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 sucient conditions that ensure that M/G is Hausdor↵.

Asucientconditioncanbegivenusingthenotionofa 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 sucient 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 allhg 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 sucient 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

G/H : H GL(T (G/H)), ! o given by

G/H(h)=(d⌧ ) , for all h H. h o 2 852 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS 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 is the canonical projection.! !

Proposition 18.15. Let (G, H) be a pair where G is a Lie group and H is a closed subgroup of G. The G/H representations : H GL(To(G/H)) and G/H ! Ad : H GL(g/h) are equivalent; this means that for every h! H, we have the commutative diagram 2 G/H Adh g/h / g/h

d⇡1 d⇡1

✏ ✏ To(G/H) / To(G/H), (d⌧h)o where ⇡ : G G/H is the canonical projection. ! 18.3. REDUCTIVE HOMOGENEOUS SPACES 853 The signiﬁcance of Proposition 18.15 is that it can be used to show that G/H has some G-invariant metric i↵ the closure of AdG/H(H)iscompactinGL(g/h).

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

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

We now consider the case where there is a subspace m of g such that g = h m,sothatg/h is isomorphic to m. Deﬁnition 18.6. Let (G, H)beapairwhereG is a Lie group and H is a closed subgroup of G.Wesaythat the homogeneous space G/H is reductive if there is some subspace m of g such that

g = h m, and

Ad (m) m for all h H. h ✓ 2 854 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Observe that unlike h,whichisaLiesubalgebraofg, the subspace m is not necessarily closed under the Lie bracket, so in general it is not a Lie algebra.

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

Since g/h is isomorphic to m,bypreviousconsiderations, the map d⇡ : g T (G/H)restrictstoanisomorphism 1 ! o between m and To(G/H)(whereo denotes the point in G/H corresponding to the coset H).

For any X g,wecanexpressd⇡ (X)intermsofthe 2 1 vector ﬁeld X⇤ introduced in Deﬁnition 18.5. 18.3. REDUCTIVE HOMOGENEOUS SPACES 855

Indeed to compute d⇡1(X), we can use the curve t exp(tX), and we have 7! d d⇡ (X)= (⇡(exp(tX))) 1 dt t=0 d = (exp(tX)H) = X⇤. dt o t=0 For every X h,sincethecurvet exp(tX)H in G/H has the constant2 value o,weseethat7!

Ker d⇡1 = h, and thus, the restriction of d⇡1 to m is an isomorphism onto T (G/H), given by X X⇤. 0 7! o Also, for every X g,sinceg = h m,wecanwrite 2 X = X + X ,forsomeuniqueX h and some unique h m h 2 X m,and m 2 d⇡1(X)=d⇡1(Xm)=Xo⇤. 856 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS

We use the isomorphism d⇡1 to transfer any inner product , m on m to an inner product , on To(G/H), andh i vice-versa, by stating that h i

X, Y = X⇤,Y⇤ , for all X, Y m; h im h 0 0 i 2 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 exp(tX) p =exp(tX)aH (with p = aH G/H, a G7!) · 2 2 is an isometry of G/H for every t R,soX⇤ is a Killing vector ﬁeld. 2

AkeypropertyofareductivespaceG/H is that there is acriterionfortheexistenceofaG-invariant metrics on G/H in terms of Ad(H)-invariant inner products on m. 18.3. REDUCTIVE HOMOGENEOUS SPACES 857

1 Recall that Adg1(g2)=g1g2g1 .Then,followingO’Neill [44] (see Proposition 22, Chapter 11), observe that

⌧ ⇡ = ⇡ Ad for all h H, h h 2

1 since h H implies that h H = H,soforallg G, 2 2 1 (⌧ ⇡)(g)=hgH = hgh H =(⇡ Ad )(g). h h

By taking derivatives at 1, we get

(d⌧ ) d⇡ = d⇡ Ad . h o 1 1 h 858 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Proposition 18.16. Let (G, H) be a pair of Lie groups deﬁning 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 the ! G representation Ad : H GL(m) (where Ad is ! h restricted to m for every h H): this means that for every h H, we have2 the commutative dia- gram 2

G Adh m / m

d⇡1 d⇡1

✏ ✏ To(G/H) / To(G/H), (d⌧h)o

where ⇡ : G G/H is the canonical projection. ! 18.3. REDUCTIVE HOMOGENEOUS SPACES 859

(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/H and Ad(H)-invariant inner products on m (inner products , such that h im u, v = Ad (u), Ad (v) , h im h h h im for all h H and all u, v m). 2 2 (3) The homogeneous space G/H has some G-invariant metric i↵the closure of AdG(H) is compact in GL(m). Furthermore, if the representation AdG : H GL(m) is irreducible, then such a met- ric is unique! up to a scalar. In particular, if H is compact, then a G-invariant metric on G/H al- ways exists. 860 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS AproofofthefollowingPropositionisgiveninO’Neill [44] (Chapter 11, Lemma 24).

Proposition 18.17. Let (G, H) be a pair of Lie groups deﬁning a reductive homogeneous space M = G/H, with reductive decomposition g = h m.Ifm has some Ad(H)-invariant inner product , m, then for any left invariant metric on G extendingh i , such h im that h and m are orthogonal, the map ⇡ : G G/H is a Riemannian submersion. !

By Proposition 16.7, a Riemannian submersion carries horizontal geodesics to geodesics. For naturally reductive homogeneous manifolds (see next), the geodesics can be determined. 18.3. REDUCTIVE HOMOGENEOUS SPACES 861 When M = G/H is a reductive homogeneous space that has a G-invariant metric, it is possible to give an expres- sion for ( Y ⇤) (where X⇤ and Y ⇤ are the vector ﬁelds rX⇤ o corresponding to X, Y m). 2

If X⇤,Y⇤,Z⇤ are the Killing vector ﬁelds associated with X, Y, Z m,thenitcanbeshownthat 2

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

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

By evaluating at o and using the fact that Xo⇤ =(Xm⇤ )o for any X g,weobtain 2

2 ( Y ⇤) ,Z⇤ + ([X, Y ]⇤ ) ,Z⇤ h rX⇤ o o i h m o o i = ([Z, X]⇤ ) ,Y⇤ + ([Z, Y ]⇤ ) ,X⇤ . h m o o i h m o o i 862 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Consequently, 1 ( Y ⇤) = ([X, Y ]⇤ ) + U(X, Y )⇤, rX⇤ o 2 m o o where [X, Y ]m is the component of [X, Y ]onm and U(X, Y )isdeterminedby

2 U(X, Y ),Z = [Z, X] ,Y + X, [Z, Y ] , h i h m i h mi for all Z m. 2

Here, we are using the isomorphism X X⇤ between m 7! 0 and To(G/H)andthefactthattheinnerproductonm is chosen so that m and To(G/H)areisometric 18.3. REDUCTIVE HOMOGENEOUS SPACES 863 Since the term U(X, Y )clearlycomplicatesmatters,itis natural to make the following deﬁnition, which is equiva- lent to requiring that U(X, Y )=0forallX, Y m. 2 Deﬁnition 18.7. AhomogeneousspaceG/H is natu- rally reductive if it is reductive, if it has a G-invariant metric, and if

[X, Z] ,Y = X, [Z, Y ] , for all X, Y, Z m. h m i h mi 2

If G/H is a naturally reductive homogeneous space, then 1 ( Y )⇤ = ([X, Y ]⇤ ) . rX o 2 m o 864 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS

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

We can now ﬁnd the geodesics on a naturally reductive homogenous space.

Indeed, if M =(G, H)isareductivehomogeneousspace and M has a G-invariant metric, then there is an Ad(H)- invariant inner product , on m. h im Pick any inner product , on h,anddeﬁneaninner h ih product on g = h m by setting h and m to be orthogonal. Then, we get a left-invariant metric on G for which the elements of h are vertical vectors and the elements of m are horizontal vectors. 18.3. REDUCTIVE HOMOGENEOUS SPACES 865 Observe that in this situation, the condition for being naturally reductive extends to left-invariant vector ﬁelds on G induced by vectors in m.

Since (dL ) : g T G is a linear isomorphism for all g 1 ! g g G,thedirectsumdecompositiong = h m yields a 2 direct sum decomposition T G =(dL ) (h) (dL ) (m). g g 1 g 1 Given a left-invariant vector ﬁeld XL induced by a vector X g,ifX = X + X is the decomposition of X onto 2 h m h m,weobtainadecomposition L L L X = Xh + Xm, into a left-invariant vector ﬁeld XL hL and a left- h 2 invariant vector ﬁeld XL mL,with m 2

L L Xh (g)=(dLg)1(Xh),Xm =(dLg)1(Xm). 866 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS

Since the (dLg)1 are isometries, if h and m are orthogonal, L L so are (dLg)1(h)and(dLg)1(m), and so Xh and Xm are orthogonal vector ﬁelds.

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

[XL,ZL] ,YL = XL, [ZL,YL] , h m i h mi for all XL,YL,ZL mL. 2

Recall that the left action of G on G/H is given by g 1 · g2H = g1g2H,andthato 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 m, the geodesic d⇡1(X) through o is given by 2

d⇡ (X)(t)=⇡ exp(tX)=exp(tX) o, for all t R. 1 · 2 18.3. REDUCTIVE HOMOGENEOUS SPACES 867 Proposition 18.18 shows that the geodesics in G/H are given by the obits of the one-parameter groups (t 7! exp tX)generatedbythemembersofm.

We can also obtain a formula for the geodesic through every point p = gH G/H. 2 We ﬁnd that the geodesic through p = gH with initial velocity (AdgX)o⇤ =(Lg) Xo⇤ is given by ⇤ t g exp(tX) o. 7! · An important corollary of Proposition 18.18 is that nat- urally reductive homogeneous spaces are complete.

Indeed, the one-parameter group t exp(tX)isdeﬁned 7! for all t R. 2 868 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS 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)givenbythe restriction of d⇡1 to m,wehave

1 R(X, Y )X, Y = [X, Y ] , [X, Y ] h i 4h m mi + [[X, Y ] ,X] ,Y , for all X, Y m. h h m i 2

Conditions on a homogeneous space that ensure that such aspaceisnaturallyreductiveareobviouslyofinterest. Here is such a condition. 18.3. REDUCTIVE HOMOGENEOUS SPACES 869 Proposition 18.19. Let M = G/H be a homoge- neous space with G a connected Lie group, assume that g admits an Ad(G)-invariant inner product , , h i and let m = h? be the orthogonal complement of h with respect to , . Then, the following properties hold: h i (1) The space G/H is reductive with respect to the de- composition g = h m. (2) Under the G-invariant metric induced by , , the homogeneous space G/H is naturally reductive.h i (3) The sectional curvature is determined by

1 R(X, Y )X, Y = [X, Y ] , [X, Y ] h i 4h m mi + [X, Y ] , [X, Y ] . h h hi 870 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Recall a Lie group G is said to be semisimple if its Lie algebra g is semisimple.

From Theorem 17.22, a Lie algebra g is semisimple i↵ its Killing form B is nondegenerate, and from Theorem 17.24, 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 group G,foranyconstantc>0, the bilinear form cB is an Ad(G)-invariant inner product on g.

Then, as a corollary of Proposition 18.19, we obtain the following result. 18.3. REDUCTIVE HOMOGENEOUS SPACES 871 Proposition 18.20. Let M = G/H be a homoge- neous space such that G is a connected compact semisimple group. Then, under any inner product , on g given by cB, where B is the Killing h i form of g and c>0 is any positive real, the space G/H is naturally reductive with respect to the decom- position g = h m, where m = h? be the orthogonal complement of h with respect to , . The sectional curvature is non-negative. h i

AhomogeneousspaceasinProposition18.20iscalleda normal homogeneous space.

Since SO(n)issemisimpleandcompactforn 3, the Stiefel manifolds S(k, n)andtheGrassmannianmani- folds G(k, n)areexamplesofhomogeneousspacessatisy- ing the assumptions of Proposition 18.20 (with an inner product induced by a scalar factor of the Killing form on SO(n)). 872 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Therefore, Stiefel manifolds S(k, n)andGrassmannian manifolds G(k, n)arenaturallyreductivehomogeneous spaces for n 3(underthereductiong = h m induced by the Killing form).

If n =2,thenSO(2) is an abelian group, and thus not semisimple. However, in this case, G(1, 2) = RP(1) ⇠= SO(2)/S(O(1) O(1)) ⇠= SO(2)/O(1), and S(1, 2) = 1 ⇥ S ⇠= SO(2)/SO(1) ⇠= SO(2). These are special cases of symmetric cases discussed in Section 18.5.

In the ﬁrst case, H = S(O(1) O(1)), and in the second case, H = SO(1). In both cases,⇥ h =(0), and we can pick m = so(2), which is trivially Ad(H)-invariant. 18.3. REDUCTIVE HOMOGENEOUS SPACES 873 In Section 18.5, we show that the inner product on so(2) given by X, Y =tr(X>Y ) h i is Ad(H)-invariant, and with the induced metric, RP(1) 1 and S ⇠= SO(2) are naturally reductive manifolds.

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

On the other hand, S(k, n)itisnotasymmetricspaceif 2 k n 2. A justiﬁcation is given in Section 18.6. Since the Grassmannian manifolds G(k, n)havemore structure (they are symmetric spaces), let us ﬁrst con- sider the Stiefel manifolds S(k, n)inmoredetail. 874 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Readers may ﬁnd material from Absil, Mahony and Sepul- chre [1], especially Chapters 1 and 2, a good complement to 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,withG = SO(n)and H = SO(n k). ⇠ Then, I 0 H = R SO(n k) , 0 R 2 ⇢✓ ◆ 00 h = S so(n k) , 0 S 2 ⇢✓ ◆ and the points of G/H ⇠= S(k, n)arethecosetsQH, with Q SO(n); that is, the equivalence classes [Q], with the2 equivalence relation on SO(n)givenby

Q Q i↵ Q = Q R, for some R H. 1 ⌘ 2 2 1 2 e e 18.3. REDUCTIVE HOMOGENEOUS SPACES 875 If we write Q =[YY], where Y consists of the ﬁrst k columns of Q and Y ?consists of the last n k columns of Q,itisclearthat[?Q]isuniquelydeterminedby Y .

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

Ik Pn,k = , 0n k,k ✓ ◆ for any Q =[YY], the unique representative Y of the equivalence class [Q? ]isgivenby

Y = QPn,k.

Observe that Y is characterized by the fact that Q = ? [YY]isorthogonal,namely,YY> + Y Y > = I. ? ? ? 876 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS We also know that the Killing form on so(n)isgivenby B(X, Y )=(n 2)tr(XY ). Now, observe that if we let

T A> m = T so(k),A Mn k,k , A 0 2 2 ⇢✓ ◆ then

00 T A 00 tr > =tr =0. 0 S A 0 SA 0 ✓✓ ◆✓ ◆◆ ✓ ◆

Furthermore, it is clear that dim(m)=dim(g) dim(h), so m is the orthogonal complement of h with respect to the Killing form. 18.3. REDUCTIVE HOMOGENEOUS SPACES 877 If X, Y m,with 2 S A T B X = > ,Y= > , A 0 B 0 ✓ ◆ ✓ ◆ observe that

S A> T B> tr =tr(ST) 2tr(A>B), A 0 B 0 ✓✓ ◆✓ ◆◆ and since S> = S,wehave

tr(ST) 2tr(A>B)) = tr(S>T ) 2tr(A>B), so we deﬁne an inner product on m by

1 1 X, Y = tr(XY )= tr(S>T )+tr(A>B). h i 2 2

We give h the same inner product. 878 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Observe that there is a bijection between the space m of n n matrices of the form ⇥ S A X = > A 0 ✓ ◆ and the set of n k matrices of the form ⇥ S = , A ✓ ◆ but the inner product given by

, =tr(> ) h 1 2i 1 2 yields

, =tr(S>T )+tr(A>B), h 1 2i without the factor 1/2infrontofS>T .Thesemetrics are di↵erent. 18.3. REDUCTIVE HOMOGENEOUS SPACES 879

The vector space m is the tangent space ToS(k, n)to S(k, n)ato =[I], the coset of the point corresponding to I.

For any other point [Q] G/H = S(k, n), the tangent 2 ⇠ space T[Q]S(k, n)isgivenby

S A> T[Q]S(k, n)= Q S so(k),A Mn k,k . A 0 2 2 ⇢ ✓ ◆ Skip upon ﬁrst reading ~

Using the decomposition Q =[YY], where Y consists of the ﬁrst k columns of Q and Y ? consists of the last n k columns of Q,wehave ?

S A> [YY] =[YS+ Y A YA>]. ? A 0 ? ✓ ◆ 880 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS If we write X = YS+ Y A,sinceY and Y are parts ? ? of an orthogonal matrix, we have Y >Y =0,Y >Y = I, ? and Y >Y = I,sowecanrecoverA from X and Y and S from? X? and Y ,by ?

Y >X = Y >(YS+ Y A)=A, ? ? ? and

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

Since A = Y >X,wealsohaveA>A = X>Y Y >X = ? ? ? X>(I YY>)X. Therefore, given Q =[YY], the matrices ?

S A> [YY] ? A 0 ✓ ◆ are in one-to-one correspondence with the n k matrices of the form YS+ Y A. ⇥ ? 18.3. REDUCTIVE HOMOGENEOUS SPACES 881 Since Y describes an element of S(k, n), we can say that the tangent vectors to S(k, n)atY are of the form

X = YS+ Y A, S so(k),A Mn k,k. ? 2 2

Since [YY]isanorthogonalmatrix,wegetY >X = S, ? which shows that Y >X is skew-symmetric.

Conversely, since the columns of [YY ]formanorthonor- ? mal basis of Rn,everyn k matrix X can be written as ⇥

S X = YY = YS+ Y A, ? A ? ✓ ◆ where S Mk,k and A Mn k,k(R), and if Y >X is 2 2 skew-symmetric, then S = Y >X is also skew-symmetric. 882 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Therefore, the tangent vectors to S(k, n)atY are the vectors X Mn,k(R)suchthatY >X is skew-symmetric. 2 This is the description given in Edelman, Arias and Smith [20].

Another useful observation is that if X = YS+ Y A is atangentvectortoS(k, n)atY ,thenthesquarenorm? X, X (in the canonical metric) is given by h i 1 X, X =tr X> I YY> X . h i 2 ⇣ ⇣ ⌘ ⌘ By polarization, we ﬁnd that the canonical metric is given by 1 X ,X =tr X> I YY> X . h 1 2i 1 2 2 ⇣ ⇣ ⌘ ⌘ 18.3. REDUCTIVE HOMOGENEOUS SPACES 883 In that paper, it is also observed that because Y has rank ? n k (since Y >Y = I), for every (n k) k matrix A, ? ? ⇥ there is some n k matrix C such that A = Y >C (every column of A must⇥ be a linear combination of? the n k columns of Y ,whicharelinearlyindependent). ? Thus, we have

YS+ Y A = YS+ Y Y >C = YS+(I YY>)C. ? ? ?

End ~

By Proposition 18.18, the geodesic through o with initial velocity

S A X = > A 0 ✓ ◆ is given by

S A (t)=exp t > P . A 0 n,k ✓ ✓ ◆◆ 884 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS This is not a very explicit formula. It is possible to do better, see Edelman, Arias and Smith [20] for details.

Let us consider the case where k = n 1, which is simpler.

If k = n 1, then n k =1,soS(n 1,n)=SO(n), H = SO(1) = 1 , h =(0)andm = so(n). ⇠ { } The inner product on so(n)isgivenby

1 1 X, Y = tr(XY )= tr(X>Y ),X,Yso(n). h i 2 2 2

Every matrix X so(n)isaskew-symmetricmatrix,and we know that every2 such matrix can be written as X = P >DP,whereP is orthogonal and where D is a block diagonal matrix whose blocks are either a 1-dimensional block consisting of a zero, of a 2 2matrixoftheform ⇥ 0 ✓ D = j , j ✓ 0 ✓ j ◆ with ✓j > 0. 18.3. REDUCTIVE HOMOGENEOUS SPACES 885

X D Then, e = P >e P = P >⌃P ,where⌃isablockdiago- nal matrix whose blocks are either a 1-dimensional block consisting of a 1, of a 2 2matrixoftheform ⇥ cos ✓ sin ✓ D = j j . j sin ✓ cos ✓ ✓ j j ◆ We also know that every matrix R SO(n)canbe written as 2

R = eX, for some matrix X so(n)asabove,with0<✓ ⇡. 2 j Then, we can give a formula for the distance d(I,Q)be- tween the identity matrix and any matrix Q SO(n). 2 886 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Since the geodesics from I through Q are of the fom

(t)=etX with eX = Q, and since the length L()ofthegeodesicfromI to eX is

1 1 L()= 0(t),0(t) 2 dt. h i Z0 We ﬁnd that

1 d(I,Q)=(✓2 + + ✓2 )2 , 1 ··· m where ✓1,...,✓m are the angles associated with the eigen- i✓ i✓m values e± 1,...,e± of Q distinct from 1, and with 0 <✓ ⇡. j 18.3. REDUCTIVE HOMOGENEOUS SPACES 887 If Q, R SO(n), then 2 1 d(Q, R)=(✓2 + + ✓2 )2 , 1 ··· m where ✓1,...,✓m are the angles associated with the eigen- i✓ i✓m 1 values e± 1,...,e± of Q R = Q>R distinct from 1, and with 0 <✓ ⇡. j 2 Remark: Since X> = X,thesquaredistanced(I,Q) can also be expressed as 1 d(I,Q)2 = min tr(X2), 2 X eX=Q | or even (with some abuse of notation, since log is multi- valued) as 1 d(I,Q)2 = min tr((log Q)2). 2 888 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS In the other special case where k =1,wehave n 1 S(1,n)=S , H = SO(n 1), ⇠ 00 h = S so(n 1) , 0 S 2 ⇢✓ ◆ and

0 u> n 1 m = u R . u 0 2 ⇢✓ ◆ Therefore, there is a one-to-one correspondence between n 1 m and R .

Given any Q SO(n), the equivalence class [Q]ofQ is uniquely determined2 by the ﬁrst column of Q,andwe n 1 view it as a point on S . 18.3. REDUCTIVE HOMOGENEOUS SPACES 889

If we let u = pu>u,weleaveitasanexercisetoprove that for anyk k

0 u X = > , u 0 ✓ ◆ we have

u> cos( u t) sin( u t) u etX = k k k k k k . 0sin( u t) u I +(cos( u t) 1) uu> 1 u u 2 k k k k k k k k @ A n 1 Consequently (under the identiﬁcation of S with the ﬁrst column of matrices Q SO(n)), the geodesic 2 through e1 (the column vector corresponding to the point n 1 o S )withinitialtangentvectoru is given by 2

cos( u t) u k k (t)= u =cos( u t)e1 +sin( u t) , sin( u t) u ! k k k k u k k k k k k

n 1 n where u R is viewed as the vector in R whose ﬁrst component2 is 0. 890 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Then, we have

u 0(t)= u sin( u t)e +cos( u t) , k k k k 1 k k u ✓ k k◆ and we ﬁnd the that the length L()(✓)ofthegeodesic from e1 to the point

u p(✓)=(✓)=cos( u ✓)e +sin( u ✓) k k 1 k k u k k is given by

✓ 1 L()(✓)= 0(t),0(t) 2 dt = ✓ u . h i k k Z0 18.3. REDUCTIVE HOMOGENEOUS SPACES 891 Since

e ,p(✓) =cos(✓ u ), h 1 i k k we see that for a unit vector u and for any angle ✓ such that 0 ✓ ⇡,thelengthofthegeodesicfrome1 to p(✓)canbeexpressedas

L()(✓)=✓ =arccos(e ,p ); h 1 i that is, the angle between the unit vectors e1 and p.This is a generalization of the distance between two points on acircle. 892 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Geodesics can also be determined in the general case where 2 k n 2; we follow Edelman, Arias and Smith [20], with one change because some point in that paper requires some justiﬁcation which is not provided.

Given a point [YY] S(k, n), and given and any tan- gent vector X = YS? +2Y A,weneedtocompute ?

S A> (t)=[YY]exp t Pn,k. ? A 0 ✓ ✓ ◆◆ We can compute this exponential if we replace the matrix by a more “regular matrix,” and for this, we use a QR- decomposition of A.Let

R A = U 0 ✓ ◆ be a QR-decomposition of A,withU an orthogonal (n k) (n k)matrixandR an upper triangular k k matrix.⇥ ⇥ 18.3. REDUCTIVE HOMOGENEOUS SPACES 893

We can write U =[U1 U2], where U1 consists of the ﬁrst k 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 I 0 S R I 0 > = > . A 0 0 U R 0 0 U ✓ ◆ ✓ 1◆✓ ◆✓ 1>◆

Then, we ﬁnd

S R> Ik (t)=[YYU1]expt . ? R 0 0 ✓ ◆✓ ◆

This is essentially the formula given in Section 2.4.2 of Edelman, Arias and Smith [20], except for the term Y U1. ? 894 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS We can easily compute the length L()(s)ofthegeodesic from o to p = esX o,foranyX m. · 2 Indeed, for any

S A> X = m, A 0 2 ✓ ◆ we know that the geodesic from o with initial velocity X is (t)=etX o,sowehave · s tX tX 1 L()(s)= (e )0, (e )0 2 dt, h i Z0 but we already did this computation and found that

2 2 1 (L()(s)) = s tr(X>X) 2 ✓ ◆ 2 1 = s tr(S>S)+tr(A>A) . 2 ✓ ◆ 18.3. REDUCTIVE HOMOGENEOUS SPACES 895 We can compute these traces using the eigenvalues of S and the singular values of A.

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

1 L()(s)=s(✓2 + + ✓2 + 2 + + 2)2 . 1 ··· m 1 ··· k 896 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS We conclude this section with a proposition that shows that under certain conditions, G is determined by m and H.

Apointp M = G/H is called a pole if the exponential map at p is2 a di↵eomorphism. The following proposition is 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 M, 2 there is a di↵eomorphism m H ⇠= G given by the map (X, h) (exp(X))h. ⇥ 7! 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 our goal is simply to show that the additional structure af- forded by an involutive automorphism of G yields spaces that are naturally reductive.

The theory of symmetric spaces was entirely created by one person, Elie´ Cartan, who accomplished the tour de force of giving a complete classiﬁcation of these spaces using the classiﬁcation of semisimple Lie algebras that he had 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] is also an excellent introduction. 898 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Given a homogeneous space G/K,thenewingredientis that we have an automorphism of G such that =id and 2 =id. 6

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

G = g G (g)=g , { 2 | }

and let G0 be the identity component of G (the con- nected component of G containing 1).

If G K G,thenwecanconsiderthe+1and 1 0 ✓ ✓ eigenspaces of d : g g,givenby 1 !

k = X g d (X)=X { 2 | 1 } m = X g d (X)= X . { 2 | 1 } 18.4. A GLIMPSE AT SYMMETRIC SPACES 899 An involutive automorphism of G satisfying G K G is called a Cartan involution. 0 ✓ ✓

The map d1 is often denoted by ✓.

For all X m and all Y k,wehave 2 2 B(X, Y )=0, where B is the Killing form of g.

Remarkably, k and m yield a reductive decomposition of G/K. 900 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Proposition 18.22. Given a homogeneous space G/K with a Cartan involution (G K G), if k and 0 ✓ ✓ m 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 G0 is compact, then we obtain the following remarkable result proved in O’Neill [44] (Chapter 11) and Ziller [54] (Chap- ter 6). 18.4. A GLIMPSE AT SYMMETRIC SPACES 901 Theorem 18.23. Let G be a connected Lie group and let : G G be an automorphism such that 2 = ! id, =id(an involutive automorphism), and G0 is compact.6 For every compact subgroup K of G, if G0 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 . Furthermore, for every 1 p G/K, there is an isometry sp : G/K G/K such that2 s (p)=p, d(s ) = id, and ! p p p s ⇡ = ⇡ , p as illustrated in the diagram below:

G / G ⇡ ⇡ ✏ ✏ / G/K sp G/K. 902 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Atriple(G,K,)satisfyingtheassumptionsofTheorem 18.23 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.

If M is a connected Riemannian manifold, for any p M, 2 an isometry sp such that sp(p)=p and d(sp)p = id is acalledaglobal symmetry at p.

AconnectedRiemannianmanifoldM for which there is a global symmetry for every point of M is called a symmetric space.

Theorem 18.23 implies that the naturally reductive homo- geneous space G/K deﬁned by a symmetric pair (G,K,) is a symmetric space. 18.4. A GLIMPSE AT SYMMETRIC SPACES 903

It can be shown that a global symmetry sp reverses 2 geodesics at p and that sp =id,sosp is an involution.

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

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

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

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

As a consequence, every symmetric space is a homoge- neous space of the form Isom(M)0/K,whereK is the isotropy group of any chosen point p M (it turns out that K is compact). 2 904 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS

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

(g)=s g s g G. p p 2 Then, we have

G K G. 0 ✓ ✓ Therefore, every symmetric space M is presented by a symmetric 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. 18.4. A GLIMPSE AT SYMMETRIC SPACES 905 Thus, we do not have a one-to-one correspondence be- tween symmetric spaces and homogeneous spaces with a Cartan involution.

From our point of view, this does not matter since we are more interested in getting symmetric spaces from the data (G,K,).

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

Remark: The reader may have noticed that in this sec- tion on symmetric spaces, the closed subgroup of G is denoted by K rather than H.Thisisinaccordancewith the convention that G/K usually refers to a symmetric space rather than just a homogeneous space. 906 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Since the homogeneous space G/H deﬁned by a sym- metric pair (G,K,)isnaturallyreductiveandhasa G-invariant metric, by Proposition 18.18, its geodesics coincide with the one-parameter groups (they are given by the Lie group exponential).

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

Proposition 18.24. Given any symmetric space M deﬁned by the triple (G,K,), for any X m and 2 and vector ﬁeld Y on M ⇠= G/K, we have ( Y ) =[X⇤,Y] . rX⇤ o o

Another nice property of symmetric space is that the cur- vature formulae are quite simple. 18.4. A GLIMPSE AT SYMMETRIC SPACES 907

If we use the isomorphism between m and T0(G/K)in- duced by the restriction of d⇡ to m,thenforallX, Y, Z 1 2 m 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⇤,wehave

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

2. The sectional curvature K(X⇤,Y⇤)ato is determined by

R(X⇤,Y⇤)X⇤,Y⇤ = [[X, Y ] ,X],Y . h i h h i 908 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS 3. The Ricci curvature at o is given by 1 Ric(X⇤,X⇤)= B(X, X), 2 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] (Chapter IV) and Helgason [25] (Chapter IV, Section 4).

However, beware that Ziller, Sakai and Helgason use the opposite of the sign convention that we are using for the curvature tensor (which is the convention used by O’Neill [44], Gallot, Hulin, Lafontaine [23], Milnor [37], and Ar- vanitoyeorgos [2]). 18.4. A GLIMPSE AT SYMMETRIC SPACES 909 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)is determined by R(x, y)x, y ,andtheRiccicurvature Ric(x, y)asthetraceofthemaph i v R(x, v)y. 7! 910 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS With the opposite sign convention, the sectional curva- ture K(x, y)isdeterminedby R(x, y)y, x ,andtheRicci curvature Ric(x, y)asthetraceofthemaph iv R(v, x)y. 7! Therefore, the sectional curvature and the Ricci curvature are 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 familiar spaces 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 is ahomogeneousspaceG(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 oriented subspaces of Rn.

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

Another basis (v1,...,vk)ofW is positively oriented if det(f) > 0, where f is the unique linear map f such that f(ui)=vi, i =1,...,k.

The set of of k-dimensional oriented subspaces of Rn is denoted by G0(k, n). 912 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS The group SO(n)actstransitivelyonG0(k, n), and us- ing a reasoning similar to the one used in the case where SO(n)actsonG(k, n), we ﬁnd that the stabilizer of the oriented subspace (e1,...,ek)isthesetoforthogonalma- trices of the form

Q 0 , 0 R ✓ ◆ where Q SO(k)andR SO(n k), because this time, Q has2 to preserve the2 orientation of the subspace spanned by (e1,...,ek).

Thus, the isotropy group is isomorphic to

SO(k) SO(n k). ⇥ 18.5. EXAMPLES OF SYMMETRIC SPACES 913 It follows that

G0(k, n) = SO(n)/SO(k) SO(n k). ⇠ ⇥ Let us see how both G0(k, n)andG(k, n)arepresented as symmetric spaces.

Again, readers may ﬁnd material from Absil, Mahony and Sepulchre [1], especially Chapters 1 and 2, a good com- plement to our presentation, which uses more advanced concepts (symmetric spaces). 914 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS 1. Grassmannians as Symmetric Spaces

Let G = SO(n)(withn 2), let

Ik 0 Ik,n k = , 0 In k ✓ ◆ where Ik is the k k-identity matrix, and let be given by ⇥

(P )=Ik,n kPIk,n k,P SO(n). 2 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 = SO(k) SO(n k). 0 ⇥ 18.5. EXAMPLES OF SYMMETRIC SPACES 915 Therefore, there are two choices for K: 1. K = SO(k) SO(n k), in which case we get the Grassmannian⇥G0(k, n)oforientedk-subspaces. 2. K = S(O(k) O(n k)), in which case we get the Grassmannian⇥G(k, n)ofk-subspaces.

As in the case of Stiefel manifolds, given any Q SO(n), the ﬁrst k columns Y of Q constitute a representative2 of the equivalence class [Q], but these representatives are not unique; there is a further equivalence relation given by

Y Y i↵ Y = Y R for some R O(k). 1 ⌘ 2 2 1 2

Nevertheless, it is useful to consider the ﬁrst k columns of Q,givenbyQP ,asrepresentativeof[Q] G(k, n). n,k 2 916 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Because is a linear map, its derivative d is equal to ,andsinceso(n)consistsofallskew-symmetricn n matrices, the +1-eigenspace is given by ⇥

S 0 k = S so(k),T so(n k) , 0 T 2 2 ⇢✓ ◆ and the 1-eigenspace by 0 A> m = A Mn k,k(R) . A 0 2 ⇢✓ ◆ (n k)k Thus, m is isomorphic to Mn k,k(R) = R . ⇠ It is also easy to show that the isotropy representation is given by

Ad((Q, R))A = QAR>, where (Q, R)representsanelementof S(O(k) O(n k)), and A represents an element of m. ⇥ 18.5. EXAMPLES OF SYMMETRIC SPACES 917 It can be shown that this representation is irreducible i↵ (k, n) =(2, 4). 6 It can also be shown that if n 3, then G0(k, n)issimply 0 connected, ⇡1(G(k, n)) = Z2,andG (k, n)isadouble cover of G(n, k).

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

0 A 0 B > , > A 0 B 0 ⌧✓ ◆ ✓ ◆ 1 0 A> 0 B> = tr =tr(A>B). 2 A 0 B 0 ✓✓ ◆✓ ◆◆

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

0 n 1 In the special case where k =1,wehaveG (1,n)=S n 1 and G(1,n)=RP ,andthentheSO(n)-invariant n 1 n 1 metric on S (resp. RP )isthecanonicalone. 918 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Skip upon ﬁrst reading ~

For any point [Q] G(k, n)withQ SO(n), if we write Q =[YY], where2 Y denotes the2 ﬁrst k columns of Q and Y denotes? the last n k columns of Q,thetangent vectors? X T G(k, n)areoftheform 2 [Q]

0 A> X =[YY] =[Y A YA>], ? A 0 ? ✓ ◆ A Mn k,k(R). 2

Consequently, there is a one-to-one correspondence be- tween matrices X as above and n k matrices of the ⇥ form X0 = Y A,foranymatrixA Mn k,k(R). ? 2 As noted in Edelman, Arias and Smith [20], because the spaces spanned by Y and Y form an orthogonal direct ? sum in Rn,thereisaone-to-onecorrespondencebetween n k matrices of the form Y A for any matrix A ⇥ ? 2 Mn k,k(R), and matrices X0 Mn,k(R)suchthat 2

Y >X0 =0. 18.5. EXAMPLES OF SYMMETRIC SPACES 919 This second description of tangent vectors to G(k, n)at [Y ]issometimesmoreconvenient.

The tangent vectors X0 Mn,k(R)totheStiefelmanifold 2 S(k, n)atY satisfy the weaker condition that Y >X0 is skew-symmetric.

End ~

Given any X m of the form 2 0 A X = > , A 0 ✓ ◆ the geodesic starting at o is given by

(t)=exp(tX) o. · Thus, we need to compute

0 tA exp(tX)=exp > . tA 0 ✓ ◆ 920 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS This can be done using SVD.

Since G(k, n)andG(n k, n)areisomorphic,without loss of generality, assume that 2k n.Then,let ⌃ A = U V 0 > ✓ ◆ be an SVD for A,withU a(n k) (n k)orthogonal matrix, ⌃a k k matrix, and V ⇥a k k orthogonal matrix. ⇥ ⇥

Since we assumed that k n k,wecanwrite U =[U1 U2], with U1 is a (n k) k matrix and U2 an (n k) (n 2k) matrix. ⇥ ⇥ 18.5. EXAMPLES OF SYMMETRIC SPACES 921 We ﬁnd that

0 A exp(tX)=expt > A 0 ✓ ◆ cos t⌃ sin t⌃0 V 0 V > 0 = sin t⌃cost⌃0 . 0 U 0 1 0 U > ✓ ◆ 00I ✓ ◆ @ A

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

0 A X = > A 0 ✓ ◆ (with A any (n k) k matrix with n k k)isgiven by ⇥

V 0 cos t⌃ (t)= V , 0 U sin t⌃ > ✓ 1◆✓ ◆ where A = U1⌃V >,acompactSVDofA. 922 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Because symmetric spaces are geodesically complete, we get an interesting corollary. Indeed, every equivalence class [Q] G(k, n)possessessomerepresentativeofthe 2 form eX for some X m,soweconcludethatforevery orthogonal matrix Q 2 SO(n), there exist some orthog- 2 onal matrices V,V O(k)andU, U O(n k), and some diagonal matrix2 ⌃with nonnegatives2 entries, so that e e cos ⌃ sin ⌃0 V 0 (V )> 0 Q = sin ⌃cos ⌃0 . 0 U 0 1 0(U) ✓ ◆ 00I >! e @ A e This is an instance of the CS-decomposition; see Golub and Van Loan [24]. 18.5. EXAMPLES OF SYMMETRIC SPACES 923 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 ✓ ⇡/2, because if cos ✓ or i i sin ✓i is negative, we can change the sign of the ith row of V (resp. the sign of the i-th row of U)andstillobtain orthogonal matrices U 0 and V 0 that do the job.

Now, it is known that (✓1,...,✓k)aretheprincipal an- gles (or Jordan angles)betweenthesubspacesspanned the ﬁrst k columns of In and the subspace spanned by the columns of Y (see Golub and van Loan [24]). 924 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Recall that given two k-dimensional subspaces and determined by two n k matrices Y and Y of rankU k,theV ⇥ 1 2 principal angles ✓1,...,✓k between and are deﬁned recursively as follows: Let = , U = ,letV U1 U V1 V

cos ✓1 =maxu, v , u ,v h i u 2=1U, v2V =1 k k2 k k2 let u1 and v1 be any two unit vectors such that cos ✓ =2Uu ,v ,andfor2V i =2,...,k,if 1 h 1 1i i = i 1 ui 1 ? and i = i 1 vi 1 ?,let U U \{ } V V \{ }

cos ✓i =maxu, v , u i,v i h i u 2=1U , v2V =1 k k2 k k2 and let ui i and vi i be any two unit vectors such that cos ✓ 2=Uu ,v . 2V i h i ii 18.5. EXAMPLES OF SYMMETRIC SPACES 925

The vectors ui and vi are not unique, but it is shown in Golub and van Loan [24] that (cos ✓1,...,cos ✓k)arethe singular values of Y1>Y2 (with 0 ✓1 ✓2 ... ✓k ⇡/2).

We can also determine the length L()(s)ofthegeodesic (t)fromo to p = esX,foranyX m,with 2 0 A X = > . A 0 ✓ ◆ The computation from Section 18.3 remains valid and we obtain

2 2 1 2 (L()(s)) = s tr(X>X) = s tr(A>A). 2 ✓ ◆

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

1 L()(s)=s(✓2 + + ✓2)2 . 1 ··· k 926 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS In view of the above discussion regarding principal angles, we conclude that if Y1 consists of the ﬁrst k columns of an orthogonal matrix Q1 and if Y2 consists of the ﬁrst k columns of an orthogonal matrix Q2 then the distance between the subspaces [Q1]and[Q2]isgivenby

1 d([Q ], [Q ]) = (✓2 + + ✓2)2 , 1 2 1 ··· k where (cos ✓1,...,cos ✓k)arethesingularvaluesofY1>Y2 (with 0 ✓ ⇡/2); the angles (✓ ,...,✓ )arethe i 1 k principal angles between the spaces [Q1]and[Q2].

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

d ([Q ], [Q ]) = Y Y > Y Y > . p2 1 2 1 1 2 2 2 18.5. EXAMPLES OF SYMMETRIC SPACES 927

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

dp2([Q1], [Q2]) = sin ⇥ =maxsin ✓i. k k1 1 i k This metric is derived by embedding the Grassmannian in the set of n n projection matrices of rank k,andthen using the 2-norm.⇥

Other metrics are proposed in Edelman, Arias and Smith [20]. 928 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS We leave it to the brave readers to compute [[X, Y ],X],Y , where h i

0 A 0 B X = > ,Y= > , A 0 B 0 ✓ ◆ ✓ ◆ and check that

[[X, Y ],X],Y = BA> AB>,BA> AB> h i h i + A>B B>A, A>B B>A , h i which shows that the sectional curvature is nonnnegative.

n 1 n 1 When k =1,whichcorrespondstoRP (or S ), we get a metric of constant positive curvature. 18.5. EXAMPLES OF SYMMETRIC SPACES 929 2. Symmetric Positive Deﬁnite Matrices

Recall that the space SPD(n)ofsymmetricpositivedef- inite matrices (n 2) appears as the homogeneous space + + GL (n, R)/SO(n), under the action of GL (n, R)on SPD(n)givenby

A S = ASA>. ·

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

1 (S)=(S>) .

It is immediately veriﬁed that

G = SO(n). 930 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS + Then, we have gl (n)=gl(n)=Mn(R), k = so(n), and m = S(n), the vector space of symmetric matrices. We deﬁne an Ad(SO(n))-invariant inner product on gl+(n) by

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

tr(XY )ifX, Y k 2 X, Y = tr(XY )ifX, Y m h i 8 2 <>0ifX m,Y k. 2 2 :> We leave it as an exercise (see Petersen [45], Chapter 8, Section 2.5) to show that

[[X, Y ],X],Y = tr([X, Y ]>[X, Y ]), for all X, Y m. h i 2 18.5. EXAMPLES OF SYMMETRIC SPACES 931 This shows that the sectional curvature is nonpositive. It can also be shown that the isotropy representation is given by

1 A(X)=AXA = AXA>, for all A SO(n)andallX m. 2 2 Recall that the exponential exp: S(n) SDP(n)isa bijection. !

Then, given any S SPD(n), there is a unique X m such that S = eX,andtheuniquegeodesicfrom2 I to2 S is given by (t)=etX.

Let us try to ﬁnd the length L()=d(I,S)ofthis geodesic. 932 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS As in Section 18.3, we have

1 1 L()= 0(t),0(t) 2 dt, h i Z0 but this time, X m is symmetric and the geodesic is unique, so we have2 1 1 tX tX 1 2 2tX 1 L()= (e )0, (e )0 2 dt = (tr(X e ))2 dt. h i Z0 Z0

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

2 2tX 2 2t⇤ tr(X e )=tr(P >⇤ PP>e P ) =tr(⇤2e2t⇤) = 2e2t1 + + 2 e2tn. 1 ··· n 18.5. EXAMPLES OF SYMMETRIC SPACES 933 Therefore,

1 1 d(I,S)=L()= (( et1)2 + +( etn)2)2 dt. 1 ··· n Z0

X Actually, since S = e and S is SPD, 1,...,n are the logarithms of the eigenvalues 1,...,n of X,sowehave

1 t log 1 2 d(I,S)=L()= ((log 1e ) + 0 ··· Z 1 t log n 2 +(logne ) )2 dt.

Unfortunately, there doesn’t appear to be a closed form formula for this integral. 934 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS The symmetric space SPD(n)containsaninteresting submanifold, namely the space of matrices S in SPD(n) such that det(S)=1.

This the symmetric space SL(n, R)/SO(n), which we suggest 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)isgivenby

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

X, Y =tr(XY ). h i 18.5. EXAMPLES OF SYMMETRIC SPACES 935 Therefore, we see that the restriction of the Killing form of sl(n)tom = S(n) sl(n)ispositivedeﬁnite,whereas \ it is negative deﬁnite on k = so(n).

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

On the other hand, the Grassmannians are examples of symmetric spaces of compact type (for n 3). In the next section, we take a quick look at these special types of symmetric spaces. 936 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS 3. Compact Lie Groups

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

1 (h ,h ) h = h hh . 1 2 · 1 2 The stabilizer of (1, 1) is clearly K =H = (h, h) h H . { | 2 } It is easy to see that the map

1 (g ,g )K g g 1 2 7! 1 2 is a di↵eomorphism between the coset space G/K and H (see Helgason [25], Chapter IV, Section 6). 18.5. EXAMPLES OF SYMMETRIC SPACES 937 ACartaninvolution is given by

(h1,h2)=(h2,h1), and obviously G = K =H.

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

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

For every (h ,h ) g,wehave 1 2 2 h + h h + h h h h h (h ,h )= 1 2, 1 2 + 1 2, 1 2 1 2 2 2 2 2 ✓ ◆ ✓ ◆ which gives the direct sum decomposition

g = k m. 938 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS The natural projection ⇡ : H H H is given by ⇥ ! 1 ⇡(h1,h2)=h1h2 , 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 given by ! (X, X) 2X. 7! Given any bi-invariant metric , on H,deﬁneamet- h i ric on m by

(X, X), (Y, Y ) =4X, Y . h i h i 18.5. EXAMPLES OF SYMMETRIC SPACES 939 The reader should check that the resulting symmetric space 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 conclude with a short discussion about the type of symmetric spaces. 940 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS 18.6 Types of Symmetric Spaces

If (G,K,)isasymmetricspacewithCartaninvolution ,andwith

g = k m, where k (the Lie algebra of K)istheeigenspaceofd1 as- sociated with the eigenvalue +1 and m is is the eigenspace associated with the eigenvalue 1. If B is the Killing form of g,itturnsoutthattherestric- tion of B to k is always negative deﬁnite.

Thus, it is natural to classify symmetric spaces depending on 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 the Killing form of g and k =(0), then the restriction of 6 B to k is negative deﬁnite.

Deﬁnition 18.8. Let M =(G,K,)beasymmetric space (K compact) with Cartan involution and Killing form B.ThespaceM is said to be of (1) Euclidean type if B =0onm. (2) Compact type if B is negative deﬁnite on m. (3) Noncompact type if B is positive deﬁnite on m. 942 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Proposition 18.26. Let M =(G,K,) be a symmet- ric space (K compact) with Cartan involution and Killing form B. The following properties hold: (1) M is of Euclidean type i↵ [m, m]=(0). In this case, M has zero sectional curvature. (2) If M is of compact type, then g is semisimple and both G and M are compact. (3) If M is of noncompact type, then g is semisimple and 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)andG0(k, n)aresymmet- ric spaces of compact type, and SL(n, R)/SO(n)isof noncompact type. 18.6. TYPES OF SYMMETRIC SPACES 943 + Since GL (n, R)isnotsemisimple, + SPD(n) ⇠= GL (n, R)/SO(n)isnotasymmetricspace of noncompact type, but it has many similar properties.

For example, it has nonpositive sectional curvature and n(n 1)/2 because it is di↵eomorphic to S(n) ⇠= R ,itis simply connected.

Here is a quick summary of the main properties of sym- metric spaces of compact and noncompact types. Proofs can be found in O’Neill [44] (Chapter 11) and Ziller [54] (Chapter 6). 944 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Proposition 18.27. Let M =(G,K,) be a symmet- ric space (K compact) with Cartan involution and Killing 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 a ﬁnite abelian group. (2) If M is of noncompact type, then M is simply con- nected, and M has nonpositive sectional curvature and negative Ricci curvature. Furthermore, M is di↵eomorphic to Rn (with n =dim(M)) and G is di↵eomorphic to K Rn. ⇥

There is also an interesting duality between symmetric spaces of compact type and noncompact type, but we will not discuss it here. 18.6. TYPES OF SYMMETRIC SPACES 945 We conclude this section by explaining what the Stiefel manifolds S(k, n)arenotsymmetricspacesfor2 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

I 0 0 I I = p ,J= n , p,q 0 I n I 0 ✓ q◆ ✓ n ◆

2 with 2 p + q and n 1. Observe that Ip,q = Ip+q and J 2 = I . n 2n 946 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS It is shown in Helgason [25] (Chapter X, Section 2 and Section 5) that, up to conjugation, the only involutive automorphisms of so(n)aregivenby

1. ✓(X)=Ip,qXIp,q,inwhichcasetheeigenspacek of ✓ associated with the eigenvalue +1 is S 0 k = S so(k),T so(n k) . 1 0 T 2 2 ⇢✓ ◆ 2. ✓(X)= JnXJn,inwhichcasetheeigenspace k of ✓ associated with the eigenvalue +1 is S T k2 = S so(n),T S(n) . TS 2 2 ⇢✓ ◆ 18.6. TYPES OF SYMMETRIC SPACES 947 However, in the case of the Stiefel manifold S(k, n), the Lie subalgebra k of so(n)associatedwithSO(n k)is 00 k = S so(n k) , 0 S 2 ⇢✓ ◆ and if 2 k n 2, then k = k and k = k . 6 1 6 2 Therefore, the Stiefel manifold S(k, n)isnotasymmetric space if 2 k n 2. This also has to do with the fact that in this case, SO(n k)isnotamaximalsubgroupofSO(n). 948 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Bibliography

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