= ( )= { 2 ( ) | 6 = } [[ ] ]+[[ ] ]+[[ ] ]= [ ]=

Chapter 5

B

L

T

1. Categories of Lie groups and Lie algebras

A C^•manifold G is a Lie group if G has a group structure and the group law

G⇥^G !^G; (g, h) 7! ^{gh 1are C•.}

For g 2 G, we denote the left multiplication map h 7! ^{gh by L}g and right multiplication h 7! ^{hg by R}g. We have the induced map on tangent spaces:

dL_g : T_hG !^TghG; dR_g : T_hG !^ThgG.

A vector field X 2 ^C•(TG) is left invariant if X_gh = dL_gX_hfor all g, h 2 G. The Lie algebra g= Lie G of G is the vector space of all left invariant vector fields (l.i.v.f.) under bracket operation. Namely, as differential operators, for f 2 ^C•(G):

[X, Y]f :=X(Y f) Y(X f).

Since a l.i.v.f. X is determined by its value X_eat the identity e2 ^G, we identify

g⇠=T_eG.

Abstractly, a vector space L over a field F (with charF 6= ^{2) with} an F-bilinear map [,] : L⇥^L ! L is called a Lie algebra (over F) if [x, y] = [y, x]and

[[x, y], z] + [[y, z], x] + [[z, x], y] =0. (Jacobi identity) . It is clear that the bracket of vector fields has this property.

Example 5.1. Consider the general linear group

G=GL(n, R) = {^g2 ^Mn⇥n(^R)| ^{det g}6=⁰}^.

163

From Cramer’s rule, we see that g7! ^{g 1is C•} hence that G is a Lie group.

As an open subset of M_n⇥n(^R) ⇠=^Rn2, we have T_eG= M_n⇥n(^R). The matrix algebra has a natural Lie algebra structure gl(n, R) de- fined by

[A, B]:= AB BA.

Theorem 5.2. gl(n, R)coincides with Lie G.

PROOF. From (gh(t))⁰ = gh⁰(t), we see that (L_g)_⇤A = gA for g 2 ^{G, A} 2 ^TeG. Thus if ˜A is the l.i.v.f. with ˜Ae = A, then ˜Ag = gA. Let G ,! ^Rn2 with coordinates(x_ij)ⁿ_i,j=1being the entries of the corresponding matrix g. Then a tangent vector A = (a_ij) 2 ^TeG and A are equivalent to˜

A=

Â

i,j

a_ij ∂

∂x_{ij e} and A˜_g =

Â

i,j

(gA)_ij ^∂

∂x_{ij g} respectively. From

Â

∂

∂x_ij(x_kmb_ml) =

Â

m d_kid_mjb_ml =d_kib_jl, we compute

[A, ˜B^˜ ]_e =

Â

i,j,k,l

✓ a_ij ∂

∂x_ij (gB)_kl ^∂

∂x_kl b_ij ∂

∂x_ij (gA)_kl ^∂

∂x_kl

◆

g=e

=

Â

i,j,l

a_ijb_jl ∂

∂x_{il e} b_ija_jl ∂

∂x_{il e} =

Â

i,l

(AB BA)_il ^∂

∂x_{il e}.

This corresponds to AB BA precisely. ⇤

A Lie subgroup H < G is itself a Lie group such that H is both a subgroup and an immersion. We allow H ⇢G to be not closed.

Example 5.3. Subgroups of matrix groups are the main sources of Lie groups.

(i) Let SL(n, R) = {^g 2 ^GL(n, R)| ^{det g} = 1} be the special linear group. Consider a smooth curve t7! ^g(t)with g(0) = e and det g(t) = 1. Then we compute tr g⁰(0) = 0. So its Lie algebra is given by sl(n, R) = {^A2 ^Mn⇥n(^R)|^{tr A}=0}^.

1. CATEGORIES OF LIE GROUPS AND LIE ALGEBRAS 165

(ii) Let O(n, R) = {^g 2 ^GL(n, R)|^gTg = e} be the orthogonal group. Consider a smooth curve t 7! ^g(t)with g(0) = e and g(t)^Tg(t) = e. Then we compute g⁰(0)^T +g⁰(0) = 0. So its Lie algebra is given by:

o(n, R) = {^A2 ^Mn⇥n(R)|^AT = A}^.

(iii) Let SO(n, R) = {^g 2 ^O(n, R)| ^{det g} = 1} be the special orthogonal group . It is clear that O(n, R)has two connected components and SO(n, R)is the identity component, so

so(n, R) = o(n, R).

(iv) Let Sp(2n, R) = {^g 2 ^M2n⇥2n(^R)|^gTJg = J} be the sym- plectic group, where

J = ^{0 In} I_n 0

! . Its Lie algebra sp(2n, R)is given by:

{^A2 ^M2n⇥2n(^R)|^ATJ = JA}^.

(v) We have similar complex Lie groups GL(n, C), SL(n, C), O(n, C), SO(n, C)and Sp(n, C). Indeed they are defined by algebraic equations with integer coefficient, so they can take values in any field. The corresponding Lie algebras gl(n, C), sl(n, C), so(n, C)and sp(n, C)are complex Lie algebras.

(vi) Let U(n) = {^g 2 ^GL(n, C)|^g⇤g = e} be the unitary group.

Consider a smooth curve t 7! ^g(t)with g(0) =e and g(t)^⇤g(t) = e. Then we compute g⁰(0)^⇤+g⁰(0) = 0. So its Lie algebra is given by u(n) = {^A 2 ^Mn⇥n(^C)|^A⇤ = A}. Notice that u(n)is a real Lie algebra.

(vii) Let SU(n) ={^g 2^U(n)| ^{det g}=1}be special unitary group.

su(n) =sl(n, C)\^u(n).

All these subgroups can be realized as the subgroup preserving certain additional structure. For “S”, g preserves volume. For “O”, g preserves the Euclidean inner product. For “Sp”, g preserves the non-degenerate symplectic form

x^TJy= (x1y_n+1 x_n+1y1) +· · · + (^xny_2n x_2ny_n). And for “U”, g preserves the Hermitian inner product.

sl(n, C), so(2n, C), sp(n, C)and so(2n+1, C)are known as classi- cal complex semi-simple Lie algebras of type A_n, B_n, C_n and D_n respec- tively. (To be explained later).

Theorem 5.4. Given a Lie group G. There is an one to one correspondence between connected Lie subgroups of G and Lie subalgebras of g.

PROOF. This follows from the Frobenius Theorem (cf. theorem1.40).

For a basis X_iof a Lie subalgebra h of g, we defined a subspace dis- tributionHgwhich is spanned by X_ig for all g2 G. The distribution H = ^Sg2GHg is integrable. Indeed, for any two C^• vector fields V =_{Â fi}X_iand W =_{Â gi}X_i, we compute

[V, W] =

Â

Â

Â

We then take H to be the maximal integral submanifold passing through e2 ^G.

To check that H is a group, let g 2 H. The map L_g maps the manifold H to gH. The left invariance says that dL_gHh =Hgh, hence gH is also an integral submanifold. Now H and gH both contain the element g, hence the maximality (uniqueness) implies that H = gH.

This implies that H is closed under multiplication and also g ¹ 2 ^H (since e 2 H). So H is a subgroup of G.

Finally, H is a Lie groups simply because the map H⇥^H ! ^H sending(g, h)to gh ¹is the restriction of the given C^•map G⇥^G!

G. ⇤

Remark 5.5. For any Lie group G, the tangent bundle TG is a trivial vector bundle with global frame given by any basis of g.

More generally, a Lie group homomorphism r : G! ^{H is a C•map} which is also a group homomorphism. The tangent map dr : TG ! TH is compatible with l.i.v.f.’s. To see this, r(gg⁰) = r(g)r(g⁰)means r L_g= L_r(g) r, so

dr dL_g =dL_r(g) dr.

2. EXPONENTIAL MAP 167

Thus dr : g ! h. dr is indeed a Lie algebra homomorphism in the sense that dr[X, Y] = [dr(X), dr(Y)], which is easily verified from the definitions.

2. Exponential map

We call a nontrivial Lie group homomorphism R ! G a one pa- rameter subgroup, even though it may not be injective. The exponential map links Lie algebras with Lie groups through the consideration of all one parameter subgroups. Before treating the abstract setting, we look at the case for matrix groups.

Example 5.6. For A 2 ^Mn⇥n(^C), t 2 ^C, we define the absolutely convergent series

e^tA =1+tA+^t

2

2!A²+· · · +^t_k!^kAk+· · · ^.

It is easily checked that if AB =BA then e^Ae^B =e^A+B. Hence e^Ahas inverse e ^A and so e^A 2 ^GL(n, C). Moreover g(t) = e^tA is the one parameter subgroup with

g⁰(t) =e^tAA=dL_g(t)A = A_g(t).

That is, e^tA is the integral curve of the l.i.v.f. determined by A 2 gl(n, C).

The discussion works for C being replaces by R. Also if we take A be in a Lie subalgebra, the e^A lies in the corresponding Lie sub- group. This follows from the previous theorem. But we can also see how it works explicitly: For example,

tr A=0 =) ^{det eA} =e^{tr A} =1.

Also

A^⇤ = A =) (e^A)^⇤e^A=e^A⇤e^A =e ^Ae^A =I_n.

Now we turn to a general Lie group G. Let X 2 g. Since RX <g is a one dimensional Lie subalgebra, by the previous theorem its inte- gral curve is a one dimensional subgroup H. By taking the universal cover R ! H if necessary, we get a one parameter subgroup which

we denote by t 7! exp tX. We shall give a direct proof of this with stronger conclusions.

Let f_t be the flow generated by X. That is, f_t(g) is the curve with f₀(g) = g and

d

dtf_t(g) = X_ft(g).

Theorem 5.7. The range of t is R for all g 2 G. Moreover, f_t : G! ^{G is a} one-parameter group of diffeomorphisms as right translations f_t =R_ft(e).

PROOF. Consider the curve gf_t(e). Since gf0(e) = g and d

dt gf_t(e) =dL_g dL_ft(e)X_e

=dL_gft(e)X_e

= X_gft(e), we conclude that f_t(g) = gf_t(e) = R_ft(e)g.

By substituting g = f_s(e) we find fs(e)f_t(e) = f_t(f_s(e)) = f_t+s(e). This shows that for g = e, the range of t can be extended to all R and f_t(e)is a one parameter subgroup. The theorem is proved by using the relation f_t(g) = gf_t(e)again. ⇤

Now we define the exponential map exp : g!^G

by exp tX =f_t(e)where f_t is the flow generated by X. Since (d exp)0(X) = ^d

dt _t=0exp tX =X, we get(d exp)0 =Idgand exp is invertible near 02 ^g.

Corollary 5.8. If H <G is a Lie subgroup, then H is generated by exp h.

However, exp is not necessarily surjective, hence exp g is not nec- essarily a group.

Exercise 5.1. Let X 2^sl(2, R)and d=^p|^{det X}|^{. Then} (i) e^X = (cosh d)I2+¹_d(sinh d)X if det X <0.

(ii) e^X = (cos d)I₂+¹_d(sin d)X if det X>0.

3. ADJOINT REPRESENTATION 169

(iii) e^X = I2+X if det X=0.

Let g_a = ^{a 0} 0 a ¹

!

2 ^SL(2, R). Then g_a lies in a unique one pa- rameter subgroup if a >0. g_a lies in infinitely many one parameter subgroup if a= 1. If a 6= ^{1 and a}<0, then g_a 62 ^{exp sl}(2, R).

3. Adjoint representation

3.0.1. Three adjoints I_g, Ad_gand ad_X. For g2 ^{G, let I}g : G !^{G be} the inner automorphism I_g(h) = L_gR_g 1(h) = R_g 1L_g(h) = ghg ¹. Since I_g(e) = e, we get its differential

Adg :=dIg : g!^g

as a Lie algebra automorphism. From dI_gg0 =d(I_g I_g0) = dI_g dI_g0, we get the adjoint representations of Lie group G

Ad : G !^{Aut g} and the adjoint representation of Lie algebra g

ad :=d(Ad) : g!^{End g.}

For G a matrix group, g is a matrix Lie algebra and it is clear that Ad_g(Y) = gYg ¹. For g(t)a curve with g(0) = e and g⁰(0) = X we then compute

ad_X(Y) = (g(t)Yg(t) ¹)⁰(0) = XY YX = [X, Y]. This property holds true in general:

Theorem 5.9. For X, Y2 ^g,

ad_XY = [X, Y].

PROOF. Let f 2 ^C•(G) and f, y be the flows generated by X, Y.

Then

(ad_XY)f = ^d

dt _t=0(Ad_{exp tX}Y)f

= ^d dt _t=0

d

ds _s=0f(I_{exp tX}(exp sY))

= ^d ds _s=0

d

dt _t=0f(exp tX·^{exp sY}·^exp( tX))

= ^d ds

d

dt f f _t y_s f_t(e) (0, 0)

= ^d

dsd f( X_ys(e)) +d(f y_s)X_e

s=0

= ^d

ds _s=0X_ys(e)f +X_e✓ d

ds _s=0f y_s

◆

= ^d

ds _s=0(X f) y_s(e) +X_eY f

= Y_eX f +X_eY f = [X, Y]_ef .

⇤

Remark 5.10. Readers with experience in differential geometry may observe that the proof is identical with the one for Lie derivative L_XY= [X, Y]. Indeed,

ad_XY= ^d

dt _t=0(Ad_{exp tX}Y)

= ^d

dt _t=0dR_{exp( tX)}dL_{exp tX}Y

= ^d

dt _t=0df _tY =L_XY

by the left invariance of Y and the definition of L_XY.

It is harder to get explicit formula for Ad_gin the abstract setting.

We have such a formula in two special cases, both are based on the

3. ADJOINT REPRESENTATION 171

commutative diagram

G r //

OO

exp

HOO exp

g dr // h

To see this, simply notice that r exp tX and exp dr(tX) are both one parameter subgroups in H with the same tangent vector dr(X) at t =0.

By applying the diagram to r = I_g, we get:

exp(Ad_gX) = g(exp X)g ¹. (For matrix groups this is obvious).

By applying the diagram to H = Aut g, r = Ad and g = exp X, we get

Ad_{exp X}Y=e^adXY.

With these preparation, we give some applications of the adjoint representation:

3.0.2. Center of a Lie group. A Lie algebra is called abelian if[X, Y] = 0 for all X, Y. We denote Z(G)by the center of G.

Proposition 5.11. Let G be a connected Lie group, then Z(G) =Ker Ad.

In particular, G is abelian if and only if g is abelian.

PROOF. If g is in the center, then for all t 2 ^{Rand X} 2^g, exp tX =g(exp tX)g ¹=exp Ad_gtX=exp tAd_gX.

Hence X =Ad_gX for all X. That is, Adg =idg.

Conversely, g2 Ker Ad implies that exp X =g(exp X)g ¹. Hence g commutes with all elements in a neighborhood of e in G. By the connectedness of G we conclude that g commutes with every ele-

ments in G. ⇤

Corollary 5.12. [X, Y] = 0 implies that exp X·^{exp Y} =exp(X+Y).

PROOF. Let h be the two dimensional abelian Lie subalgebra of g spanned by X and Y. Consider the Lie group H generated by exp h.

The proposition show that H is abelian and so the curve g(t) = exp tX·exp tY is an one parameter subgroup. Since g⁰(0) = X+Y, we conclude that exp tX·^{exp tY}=exp t(X+Y). ⇤

Corollary 5.13. If G is a connected Lie groups with trivial center, then Ad : G ,!^{Aut g}=GL(g)

is a faithful representation. In particular, G is a matrix subgroup.

3.0.3. Normal Lie subgroups. A subspace h of g is a Lie ideal if[h, g]⇢ h. In this case we denote by hCg. It is clear that h is at least a subal- gebra.

Proposition 5.14. Let H <G be a connected Lie subgroup of a connected Lie group. Then

HC^G() ^{h :}=Lie H C ^g.

PROOF. Let g =exp X with X 2^{gand Y}2 ^h, If h is a Lie ideal of g, then

g(exp Y)g ¹ =exp Ad_gY

=exp(e^adXY)

=exp

✓

I+ad_X+ ¹

2!ad²_X+· · ·

◆ Y 2 ^{exp h}⇢^H.

Since H is generated by h, this proves that H is normal.

Conversely, if H is normal, then the above computation shows that

g(t):=exp(e^adtXY)2 ^H.

Hence h 3^g0(0) = ad_XY = [X, Y]and h is a Lie ideal. ⇤ 3.1. Fundamental correspondences.

4. DIFFERENTIAL GEOMETRY ON LIE GROUPS 173

3.1.1. Equivalence of categories.

Theorem 5.15. Let G and H be connected Lie groups with Lie algebras g and h, If G is simply connected, then there is a one to one correspondence be- tween Lie group homomorphisms G ! H and Lie algebra homomorphisms g!^h.

IDEA OF PROOF. The is proved by exploring the Frobenius theo- rem on the product group G⇥H in a manner similar to the subgroup case.

Indeed a morphism r : G ! H is equivalent to a subgroup G ⇢ ^G⇥H (graph of r) such that p_G : G⇥^H ! G maps G onto G bijectively.

The given map g !^h gives rise to a Lie subalgebra of g h and by the subgroup case we have proved, the corresponding Lie sub- group exists. The remaining problem is to prove the bijectivity of G

onto G when G is simply connected. ⇤

Exercise 5.2. Complete the remaining problem of Theorem5.15.

3.1.2. Ado’s imbedding theorem.

Theorem 5.16. Every (finite dimensional) Lie algebra can be regarded as a Lie subalgebra of some gl(n, R). Hence every simpley connected Lie group is a subgroup of GL(n, R). Moreover, every compact Lie group can be imbedded as a closed subgroup of some O(n, R).

For a proof, see [Bou98], chapter I.

4. Differential geometry on Lie groups

4.1. Levi-Civita connection. Any inner producth^,ieon T_eG=g uniquely determined a left invariant (Riemannian) metric on G by left translations. Namely for v, w2 ^TgG,

h^{v, w}ig :=h^dL_g ^{1v, dL}_g ^1wie.

A bi-invariant metric is a metric which is both left and right invariant.

We will shortly determine all Lie groups which admit bi-invariant metrics.

Proposition 5.17. (i)For any left invariant metrich^,ion G, and X, Y 2 g, the Levi-Civita connection is given by

rXY = ¹₂([X, Y] ad^⇤_XY ad_Y^⇤X).

(ii) Ifh^,iis bi-invariant, thenh^adZX, Yi + h^{X, ad}ZYi =0 for X, Y, Z2 g. In particular,rXY = ¹₂[X, Y].

Moreover, R(X, Y)Z= ¹₄[[X, Y], Z]and R(X, Y, X, Y) = ¹₄|[^{X, Y}]|² 0.

PROOF. Recall that the Levi-Civita connection is the unique first order differential operatorrX : C^•(TM) ! ^C•(TM) with rXY rYX = [X, Y] (torsion free) and Xh^{Y, Z}i = hrXY, Zi + h^X,rYZi (metrical). For any three vector fields X, Y, Z 2 ^C•(TM), a cyclic computation leads to

2hrXY, Zi = ^Xh^{Y, Z}i +^Yh^{Z, X}i ^Zh^{X, Y}i

h^Z,[Y, X]i h^Y,[X, Z]i h^X,[Y, Z]i^.

If X, Y, Z 2 g, all the inner products are constant in G. This leads to (i).

For (ii), the bi-invariance implies in particular that for X, Y, Z 2^g,

⌦Ad_{exp tZ}X, Ad_{exp tZ}Y↵

=h^{X, Y}i^.

Take differentiation at t = 0 leads toh^adZX, Yi + h^{X, ad}ZYi = ^{0. In} the above formula, only the term h^Z,[Y, X]i is left, hencerXY =

12[X, Y].

By the definition of the Riemann curvature operator, R(X, Y)Z =rXrYZ rYrXZ r[X,Y]Z

= ¹₄[X,[Y, Z]] ¹₄[Y,[X, Z]] ¹₂[[X, Y], Z] = ¹₄[[X, Y], Z], where the Jacobi identity is used to rewrite the second term. Finally,

R(X, Y, Z, W) :=h^R(X, Y)W, Zi = ¹₄h[[^{X, Y}], W], Zi

= ¹₄h[^W,[X, Y]], Zi = ¹₄h[^{X, Y}],[W, Z]i = ¹₄h[^{X, Y}],[Z, W]i^, where the adW invariance ofh^,i^{is used.} ⇤

It is also straightforward to deduce from (i):

4. DIFFERENTIAL GEOMETRY ON LIE GROUPS 175

Corollary 5.18. For left invariant metrics,

R(X, Y, X, Y) =|^ad⇤XY+ad_Y^⇤X|² h^ad⇤XX, ad_Y^⇤Yi

34|[^{X, Y}]|^{2 1}₂h[[^{X, Y}], Y], Xi ¹₂h[[^{Y, X}], X], Yi^.

Exercise 5.3. Show the Corollary5.18by Proposition5.17(i).

4.1.1. Lie groups with bi-invariant metrics.

Theorem 5.19. A connected Lie group G with a bi-invariant metric is complete, the exponential map is surjective and its one parameter subgroups coincides with geodesics through e2 ^G.

PROOF. By Proposition 5.17, for any l.i.v.f. X, rXX = ¹₂[X, X] = 0. Hence one parameter subgroups are the same as geodesics through e 2 G. This implies that geodesics through e can be extended infin- itely, so G is complete by the Hopf-Rinow theorem. In particular, the two exponential maps exp and exp_e(in Riemannian geometry) coin-

cide and are surjective. ⇤

Corollary 5.20. If G has a bi-invariant metric, then any Lie group immer- sion H !G is totally geodesic.

Corollary 5.21. There is no bi-invariant metrics on SL(2, R).

Exercise 5.4. When G is compact, the bi-invariant metrics always exist. For example, for G ⇢ ^O(n, R) ⇢ ^{Sn2 1}(p

n), the Euclidean metrich^{A, B}i = ^{tr ABT} is bi-invariant.

Example 5.22. The Euclidean metric on Rⁿ is clearly bi-invariant.

These examples turns out to be basically all the examples:

Theorem 5.23. A simply connected Lie group G which admits a bi-invariant metric takes the form G =^Rn⇥H for H compact and n 2 ^Z 0.

PROOF. Let zC^g be the center, which is clearly an ideal. Then h :=z^? <gis also an ideal: For a2 ^{z?, b} 2^{g, and c} 2 ^z,

h[^{b, a}], ci = h^a,[b, c]i =⁰=) [^{b, a}]2 ^z?.

(This holds true for any ideal z.) Since G is simply connected, the decomposition g = z h leads to G = Z⇥H with Lie Z = z and Lie H =h.

The center ZCG is simply connected and abelian, hence Z ⇠=^Rn for some n. Let e₁, . . . , e_h 2 ^hbe an orthonormal basis. For any X 2^h, the group H with the induced bi-invariant metric has Ricci curvature

Ric(X, X) = ¹₄

Â

i=1|[^{X, e}i]|² >0.

By translation, this show that the Ricci curvature has a positive lower bound on H. Hence by the theorem of Bonnet-Meyer H must be

compact. ⇤

5. Homogeneous spaces

5.1. General homogeneous spaces. Let H < G be a closed Lie subgroup. Then the coset space G/H ={^gH|^g 2 ^G}has a natural C^• manifold structure such that the projection map p : G ! ^G/H is C^•. G acts transitively on G/H by left translations. Also the sta- bilizer (also called isotropy subgroup) G_[gH] ⇠= H at each point[gH]. Conversely, given a transitive C^•action G⇥^M! ^{M on a C• mani-} fold M. Let H =G_m0 for some m0 2 M. Then G/H ⇠= M. A space of the form G/H is called a homogeneous space. If HCG then G/H is a also Lie group.

Example 5.24. Here are some standard examples:

(i) O(n)⇥^{Sn 1} ! ^{Sn 1} is transitive and O(n)_en ⇠= O(n 1). So S^{n 1}⇠=O(n)/O(n 1).

(ii) U(n)⇥^{S2n 1} ! ^{S2n 1} is transitive and U(n)_en ⇠=U(n 1). So S^{2n 1} ⇠=U(n)/U(n 1). Similarly, S^{2n 1} ⇠=SU(n)/SU(n 1). In particular, S¹ ⇠=U(1)and S³ ⇠=SU(2)are Lie groups.

5. HOMOGENEOUS SPACES 177

(iii) Real projective space: RP^{n 1}=S^{n 1}/{±¹}^{. So}

RP^{n 1} ⇠=O(n)/O(n 1)⇥ {±¹} ⇠=SO(n)/O(n 1). (iv) Complex projective space: CP^{n 1} = (^Cn\{⁰})^{/C⇥. So}

CP^{n 1}⇠=S^{2n 1}/S¹ ⇠=U(n)/U(n 1)⇥^U(1) ⇠=SU(n)/U(n 1). (v) Stiefel manifold of k-frames: GL(n, R)⇥ ^˜Vn,k ! ^˜Vn,k is tran-

sitive where ˜V_n,k is the set of all k frames in Rⁿ. For S = {^e1, . . . , e_k}^,

G_S =

( I A 0 B

!

2 ^GL(n, R) )

.

So ˜V_n,k ⇠= GL(n, R)/G_S. For V_n,k the set of all orthonormal k-frames,

V_n,k ⇠=O(n)/O(n k) ⇠=SO(n)/SO(n k). For complex Stiefel manifold V_n,k^C of k-frames in Cⁿ,

V_n,k^C ⇠=U(n)/U(n k) ⇠=SU(n)/SU(n k).

(vi) Grassmannian manifolds: Let G_n,kbe the set of all k-dimensional subspaces in Rⁿ, then G_n,k ⇠=V_n.k/O(k) ⇠=O(n)/O(n k)⇥ O(k) and dim G_n,k = k(n k). Similarly for the complex Grassmannian

G^C_n,k ⇠=V_n.k^C/U(k) ⇠=U(n)/U(n k)⇥^U(k).

It is a complex manifold with dimCG_n,k^C = k(n k). Grass- mannians generalizes projective spaces. They are very im- portant for the study of vector bundles.

(vii) Poincar´e’s upper half plane: Let H = {^z 2 ^C|^{Im z} > 0}^. SL(2, R)acts on H transitively by

a b c d

!

z = ^az+b cz+d,

The stabilizer at i is SO(2, R), so H⇠=SL(2, R)/SO(2, R). H is non-compact and analytically isomorphic to the unit disk,

an example of the bounded symmetric domains. The double coset space

G\^H⇠=G\^SL(2, R)/SO(2, R)

with G < SL(2, R) contains all Riemann surfaces of genus g 2 (uniformization theorem). If G < SL(2, Z)is an arith- metic subgroup, then it represents certain moduli spaces of elliptic curves.

5.2. Riemannian homogeneous spaces. For further study, we need notions and results from differential geometry. Let (M, ds²) be a Riemannian manifold. That is, ds²is a family of inner products h^,i_x^{on T}xM varying smoothly in x2 M. An isometry g : M ! ^{M is} a C^• map such that g^⇤ds² = ds². Equivalently, h^dg(v), dg(w)i_g(x) = h^{v, w}i_xfor all v, w 2 ^TxM. It is known that the full isometry group

G ⌘^O(M, ds²) :={^g2 ^C•(M, M)|^g⇤ds2 =ds²}

is a Lie group. For each x 2 ^{M, G}x induces a linear representation r : G_x ! ^O(T_xM). Since an isometry maps geodesics to geodesics, r(h) determines h through the geodesic exponential map exp_x : U ⇢ T_xM! M and thus r is injective. In particular, each isotropy group G_xis compact.

A connected Riemannian manifold(M, ds²)is Riemannian homo- geneous if for any two points x, y2 M, there exists an isometry g such that g(x) = y. In this case, we have a transitive action G⇥^M ! ^M and M ⇠= G/G_x. In particular, M is homogeneous with compact isotropy.

Proposition 5.25. A Riemannian homogeneous space is complete.

Exercise 5.5. Prove the Proposition5.25

A natural question arises: When is a general homogeneous space M ⇠= G/H Riemannian homogeneous? That is we are searching for met- rics on G/H such that G acts on it as isometries. Such a metric is called a G-invariant metric, which may not always exist. Also there could

5. HOMOGENEOUS SPACES 179

be different ways to represent M as a group quotient. Thus we need to clarify these issues first.

In considering the homogeneous structure we may assume that G acts on G/H effectively in the sense that any g 2 ^G\{^e} ^{acts non-} trivially. Indeed,

g[kH] = [kH] () ^{k 1gk} 2 ^H () ^g2 ^{kHk 1.}

Hence g acts trivially if and only if g 2 ^Tk2GkHk ¹ =: H0. It is clear that H0is the largest subgroup of H with H0C^{G. Thus}

G/H ⇠= ^G/H0

H/H0 =: G1/H1

has an effective G1action.

Denote G ! ^{G/H by g} 7! ^{¯g :}= gH. There is a natural identi- fication T_¯eG/H = g/h. Since Ad_H and adh act on g and leave the subspace h invariant, we get the natural adjoint actions on g/h in- duced from p : g!^g/h.

Lemma 5.26. For h2 ^{H, dL}h ⌘^Adhmodulo h on T_¯eG/H.

PROOF. Differentiate the equation h exp(tX)H =h exp(tX)h ¹H.

⇤ Proposition 5.27. A G-invariant metric on the homogeneous space M = G/H is equivalent to an inner product h^,i ^{on g/h} ⇠= T_¯eM which is Ad_H-invariant. If H is connected, this is equivalent to “adh-invariance”:

Namely, for A2 ^{h, X, Y} 2 ^g/h,

h^adAX, Yi + h^{X, ad}AYi =^0.

PROOF. The necessity of Ad_H invariance onh^,ifollows from the above lemma. To see its sufficiency, we simply define for v, w 2 T_¯gG/H

h^{v, w}i¯g :=h^dL_g ^{1v, dL}_g ^1wi^.

Then h^{v, w}i_gh = h^dL_h ^1dL_g ^{1v, dL}_h ^1dL_g ^1wi = h^dL_g ^{1v, dL}_g ^1wi = h^{v, w}i¯g. Hence the left invariant metric on G/H is well defined.

The remaining statement on adhis left as an exercise. ⇤

Exercise 5.6. Show the remaining statement on adhin Proposition5.27.s

Theorem 5.28. Assume that G acts on M = G/H effectively. Then M admits a G invariant metric if and only if Ad_H ⇢ ^GL(g) has compact closure.

Moreover, G invariant metrics on G/H are precisely left invariant met- rics on G which is also H bi-invariant.

PROOF. ()) Write G/H = G^⇤/H^⇤ with G^⇤ = O(M, ds²), H^⇤ = G^⇤_¯e. Then G ! ^G⇤, and hence g ! ^g⇤, is injective. We know that im Ad_H⇤ ⇢^GL(g^⇤)is compact since H^⇤is. To realize it inside the or- thogonal group we simply pick an arbitrary inner product on g^⇤and average it by this compact image so that the resulting inner product h^,i^{⇤ on g⇤ is Ad}H^⇤-invariant. (This is the same procedure to con- struct bi-invariant metrics on a compact Lie group.) Leth^,i = h^,i|g^⇤. Then it is clear that the image Ad_H ⇢^O(g,h^,i)^.

((^{) If Ad}Hhas compact closure K ⇢^GL(g), starting with any in- ner product on g the averaging procedure over K again produces an Ad_H-invariant inner product h^,i on g. Let p := h^? which is isomorphic onto g/h under p. It is clear that Ad_H(p) ⇢ ^{p since} h^AdH(h^?), hi = h^{h?, Ad}Hhi = ^{0. Thus} h^,i|p defines the desired

Ad_H-invariant inner product on g/h. ⇤

6. Symmetric spaces

6.1. Local and global symmetric spaces. A connected Riemann- ian manifold(M, ds²) is a symmetric space if for all x 2 M there is an isometry s_x : M! M such that x is an isolated fixed point of s_x and ds_x : T_xM ! ^TxM sends v ! v. It is locally symmetric if s_x exists only locally.

To construct local isometry, consider the map ˆs_x which reverses geodesics g with g(0) = x:

ˆsx(g(t)) =g( t).

6. SYMMETRIC SPACES 181

This coincides with s_x when (M, ds²) is locally symmetric, because local isometry maps geodesics to geodesics and geodesics are deter- mined by initial conditions g(0)and g⁰(0).

Proposition 5.29. Symmetric spaces are Riemannian homogeneous.

PROOF. Let G = O(M, ds²). In particular it contains the sub- group generated by the symmetries s_x, x 2 M. We only need to show that G acts on M transitively. For any x, y which are joined by a geodesic g with g(0) = x, g(T) = y, let s_z be the isometry with z =g(T/2). Clearly s_z(x) = y.

In general, x and y can be joined by a sequence of broken geodesics g_i. Then we take the isometry to be the composite of those s_zi’s. ⇤ Proposition 5.30. In terms of curvature,(M, ds²)is locally symmetric if and only if thatr^R =0, that is the curvature tensor is parallel.

PROOF. Indeed, “)” is easy: For any tensor T of even degree, rT is of odd degree. Since s_xis a local isometry, we get

r^T =s^⇤_x(r^T) = r^T,

hence r^T = 0. “(” is a consequence of the Cartan Theorem (cf.

theorem3.47). ⇤

Corollary 5.31. Simply connected locally symmetric spaces are symmetric.

This follows fromr^R =0 and the Cartan-Ambrose-Hicks Theorem (cf. theorem3.48).

Theorem 5.32. A connected Lie group G with a bi-invariant metric, e.g., for G compact times Euclidean, is a G⇥G symmetric space.

PROOF. Let G⇥G act on G by (g, h)a = gah ¹. Then G ⇠= G⇥G/G, with the stabilizer at e 2 G being the diagonal group iso- morphic to G. We claim that the map

s_g : h7! ^{gh 1g} defined the symmetry at g.

We check this for s_e : h 7! ^{h 1}first. Indeed, near e 2 ^{G the map} s_e is given by exp X 7! ^exp( X). From this we see that s_e reverses one parameter subgroups and ds_e = Id_TeG.

To show that s_e is an isometry, consider any point g 2 ^{G and} a vector v = dLgX 2 ^TgG with X 2 ^TeG. Then v = g⁰(0) where g(t) = g exp tX. Then s_eg(t) = exp( tX)g ¹ = R_g 1exp( tX). Hence

(ds_e)_gv= (ds_e)_gg⁰(0) = dR_g 1X.

With w = dL_gY, we compute by using bi-invariance of the metric that

⌦(dse)_gv,(dse)_gw↵

=^D dR_g 1X, dR_g 1YE

=h^{X, Y}i = h^{v, w}i^. For general g 2 ^{G, s}g = L_gR_gs_e is the composite of three isome- tries, hence s_gis also an isometry.

It remains to check that (ds_g)_g = Id_TgG. As before let v = dL_gX 2 ^TgG. g(t) = g exp tX. Then s_gg(t) = g exp( tX)g ¹g = g exp( tX). Hence

(ds_g)_gv= (ds_g)_gg⁰(0) = dL_gX= v.

This completes the proof that G is symmetric. ⇤ 6.2. Symmetric spaces via Lie algebras. When is a homogeneous space M = G/H symmetric? This will be reduced to a problem on Lie algebras. Recall that s 2 Aut G ia an involution if s 6= ^IdG and s²=Id_G.

Theorem 5.33. (Basic structure theorem for symmetric spaces).

(a) Let M=G/H be a symmetric space with G =O(M, ds²), then s : G!^{G; g}7! ^s(g) = sxgsx

is an involution of G and K = G^s is a closed subgroup contain- ing H such that K = H . H contains no non-trivial normal subgroup of G.

(b) Conversely, let G be a Lie group with an involution s. Let K =G^s and fix a G-invariant metric h^,i ^{on M} = G/K. Let ¯s be the

6. SYMMETRIC SPACES 183

diffeomorphism on M induced from s. Ifh^,iis ¯s-invariant then M is symmetric.

(c) A simply connected Lie group G with an involution s is equivalent to a Z₂graded decomposition g=h pin the sense that

[h, h]⇢^h, [h, p] ⇢^p, [p, p] ⇢^h.

Given s, the subalgebra h and the subspace p are the±1 eigenspace of ds : g !^grespectively.

PROOF. For (a), s is an involution since

s(gh) =sxghsx = (sxgsx)(sxhsx) = s(g)s(h)

and s²(g) = s(s_xgs_x) = s_x(s_xgs_x)s_x = g. One can check that K\^H is open and closed in K, hence F = H . H contains no non-trivial normal subgroup of G since otherwise the action of G on M is not effective.

For (b),h^,iis ¯s-invariant means that ¯s is an isometry on M. Since (ds_e)² = idg, we have the ±1 eigenspace decomposition g = k p and T¯eM ⇠= p. So d¯s¯e = id_T¯eM. Thus s¯e := ¯s is the symmetry at ¯e. We noticed that a Riemannian homogeneous space which is symmetric at one point is then symmetric everywhere. Indeed, the symmetry at ¯g is given by

s¯g :=L_g ¯s L_g 1 =L_g s L_g 1 (mod K).

It is clear that s_¯g is well defined, s_¯g(¯g) = ¯g, s²_¯g = id_M and s_¯g is an isometry. The property(ds_¯g)_¯g = id can be easily checked as in the Lie group case.

For (c), let v2 ^{hand w} 2^{p. Then}

ds[v, w] = [ds(v), ds(w)] = [+v, w] = [v, w].

Hence[h, p]⇢p. The proofs of the other two inclusions are similar.

Conversely, given Z2 graded decomposition g = h p, define a Lie algebra morphism T : g ! ^{g with T}|h = id and T|^p = id.

Since G is simply connected, this gives rise to a Lie group morphism

s : G ! ^{G. Since d}(s²) = ds ds = T T = idg, we conclude that s²=id_Gby the unique correspondence between morphisms. ⇤ Exercise 5.7. Show that in K\H in Theorem5.33is open.

So the problem on constructing symmetric spaces is reduced to finding a Z2 decomposition g = h pwith compatible inner prod- ucth^,ion p. Combining with Proposition5.27and Theorem5.28, the corresponding metric on G/H can be constructed from a left invari- ant metric on G which is bi-invariant on H. Examples are provided by the semi-simple Lie groups.

6.3. Examples via semi-simple Lie groups. Let g be a Lie alge- bra over F =^Ror C. Define the Killing form

B(X, Y) = tr(ad X ad Y); B : g⇥^g !^F.

It is the main source to provide adjoint invariant quadratic forms:

Lemma 5.34 (Exercise). B is ad-invariant: B(ad_ZX, Y) +B(X, ad_ZY) = 0.

Exercise 5.8. Show Lemma5.34

We say that g is semi-simple if B is non-degenerate, g is simpleif g is not abelian and g contains no proper Lie ideals.

Theorem 5.35. gis semi-simple if and only if g is a direct sum of simple ideals.

The proof for the “only if” part is similar to the proof of Theorem 5.23 by using B in place of the bi-invariant metric. The “if” part follows from the Killing-Cartan criterion which will not be presented here.

We say G is semi-simple (simple) if g is semi-simple (simple). For G simple, every bi-invariant metrich^,iis determined by its value at e and proportional to the Killing form.

Example 5.36. We give two main series of examples of symmetric spaces G/H that arise from semi-simple Lie groups G. Notice that we had seen that H may always be assumed to be compact.

6. SYMMETRIC SPACES 185

(i) Type I: G is compact and B is negative definite. E.g.

SO(2n)/U(n), SO(p+q)/SO(p)⇥^SO(q), SU(2n)/SO(n), SU(p+q)/SU(p)⇥^U(q),

Sp(n)/U(n), Sp(p+q)/Sp(p)⇥^Sp(q).

These includes spheres, projective spaces and Grassmanni- ans.

(ii) Type II: G is non-compact and B is indefinite.

In this case there is a maximal compact subalgebra h and Z₂ decomposition g = h p (the Cartan decomposition).

Moreover, B is negative definite on h and positive definite on p.

E.g.

SO(p, q)/SO(p)⇥^SO(q),

with respect to the indefinite inner product Â_i^p₌₁x²_i Â_j^p₌⁺_p^q₊₁x²_j. For q =1, we get the Poincar´e upper half space H^p.

Similarly

SU(p, q)/U(p)⇥^SU(q),

with respect to the indefinite Hermitian inner product Â_i^p₌₁|^zi|² Â^p_j=⁺_p^q₊₁|^zj|^{2. For q} = 1, we get the unit ball in C^p. Other ex- amples are SL(n, R)/SO(n, R) (for n = 2 we had seen that this gives Poincar´e upper half plane), SO(n, C)/SO(n, R), SL(n, C)/SU(n).

The main theory of Cartan says that any simply connected sym- metric space may be decomposed into a product of three factors

M= M0⇥^M+⇥^{M ,}

where M₀is a Euclidean space, M₊is of compact type and M is of non-compact type. Both M+ and M may be further decomposed into irreducible factors and each factor can be constructed from cer- tain semi-simple Lie algebras in a way similar to the above examples.

The details can be found in Helgason’s classic text.

7. Curvature for symmetric spaces

7.1. Riemannian submersion. A map f : (M, ¯g^¯ ) ! (^{M, g}) is a Riemannian submersion if it is a C^• submersion f : ¯M ! ^{M and} d f : T^hM¯ ! TM is an isometry, where T ¯M = T^vM¯ T^hM is the¯ orthogonal decomposition defined by: T^v_¯pM :¯ =ker d f_¯pis the vertical tangent space which is also the tangent space of the fiber submanifold M¯_p := f ¹(p) with p= f(¯p), and T^h_¯pM :¯ = (T^v_¯pM¯)^? is the horizontal tangent space.

If f(¯p) = p and X 2 ^TpM, then there is a unique horizontal lift

¯X 2 ^T¯pM such that d f¯ ¯p ¯X = X. Under such a lifting, one may relate the Levi-Civita connection and Riemannian curvature tensor on M in terms of those on ¯M. This is particularly useful in dealing with Riemannian homogeneous spaces or symmetric spaces of the form G ! ^M =G/H. The following simple relations, due to O’Neill, can be found in most textbook in Riemannian geometry. The proofs are left as exercises.

Theorem 5.37. Let f : ¯M !M be a Riemannian submersion. Then (a) ¯r¯X ¯Y=rXY+¹₂[¯X, ¯Y]^vfor any vector fields X, Y on M and any

lifts ¯X, ¯Y. The vertical component[¯X, ¯Y]^vis tensorial in X and Y.

(b) For any X, Y, Z, W 2 ^TpM,

R(X, Y, Z, W) = ¯R( ¯X, ¯Y, ¯Z, ¯W) + ¹₂h[^{¯X, ¯Y}]^v,[¯Z, ¯W]^vi +¹₄h[^{¯X, ¯Z}]^v,[¯Y, ¯W]^vi ¹₄h[^{¯X, ¯W}]^v,[¯Y, ¯Z]^vi^. (c) R(X, Y, X, Y) = ¯R(¯X, ¯Y, ¯X, ¯Y) +³₄|[^{¯X, ¯Y}]^v|^2.

Combining with the curvature formula for Lie groups, we may achieve

Theorem 5.38. (a) Let G be a compact semi-simple Lie group with an involution s (s² = id). Let g = h p be the ± ^eigenspace decomposition. Then B defines a bi-invariant metric on G and G/H is a symmetric space with curvature

R(X, Y, X, Y) =|[^{X, Y}]|^2.

7. CURVATURE FOR SYMMETRIC SPACES 187

(b) Let G be a non-compact semi-simple Lie group and g = h p be a Z2 decomposition as in Example 5.36 (ii). Then B|^{p defines} an invariant metric on G/H and make it a symmetric space with curvature

R(X, Y, X, Y) = |[^{X, Y}]|^2.

PROOF. We only give the proof for (b). The proof for (a) is similar and easier.

For the Riemannian submersion G! G/H with the left invariant metric on G defined by

h^,i|^h = B|^h, h^,i|^p= B|^p,

we have T_e^vG = h and T_e^hG = p. Let X, Y, Z 2 ^T[H]G/H ⇠= p. By Corollary5.18and Theorem5.37(c), we get

R(X, Y, X, Y) =|^ad⇤XY+ad_Y^⇤X|² h^ad⇤XX, ad^⇤_YYi

34|[^{X, Y}]^p|^{2 1}₂h[[^{X, Y}], Y], Xi ¹₂h[[^{Y, X}], X], Yi^. Since[p, p] ⇢^h,[X, Y]^p =0. Also[X, Z] 2 ^himplies thath^ad⇤XY, Zi = h^Y,[X, Z]i = 0, hence ad^⇤_XY 2 h. Now for T 2 h, the left invariant metrich^,ion G is also right invariant under H (i.e. ad-invariance of B) says that

h[^{T, X}], Yi + h^X,[T, Y]i = ^0,

which is equivalent toh^ad⇤XY+ad^⇤_YX, Ti =0, hence ad^⇤_XY+ad^⇤_YX = 0. By setting X =Y we get also ad^⇤_XX =0. So only the last two terms remained in the curvature formula.

Since[[X, Y], Y],[[Y, X], X] 2 p, we compute by ad-invariance of B

R(X, Y, X, Y) = ¹₂B([[X, Y], Y], X) ¹₂B([[Y, X], X], Y)

= ¹₂B([X, Y],[X, Y]) +¹₂B([Y, X],[Y, X]).

Since[X, Y]2 h, this gives |[^{X, Y}]|². The proof is complete. ⇤

8. Topology of Lie groups and symmetric spaces

For a group action G on a manifold M, a differential form w 2 L^p(M)is an invariant form if g^⇤w =w for all g2 ^G.

Theorem 5.39. Let M = G/H be a symmetric space with compact G, Then

H^⇤(M, R) ⇠= A^⇤_inv(M) = H^⇤(M).

PROOF. By the de Rham theorem H^⇤(M, R) ⇠=H_dR^⇤ (M, R), hence we need to show that every invariant form is also closed and every closed differential form is equivalent to an unique invariant form.

Step 1. w 2 ^Apinv(M) ) ^dw =0. We show first that ˆw := s^⇤_xw 2 A_inv^p (M)for any x 2 M. For this, recall s_xg= s(g)s_x, where s is the involution. So

g^⇤ ˆw =g^⇤s^⇤_xw =s^⇤_xs(g)^⇤w =s^⇤_xw = ˆw.

From ds_x = Id on T_xM we get ˆw|x = ( 1)^pw|x. Together with the invariance of w and ˆw, this implies that ˆw = ( 1)^pw.

Now dw and d ˆw are also invariant forms (since d commutes with g^⇤) and s^⇤_xdw = d(s^⇤_xw) = d ˆw 2 ^Ainv^p+1(M). So similarly d ˆw = ( 1)^p+1dw. But we also have d ˆw = ( 1)^pdw, hence we conclude dw =0.

Step 2. dw =0) ^w ⇠ ^˜w 2 ^Ainv^p (M). We prove this step for any homogeneous space with compact G. On a Lie group G, pick up any left invariant metric, its volume form give rise to invariant measure dµ which can be normalized to have total volume 1 if G is compact.

For any g 2 ^{G, g⇤w} ⇠ ^w since the map g : M ! ^{M is homo-} topic to identity. This holds true for any affine linear combination:

 µig^⇤_iw ⇠^w with  µi =1. Taking limits (by definition of Riemann sum) we find that

˜w := Z

Gg^⇤wdµg ⇠^w.

˜w2 ^L_inv^p (M)since for any h2 ^G, h^⇤w =h^⇤Z

Gg^⇤wdµg = Z

Gh^⇤g^⇤wh^⇤dµg = Z

G(gh)^⇤wdµ_gh= ˜w.

8. TOPOLOGY OF LIE GROUPS AND SYMMETRIC SPACES 189

Step 3. We show that an exact invariant form must be zero. Fix a G invariant metric on M. We recall the Hodge star operator ⇤ ^: L^pT_x^⇤M ! ^{Ln pT}x^⇤M. The G invariance implies that ⇤^g⇤ = g^⇤⇤ ^for any g 2 ^{G. Hence w} 2 ^A_inv^p (M) ) ⇤^w 2 ^A_inv^p (M). In particular d(⇤^w) = 0 by step 1.

For w, h2 ^Ap(M),

h^{w, h}i = Z

Mw^ ⇤^h

is a inner product on A^p(M). Now for w = dh an exact invariant p form,

h^{w, w}i = Z

Mdh^ (⇤^w) = Z

Md(h^ (⇤^w)) ( 1)^p Z

Mh^^d(⇤^w) =0 by Stokes theorem and d(⇤^w) =0. Hence w=0 as desired.

Step 4. It remains to show that invariant forms are precisely har- monic forms. If w 2 ^A_inv^p (M), we have just seen that

dw=0 and d^⇤w = ( 1)^{p(n p)}⇤^d⇤^w =0.

Thus 4^w = (dd^⇤+d^⇤d)w = 0. If we assume the Hodge theorem which says that H_dR^⇤ (M, R) ⇠=H^⇤(M), then harmonic forms must be invariant forms.

We would like to give a direct proof: First notice that4^h = 0 if and only if that dh =0 and d^⇤h =0, which is seen from the identity

h4^{h, h}i = ||^dh||²+||^d⇤h||^2.

Let 4^w = 0. For any X 2 g, we compute by Cartan’s homotopy formula

L_Xw =i_Xdw+di_Xw =di_Xw.

The invariance of the metric implies that4commutes with L_X, hence L_Xwis also harmonic. Soh^LXw, L_Xwi = h^LX, di_Xwi = h^d⇤LXw, i_Xwi = 0 and then L_Xw =0. By definition of Lie derivatives this implies that w is an invariant form. The proof is completed. ⇤