =��Y1Y

>1 � Y2Y

>2

��2.


If we write ⇥ = diag(✓1 . . . , ✓k), then it is shown that

dp2([Q1], [Q2]) = ksin⇥k1 = max1ik

sin ✓i.

This metric is derived by embedding the Grassmannianin the set of n⇥n projection matrices of rank k, and thenusing the 2-norm.

Other metrics are proposed in Edelman, Arias and Smith[20].


We leave it to the brave readers to compute h[[X, Y ], X ], Y i,where

X =

✓0 �A>

A 0

◆, Y =

✓0 �B>

B 0

◆,

and check that

h[[X, Y ], X ], Y i = hBA> � AB>, BA> � AB>i+ hA>B � B>A, A>B � B>Ai,

which shows that the sectional curvature is nonnnegative.

When k = 1, which corresponds to RPn�1 (or Sn�1), weget a metric of constant positive curvature.


2. Symmetric Positive Definite Matrices

Recall that the space SPD(n) of symmetric positive def-inite matrices (n � 2) appears as the homogeneous spaceGL+(n,R)/SO(n), under the action of GL+(n,R) onSPD(n) given by

A · S = ASA>.

Write G = GL+(n,R), K = SO(n), and choose theCartan involution � given by

�(S) = (S>)�1.

It is immediately verified that

G� = SO(n).


Then, we have gl+(n) = gl(n) = Mn(R), k = so(n), andm = S(n), the vector space of symmetric matrices. Wedefine an Ad(SO(n))-invariant inner product on gl

+(n)by

hX, Y i = tr(X>Y ).

If X 2 m and Y 2 k = so(n), we find that hX, Y i = 0.Thus, we have

hX, Y i =

8><

>:

�tr(XY ) if X, Y 2 k

tr(XY ) if X, Y 2 m

0 if X 2 m, Y 2 k.

We leave it as an exercise (see Petersen [45], Chapter 8,Section 2.5) to show that

h[[X, Y ], X ], Y i = �tr([X, Y ]>[X, Y ]), for all X, Y 2 m.


This shows that the sectional curvature is nonpositive.It can also be shown that the isotropy representation isgiven by

�A(X) = AXA�1 = AXA>,

for all A 2 SO(n) and all X 2 m.

Recall that the exponential exp : S(n) ! SDP(n) is abijection.

Then, given any S 2 SPD(n), there is a unique X 2 m

such that S = eX , and the unique geodesic from I to Sis given by

�(t) = etX.

Let us try to find the length L(�) = d(I, S) of thisgeodesic.


As in Section 18.3, we have

L(�) =

Z 1

0

h�0(t), �0(t)i12dt,

but this time, X 2 m is symmetric and the geodesic isunique, so we have

L(�) =

Z 1

0

h(etX)0, (etX)0i12dt =

Z 1

0

(tr(X2e2tX))12dt.

Since X is a symmetric matrix, we can write

X = P>⇤P,

with P orthogonal and ⇤ = diag(�1, . . . , �n), a real di-agonal matrix, and we have

tr(X2e2tX) = tr(P>⇤2PP>e2t⇤P )

= tr(⇤2e2t⇤)

= �21e

2t�1 + · · · + �2ne

2t�n.


Therefore,

d(I, S) = L(�) =

Z 1

0

((�1et�1)2 + · · · + (�ne

t�n)2)12dt.

Actually, since S = eX and S is SPD, �1, . . . , �n are thelogarithms of the eigenvalues �1, . . . , �n of X , so we have

d(I, S) = L(�) =

Z 1

0

((log �1et log �1)2 + · · ·

+ (log �net log �n)2)

12dt.

Unfortunately, there doesn’t appear to be a closed formformula for this integral.


The symmetric space SPD(n) contains an interestingsubmanifold, namely the space of matrices S in SPD(n)such that det(S) = 1.

This the symmetric space SL(n,R)/SO(n), which wesuggest denoting by SSPD(n). For this space, g = sl(n),and the reductive decomposition is given by

k = so(n), m = S(n) \ sl(n).

Now, recall that the Killing form on gl(n) is given by

B(X, Y ) = 2ntr(XY ) � 2tr(X)tr(Y ).

On sl(n), the Killing form is B(X, Y ) = 2ntr(XY ), andit is proportional to the inner product

hX, Y i = tr(XY ).


Therefore, we see that the restriction of the Killing formof sl(n) to m = S(n)\ sl(n) is positive definite, whereasit is negative definite on k = so(n).

The symmetric space SSPD(n) ⇠= SL(n,R)/SO(n) isan example of a symmetric space of noncompact type.

On the other hand, the Grassmannians are examples ofsymmetric spaces of compact type (for n � 3). In thenext section, we take a quick look at these special typesof symmetric spaces.


3. Compact Lie Groups

If H be a compact Lie group, then G = H ⇥ H acts onH via

(h1, h2) · h = h1hh�12 .

The stabilizer of (1, 1) is clearlyK = �H = {(h, h) | h 2 H}.

It is easy to see that the map

(g1, g2)K 7! g1g�12

is a di↵eomorphism between the coset space G/K and H(see Helgason [25], Chapter IV, Section 6).


A Cartan involution � is given by

�(h1, h2) = (h2, h1),

and obviously G� = K = �H .

Therefore, H appears as the symmetric space G/K, withG = H ⇥ H , K = �H , and

k = {(X, X) | X 2 h}, m = {(X, �X) | X 2 h}.

For every (h1, h2) 2 g, we have

(h1, h2) =

✓h1 + h2

2,h1 + h2

2

◆+

✓h1 � h2

2, �h1 � h2

2

◆

which gives the direct sum decomposition

g = k � m.


The natural projection ⇡ : H ⇥ H ! H is given by

⇡(h1, h2) = h1h�12 ,

which yields d⇡(1,1)(X, Y ) = X � Y (see Helgason [25],Chapter IV, Section 6).

It follows that the natural isomorphism m ! h is givenby

(X, �X) 7! 2X.

Given any bi-invariant metric h�, �i on H , define a met-ric on m by

h(X, �X), (Y, �Y )i = 4hX, Y i.


The reader should check that the resulting symmetricspace is isometric to H (see Sakai [49], Chapter IV, Ex-ercise 4).

More examples of symmetric spaces are presented in Ziller[54] and Helgason [25].

To close our brief tour of symmetric spaces, we concludewith a short discussion about the type of symmetric spaces.


18.6 Types of Symmetric Spaces

If (G, K, �) is a symmetric space with Cartan involution�, and with

g = k � m,

where k (the Lie algebra of K) is the eigenspace of d�1 as-sociated with the eigenvalue +1 andm is is the eigenspaceassociated with the eigenvalue �1.

If B is the Killing form of g, it turns out that the restric-tion of B to k is always negative definite.

Thus, it is natural to classify symmetric spaces dependingon the behavior of B on m.

18.6. TYPES OF SYMMETRIC SPACES 941

Proposition 18.25. If (G, K, �) is a symmetric space(K compact) with Cartan involution � and if B is theKilling form of g and k 6= (0), then the restriction ofB to k is negative definite.

Definition 18.8. Let M = (G, K, �) be a symmetricspace (K compact) with Cartan involution � and Killingform B. The space M is said to be of

(1) Euclidean type if B = 0 on m.

(2) Compact type if B is negative definite on m.

(3) Noncompact type if B is positive definite on m.


Proposition 18.26. Let M = (G, K, �) be a symmet-ric space (K compact) with Cartan involution � andKilling form B. The following properties hold:

(1) M is of Euclidean type i↵ [m,m] = (0). In thiscase, M has zero sectional curvature.

(2) If M is of compact type, then g is semisimple andboth G and M are compact.

(3) If M is of noncompact type, then g is semisimpleand both G and M are non-compact.

Symmetric spaces of Euclidean type are not that inter-esting, since they have zero sectional curvature.

The Grassmannians G(k, n) and G0(k, n) are symmet-ric spaces of compact type, and SL(n,R)/SO(n) is ofnoncompact type.


Since GL+(n,R) is not semisimple,SPD(n) ⇠= GL+(n,R)/SO(n) is not a symmetric spaceof noncompact type, but it has many similar properties.

For example, it has nonpositive sectional curvature andbecause it is di↵eomorphic to S(n) ⇠= Rn(n�1)/2, it issimply connected.

Here is a quick summary of the main properties of sym-metric spaces of compact and noncompact types. Proofscan be found in O’Neill [44] (Chapter 11) and Ziller [54](Chapter 6).


Proposition 18.27. Let M = (G, K, �) be a symmet-ric space (K compact) with Cartan involution � andKilling form B. The following properties hold:

(1) If M is of compact type, then M has nonnega-tive sectional curvature and positive Ricci curva-ture. The fundamental group ⇡1(M) of M is afinite abelian group.

(2) If M is of noncompact type, then M is simply con-nected, and M has nonpositive sectional curvatureand negative Ricci curvature. Furthermore, M isdi↵eomorphic to Rn (with n = dim(M)) and G isdi↵eomorphic to K ⇥ Rn.

There is also an interesting duality between symmetricspaces of compact type and noncompact type, but wewill not discuss it here.


We conclude this section by explaining what the Stiefelmanifolds S(k, n) are not symmetric spaces for 2 k n � 2.

This has to do with the nature of the involutions of so(n).Recall that the matrices Ip,q and Jn are defined by

Ip,q =

✓Ip 00 �Iq

◆, Jn =

✓0 In

�In 0

◆,

with 2 p + q and n � 1. Observe that I2p,q = Ip+q and

J2n = �I2n.


It is shown in Helgason [25] (Chapter X, Section 2 andSection 5) that, up to conjugation, the only involutiveautomorphisms of so(n) are given by

1. ✓(X) = Ip,qXIp,q, in which case the eigenspace k of ✓associated with the eigenvalue +1 is

k1 =

⇢✓S 00 T

◆ �� S 2 so(k), T 2 so(n � k)

�.

2. ✓(X) = �JnXJn, in which case the eigenspace k of ✓associated with the eigenvalue +1 is

k2 =

⇢✓S �TT S

◆ �� S 2 so(n), T 2 S(n)

�.


However, in the case of the Stiefel manifold S(k, n), theLie subalgebra k of so(n) associated with SO(n � k) is

k =

⇢✓0 00 S

◆ �� S 2 so(n � k)

�,

and if 2 k n � 2, then k 6= k1 and k 6= k2.

Therefore, the Stiefel manifold S(k, n) is not a symmetricspace if 2 k n � 2.

This also has to do with the fact that in this case,SO(n � k) is not a maximal subgroup of SO(n).

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