Q(±, q) = q(q ’ n) +

1 for ± < 0.

This result is due to [Schmidt, 1989].

Proof. Since the signature is constant on connected components we have to de-

1 1

termine it only for ± = n and ± = n ’ 1. In a basis for T M and its dual basis

for T — M the bilinear form h ∈ S 2 Tx M has a symmetric matrix. If the basis is

—

orthonormal for g we have (for At = A and C t = C)

’ Idq ’A ’B

0 A B

H = g ’1 h = = ,

Bt Bt

0 Idn’q C C

which describes a typical matrix in the space Lsym,g (Tx M, Tx M ) of all matrices

H ∈ L(Tx M, Tx M ) which are symmetric with respect to gx .

1

Now we treat the case ± = n . The standard inner product tr(HK t ) is positive

de¬nite on Lsym,g (Tx M, Tx M ), and the linear mapping K ’ K t has an eigenspace

of dimension q(n ’ q) for the eigenvalue ’1 in it and a complementary eigenspace

for the eigenvalue 1. So tr(HK) has signature q(n ’ q).

1

For the case ± = n ’1 we again split the space Lsym,g (Tx M, Tx M ) into the subspace

with 0 on the main diagonal, where γg (h, k) = tr(HK) and where K ’ K t has

±

again an eigenspace of dimension q(n ’ q) for the eigenvalue ’1, and the space

±

of diagonal matrices. There γg has signature 1 as determined in the proof of

(45.5).

45.17

46 46. The Korteweg “ De Vries equation as a geodesic equation 497

45.18. The submanifold of almost symplectic structures.

A 2-form ω ∈ „¦2 (M ) = C ∞ (M ← Λ2 T — M ) can be non degenerate only if M is

of even dimension dim M = n = 2m. Then ω is non degenerate if and only if

ω § · · · § ω = ω m is nowhere vanishing. Usually this latter 2m-form is regarded as

the volume form associated with ω, but a short computation shows that we have

m

1

m! |ω |.

vol(ω) =

tr(ω ’1 •)ω m for all • ∈ „¦2 (M ).

1

This implies m• § ω m’1 = 2

45.19. Theorem. The space „¦2 (M ) of non degenerate 2-forms is a splitting

nd

geodesically closed submanifold of (B, G± ) for each ± = 0.

Proof. We consider a geodesic c(t) = c0 e(a(t) Id +b(t)H0 ) for the metric G± in B

starting at c0 in the direction of h as in (45.11). If c0 = ω ∈ „¦2 (M ) then h ∈

nd

’1

2

„¦c (M ) if and only if H = ω h is symmetric with respect to ω, since we have

ω(HX, Y ) = ωω ’1 hX, Y = hX, Y = h(X, Y ) = ’h(Y, X) = ’ω(HY, X) =

ω(X, HY ). At a point x ∈ M we may choose a Darboux frame (ei ) such that

ω(X, Y ) = Y t JX where

0 Id

J= .

’ Id 0

Then h is skew if and only if JH is a skew symmetric matrix in the Darboux frame,

t

or JH = H t J. Since (eA )t = eA the matrix ea(t) Id +b(t)H0 then has the same prop-

erty, c(t) is skew for all t. Thus, „¦2 (M ) is a geodesically closed submanifold.

nd

45.20. Lemma. For a non degenerate 2-form ω the signature of the quadratic

form • ’ tr(ω ’1 •ω ’1 •) on Λ2 Tx M is m2 ’ m for ± > 0 and m2 ’ m + 1 for

—

± < 0.

Proof. Use the method of (45.5) and (45.17). The description of the space of

matrices can be read o¬ the proof of (45.19).

45.21. Symplectic structures. The space Symp(M ) of all symplectic structures

is a closed submanifold of (B, G± ). For a compact manifold M it is splitting by

the Hodge decomposition theorem. For dim M = 2 we have Symp(M ) = „¦2 (M ),

nd

so it is geodesically closed. But for dim M ≥ 4 the submanifold Symp(M ) is not

geodesically closed. For ω ∈ Symp(M ) and •, ψ ∈ Tω Symp(M ) the Christo¬el

form “± (•, ψ) is not closed in general.

ω

46. The Korteweg “ De Vries

Equation as a Geodesic Equation

This section is based on [Michor, Ratiu, 1997], an overview of related ideas can

be found in [Segal, 1991]. That the Korteweg “ de Vries equation is a geodesic

equation is attributed to [Gelfand, Dorfman, 1979], [Kirillov, 1981] or [Ovsienko,

Khesin, 1987]. The curvature of a right invariant metric on an (in¬nite dimensional)

Lie group was computed by [Arnold, 1966a, 1966b], see also [Arnold, 1978].

46

498 Chapter IX. Manifolds of mappings 46.3

46.1. Recall from (44.1) the principal bundle of embeddings Emb(M, N ), where

M and N are smooth ¬nite dimensional manifolds, connected and second count-

able without boundary such that dim M ¤ dim N . The space Emb(M, N ) of all

embeddings from M into N is an open submanifold of C ∞ (M, N ), which is stable

under the right action of the di¬eomorphism group. Then Emb(M, N ) is the total

space of a smooth principal ¬ber bundle with structure group the di¬eomorphism

group. The base is called B(M, N ), it is a Hausdor¬ smooth manifold modeled on

nuclear (LF)-spaces. It can be thought of as the ”nonlinear Grassmannian” of all

submanifolds of N which are of type M .

Recall from (44.24) that if we take a Hilbert space H instead of N , then B(M, H)

is the classifying space for Di¬(M ) if M is compact, and the classifying bundle

Emb(M, H) carries also a universal connection.

46.2. If (N, g) is a Riemannian manifold then on the manifold Emb(M, N ) we have

an induced weak Riemannian metric given by

g(s1 , s2 ) vol(e— g).

Ge (s1 , s2 ) =

M

Its covariant derivative and curvature were investigated in [Binz, 1980] for the case

that N = Rdim M +1 with the standard inner product, and in [Kainz, 1984] in the

general case. We shall not reproduce the general formulas here. This weak Rie-

mannian metric is invariant under the action of the di¬eomorphism group Di¬(M )

by composition from the right, thus it induces a Riemannian metric on the base

manifold B(M, N ), which can be viewed as an in¬nite dimensional non-linear ana-

logue of the Fubini-Study metric on projective spaces and Grassmannians.

46.3. Example. Let us consider the metric on the space Emb(R, R) of all embed-

dings of the real line into itself, which contains the di¬eomorphism group Di¬(R) as

an open subset. We could also treat Emb(S 1 , S 1 ), where the results are the same.

∞

f ∈ Emb(R, R), h, k ∈ Cc (R, R).

Gf (h, k) = h(x)k(x)|f (x)| dx,

R

We shall compute the geodesic equation for this metric by variational calculus. The

energy of a curve f of embeddings (without loss of generality orientation preserving)

is the expression

b b

ft2 fx dxdt.

1 1

E(f ) = Gf (ft , ft )dt =

2 2

a a R

If we assume that f (x, t, s) depends smoothly on one variable more, so that we have

a variation with ¬xed endpoints, then the derivative with respect to s of the energy

46.3

46.3 46. The Korteweg “ De Vries equation as a geodesic equation 499

is given by

b

ft2 fx dxdt

‚ ‚ 1

‚s |0 E(f ( ‚s |0 2

, s)) =

a R

b

(2ft fts fx + ft2 fxs )dxdt

1

= 2

a R

b

= ’1 (2ftt fs fx + 2ft fs ftx + 2ft ftx fs )dxdt

2

a R

b

ft ftx

=’ ftt + 2

fs fx dxdt,

fx

a R

so that the geodesic equation with its initial data is

ft ftx ∞

, f ( , 0) ∈ Emb+ (R, R), ft ( , 0) ∈ Cc (R, R)

ftt = ’2

(1)

fx

= “f (ft , ft ),

∞ ∞ ∞

where the Christo¬el symbol “ : Emb(R, R) — Cc (R, R) — Cc (R, R) ’ Cc (R, R)

is given by symmetrization

hkx + hx k (hk)x

“f (h, k) = ’ =’

(2) .

fx fx

For vector ¬elds X, Y on Emb(R, R) the covariant derivative is given by the ex-

pression Emb Y = dY (X) ’ “(X, Y ). The Riemannian curvature R(X, Y )Z =

X

( X Y ’ Y X ’ [X,Y ] )Z is then expressed in terms of the Christo¬el symbol

by the usual formula

Rf (h, k) = ’d“(f )(h)(k, ) + d“(f )(k)(h, ) + “f (h, “f (k, )) ’ “f (k, “f (h, ))

h (k x x

)

k (h x)x

hx (k )x kx (h )x f f

x x

=’ ’

+ +

2 2

fx fx fx fx

1

fxx hx k ’ fxx hkx + fx hkxx ’ fx hxx k + 2fx hkx x ’ 2fx hx k x

(3) = 3

fx

The geodesic equation can be solved in the following way: If instead of the obvious

∞ 2

framing we choose T Emb = Emb —Cc (f, h) ’ (f, hfx ) =: (f, H) then the

geodesic equation becomes Ft = ‚t (ft fx ) = fx (ftt + 2 ftfftx ) = 0, so that F = ft fx

‚ 2 2 2

x

is constant in t, or ft (x, t)fx (x, t)2 = ft (x, 0)fx (x, 0)2 . Using that one can then use

separation of variables to solve the geodesic equation. The solution blows up in

¬nite time in general.

Now let us consider the trivialization of T Emb(R, R) by right translation (this is

clearest for Di¬(R)), then we have

u : = ft —¦ f ’1 , in more detail u(y, t) = ft (f ( , t)’1 (y), t)

1

ux = (ftx —¦ f ’1 ) ,

fx —¦ f ’1

1 ftx ft

ut = ftt —¦ f ’1 ’ (ftx —¦ f ’1 ) (ft —¦ f ’1 ) = ’3 —¦ f ’1

’1

fx —¦ f fx

ut = ’3ux u.

(4)

where we used Tf (Inv)h = ’T (f ’1 ) —¦ h —¦ f ’1 .

46.3

500 Chapter IX. Manifolds of mappings 46.5

46.4. Geodesics of a right invariant metric on a Lie group. Let G be a

Lie group which may be in¬nite dimensional, with Lie algebra g. Recall (36.1) that

µ : G—G ’ G denotes the multiplication with µx left translation and µx right trans-

lation by x, and (36.10) that κ = κr ∈ „¦1 (G, g) denotes the right Maurer-Cartan

’1

form, κx (ξ) = Tx (µx ).ξ. It satis¬es (38.1) the right Maurer-Cartan equation

dκ ’ 1 [κ, κ]§ = 0. Let : g — g ’ R be a positive de¬nite bounded inner

,

2

product. Then

’1 ’1

Gx (ξ, ·) = T (µx ).ξ, T (µx ).· = κ(ξ), κ(·)

(1)

is a right invariant Riemannian metric on G, and any right invariant bounded

Riemannian metric is of this form, for suitable , .

Let g : [a, b] ’ G be a smooth curve. The velocity ¬eld of g, viewed in the right

’1

trivialization, is right logarithmic derivative δ r g(‚t ) = T (µg )‚t g = κ(‚t g) =

(g — κ)(‚t ). The energy of the curve g is given by

b b

g — κ(‚t ), g — κ(‚t ) dt.

1 1

E(g) = Gg (g , g )dt =

2 2

a a

For a variation g(t, s) with ¬xed endpoints we have then, using the right Maurer-

Cartan equation and integration by parts

b

2 ‚s (g — κ)(‚t ), g — κ(‚t ) dt

1

‚s E(g) = 2

a

b

‚t (g — κ)(‚s ) ’ d(g — κ)(‚t , ‚s ), g — κ(‚t ) dt

=

a

b

(’ (g — κ)(‚s ), ‚t (g — κ)(‚t ) ’ [g — κ(‚t ), g — κ(‚s )], g — κ(‚t ) ) dt

=

a

b

(g — κ)(‚s ), ‚t (g — κ)(‚t ) + ad(g — κ(‚t )) (g — κ(‚t )) dt

=’

a

where ad(g — κ(‚t )) : g ’ g is the adjoint of ad(g — κ(‚t )) with respect to the inner

product , . In in¬nite dimensions one also has to check the existence of this

adjoint. In terms of the right logarithmic derivative u : [a, b] ’ g of g : [a, b] ’ G,

’1

given by u(t) := g — κ(‚t ) = Tg(t) (µg(t) )g (t), the geodesic equation looks like

ut = ’ad(u) u.

(2)

46.5. The covariant derivative. Our next aim is to derive the Riemann-

ian curvature, and for that we develop the basis-free version of Cartan™s method

of moving frames in this setting, which also works in in¬nite dimensions. The

right trivialization or framing (κ, πG ) : T G ’ g — G induces the isomorphism

R : C ∞ (G, g) ’ X(G), given by RX (x) = Te (µx ).X(x). For the Lie bracket and

the Riemannian metric we have

[RX , RY ] = R(’[X, Y ]g + dY.RX ’ dX.RY ),

(1)

R’1 [RX , RY ] = ’[X, Y ]g + RX (Y ) ’ RY (X),

Gx (RX (x), RY (x)) = X(x), Y (x) .

46.5

46.6 46. The Korteweg “ De Vries equation as a geodesic equation 501

Lemma. Assume that for all X ∈ g the adjoint ad(X) with respect to the inner

exists and that X ’ ad(X) is bounded. Then the Levi-Civita

product ,

covariant derivative of the metric (1) exists and is given in terms of the isomorphism

R by

= dY.RX + 1 ad(X) Y + 1 ad(Y ) X ’ 1 ad(X)Y.

(2) XY 2 2 2

Proof. Easy computations shows that this covariant derivative respects the Rie-

mannian metric,

RX Y, Z = dY.RX , Z + Y, dZ.RX = X Y, Z + Y, XZ ,

and is torsionfree,

’ X + [X, Y ]g ’ dY.RX + dX.RY = 0.

XY Y

Let us write ±(X) : g ’ g, where ±(X)Y = ad(Y ) X, then we have

= RX + 1 ad(X) + 1 ±(X) ’ 1 ad(X)

(3) X 2 2 2

46.6. The curvature. First note that we have the following relations:

(1) [RX , ad(Y )] = ad(RX (Y )), [RX , ±(Y )] = ±(RX (Y )),

[ad(X) , ad(Y ) ] = ’ad([X, Y ]g ) .

[RX , ad(Y ) ] = ad(RX (Y )) ,

The Riemannian curvature is then computed by

(2) R(X, Y ) = [ ]’

X, ’[X,Y ]g +RX (Y )’RY (X)

Y

= [RX + 1 ad(X) + 1 ±(X) ’ 1 ad(X), RY + 1 ad(Y ) + 1 ±(Y ) ’ 1 ad(Y )]

2 2 2 2 2 2

’ R’[X,Y ]g +RX (Y )’RY (X) ’ 1 ad(’[X, Y ]g + RX (Y ) ’ RY (X))

2

’ 2 ±(’[X, Y ]g + RX (Y ) ’ RY (X)) + 1 ad(’[X, Y ]g + RX (Y ) ’ RY (X))

1

2

= ’ 1 [ad(X) + ad(X), ad(Y ) + ad(Y )]

4

+ 1 [ad(X) ’ ad(X), ±(Y )] + 4 [±(X), ad(Y ) ’ ad(Y )]

1

4

+ 1 [±(X), ±(Y )] + 1 ±([X, Y ]g ).

4 2

If we plug in all de¬nitions and use 4 times the Jacobi identity we get the following

expression

4R(X, Y )Z, U = 2 [X, Y ], [Z, U ] ’ [Y, Z], [X, U ] + [X, Z], [Y, U ]

’ Z, [U, [X, Y ]] + U, [Z, [X, Y ]] ’ Y, [X, [U, Z]] ’ X, [Y, [Z, U ]]

+ ad(X) Z, ad(Y ) U + ad(X) Z, ad(U ) Y + ad(Z) X, ad(Y ) U

’ ad(U ) X, ad(Y ) Z ’ ad(Y ) Z, ad(X) U ’ ad(Z) Y, ad(X) U

’ ad(U ) X, ad(Z) Y + ad(U ) Y, ad(Z) X .

46.6

502 Chapter IX. Manifolds of mappings 46.8

46.7. Jacobi ¬elds, I. We compute ¬rst the Jacobi equation via variations of

geodesics. So let g : R2 ’ G be smooth, t ’ g(t, s) a geodesic for each s. Let again

u = κ(‚t g) = (g — κ)(‚t ) be the velocity ¬eld along the geodesic in right trivialization

which satis¬es the geodesic equation ut = ’ad(u) u. Then y := κ(‚s g) = (g — κ)(‚s )

is the Jacobi ¬eld corresponding to this variation, written in the right trivialization.

From the right Maurer-Cartan equation we then have:

yt = ‚t (g — κ)(‚s ) = d(g — κ)(‚t , ‚s ) + ‚s (g — κ)(‚t ) + 0

= [(g — κ)(‚t ), (g — κ)(‚s )]g + us

= [u, y] + us .

From this, using the geodesic equation and (46.6.1) we get

ust = uts = ‚s ut = ’‚s (ad(u) u) = ’ad(us ) u ’ ad(u) us

= ’ad(yt + [y, u]) u ’ ad(u) (yt + [y, u])

= ’±(u)yt ’ ad([y, u]) u ’ ad(u) yt ’ ad(u) ([y, u])

= ’ad(u) yt ’ ±(u)yt + [ad(y) , ad(u) ]u ’ ad(u) ad(y)u.

Finally, we get the Jacobi equation as

ytt = [ut , y] + [u, yt ] + ust

= ad(y)ad(u) u + ad(u)yt ’ ad(u) yt

’ ±(u)yt + [ad(y) , ad(u) ]u ’ ad(u) ad(y)u

ytt = [ad(y) + ad(y), ad(u) ]u ’ ad(u) yt ’ ±(u)yt + ad(u)yt .

(1)

46.8. Jacobi ¬elds, II. Let y be a Jacobi ¬eld along a geodesic g with right

trivialized velocity ¬eld u. Then y satis¬es the Jacobi equation

+ R(y, u)u = 0

‚t y

‚t

We want to show that this leads to same equation as (46.7). First note that from

(46.5.2) we have

= yt + 2 ad(u) y + 1 ±(u)y ’ 1 ad(u)y

1

‚t y 2 2

so that we get, using ut = ’ad(u) u heavily:

yt + 1 ad(u) y + 1 ±(u)y ’ 1 ad(u)y

‚t y =

‚t ‚t 2 2 2

= ytt + 1 ad(ut ) y + 1 ad(u) yt + 1 ±(ut )y

2 2 2

+ 1 ±(u)yt ’ 1 ad(ut )y ’ 1 ad(u)yt

2 2 2

1

yt + 1 ad(u) y + 1 ±(u)y ’ 1 ad(u)y

+ 2 ad(u) 2 2 2

+ 1 ±(u) yt + 1 ad(u) y + 1 ±(u)y ’ 1 ad(u)y

2 2 2 2

’ 1 ad(u) yt + 1 ad(u) y + 1 ±(u)y ’ 1 ad(u)y

2 2 2 2

46.8

46.8 46. The Korteweg “ De Vries equation as a geodesic equation 503

= ytt + ad(u) yt + ±(u)yt ’ ad(u)yt

’ 1 ±(y)ad(u) u ’ 1 ad(y) ad(u) u ’ 2 ad(y)ad(u) u

1

2 2

+ 1 ad(u) 1

+ 1 ad(y) u + 1 ad(y)u

2 ±(y)u

2 2 2

+ 1 ±(u) 1

+ 1 ad(y) u + 1 ad(y)u

2 ±(y)u

2 2 2

’ 1 ad(u) 1

+ 1 ad(y) u + 1 ad(y)u

2 ±(y)u

2 2 2

In the second line of the last expression we use

’ 1 ±(y)ad(u) u = ’ 4 ±(y)ad(u) u ’ 1 ±(y)±(u)u

1

2 4

and similar forms for the other two terms to get:

= ytt + ad(u) yt + ±(u)yt ’ ad(u)yt

‚t y

‚t

+ 4 [ad(u) , ±(y)]u + 1 [ad(u) , ad(y) ]u + 1 [ad(u) , ad(y)]u

1

4 4

+ 4 [±(u), ±(y)]u + 1 [±(u), ad(y) ]u + 1 [±(u), ad(y)]u

1

4 4

’ 1 [ad(u), ±(y)]u ’ 1 [ad(u), ad(y) + ad(y)]u,

4 4

where in the last line we also used ad(u)u = 0. We now compute the curvature

term:

1

R(y, u)u = ’ 4 [ad(y) + ad(y), ad(u) + ad(u)]u

+ 4 [ad(y) ’ ad(y), ±(u)]u + 1 [±(y), ad(u) ’ ad(u)]u

1

4

+ 4 [±(y), ±(u)] + 1 ±([y, u])u

1

2

= ’ 1 [ad(y) + ad(y), ad(u) ]u ’ 1 [ad(y) + ad(y), ad(u)]u

4 4

+ 1 [ad(y) , ±(u)]u ’ 1 [ad(y), ±(u)]u + 1 [±(y), ad(u) ’ ad(u)]u

4 4 4

+ 4 [±(y), ±(u)]u + 1 ad(u) ad(y)u

1

2

Summing up we get

+ R(y, u)u = ytt + ad(u) yt + ±(u)yt ’ ad(u)yt

‚t y

‚t

’ 1 [ad(y) + ad(y), ad(u) ]u

2

+ 1 [±(u), ad(y)]u + 1 ad(u) ad(y)u

2 2

Finally, we need the following computation using (46.6.1):

1

= 1 ±(u)[y, u] ’ 1 ad(y)±(u)u

2 [±(u), ad(y)]u 2 2

= 2 ad([y, u]) u ’ 1 ad(y)ad(u) u

1

2

= ’ 2 [ad(y) , ad(u) ]u ’ 1 ad(y)ad(u) u.

1

2

Inserting we get the desired result:

+ R(y, u)u = ytt + ad(u) yt + ±(u)yt ’ ad(u)yt

‚t y

‚t

’ [ad(y) + ad(y), ad(u) ]u.

46.8

504 Chapter IX. Manifolds of mappings 46.10

46.9. The weak symplectic structure on the space of Jacobi ¬elds. Let

us assume now that the geodesic equation in g

ut = ’ad(u) u

admits a unique solution for some time interval, depending smoothly on the choice

of the initial value u(0). Furthermore, we assume that G is a regular Lie group

(see (38.4)) so that each smooth curve u in g is the right logarithmic derivative

(see (38.1)) of a curve evolG (u) = g in G, depending smoothly on u. Let us

also assume that Jacobi ¬elds exist on the same time interval on which u exists,

depending uniquely on the initial values y(0) and yt (0). So the space of Jacobi

¬elds is isomorphic to g — g.

There is the well known symplectic structure on the space Ju of all Jacobi ¬elds

along a ¬xed geodesic with velocity ¬eld u. It is given by the following expression

which is constant in time t:

’

σ(y, z) := y, ‚t z ‚t y, z

= y, zt + 1 ad(u) z + 1 ±(u)z ’ 1 ad(u)z

2 2 2

’ yt + 1 ad(u) y + 1 ±(u)y ’ 1 ad(u)y, z

2 2 2

= y, zt ’ yt , z + [u, y], z ’ y, [u, z] ’ [y, z], u

= y, zt ’ ad(u)z + 1 ±(u)z ’ yt ’ ad(u)y + 1 ±(u)y, z .

2 2

It is a nice exercise to derive directly from the equation of Jacobi ¬elds (46.7.1) that

σ(y, z) is indeed constant in t: plug in all de¬nitions and use the Jacobi equation

(for the Lie bracket).

46.10. Geodesics and curvature on Di¬(S 1 ) revisited. We consider again

the Lie groups Di¬(R) and Di¬(S 1 ) with Lie algebras Xc (R) and X(S 1 ) where the

Lie bracket [X, Y ] = X Y ’ XY is the negative of the usual one. For the inner

product X, Y = X(x)Y (x) dx integration by parts gives

(X Y Z ’ XY Z)dx =

[X, Y ], Z = (2X Y Z + XY Z )dx = Y, ad(X) Z ,

R R

which in turn gives rise to

ad(X) Z = 2X Z + XZ ,

±(X)Z = 2Z X + ZX ,

(ad(X) + ad(X))Z = 3X Z,

(ad(X) ’ ad(X))Z = X Z + 2XZ = ±(X)Z.

The last equation means that ’ 1 ±(X) is the skew-symmetrization of ad(X) with

2

respect to to the inner product , . From the theory of symmetric spaces one

then expects that ’ 1 ± is a Lie algebra homomorphism and indeed one can check

2

that

’ 1 ±([X, Y ]) = [’ 1 ±(X), ’ 1 ±(Y )]

2 2 2

46.10

46.12 46. The Korteweg “ De Vries equation as a geodesic equation 505

holds. From (46.4.2) we get the same geodesic equation as in (46.3.4):

ut = ’ad(u) u = ’3ux u.

Using the above relations and the curvature formula (46.6.2) the curvature becomes

R(X, Y )Z = ’X Y Z + XY Z ’ 2X Y Z + 2XY Z = ’2[X, Y ]Z ’ [X, Y ] Z.

= ’±([X, Y ])Z

If we change the framing of the tangent bundle by

hxx fx ’ hx fxx

hx

X = h —¦ f ’1 , —¦ f ’1 , —¦ f ’1 ,

X= X= 3

fx fx

and similarly for Y = k —¦ f ’1 and Z = —¦ f ’1 , then (R(X, Y )Z) —¦ f coincides with

formula (46.3.3) for the curvature.

46.11. Jacobi ¬elds on Di¬(S 1 ). A Jacobi ¬eld y on Di¬(S 1 ) along a geodesic g

with velocity ¬eld u is a solution of the partial di¬erential equation (46.7.1), which

in our case becomes

ytt = [ad(y) + ad(y), ad(u) ]u ’ ad(u) yt ’ ±(u)yt + ad(u)yt

(1)

= ’3u2 yxx ’ 4uytx ’ 2ux yt ,

ut = ’3ux u.

Since the geodesic equation has solutions, locally in time, see the hint in (46.3),

and since Di¬(S 1 ) and Di¬(R) is a regular Lie group (see (43.1)), the space of all

Jacoby ¬elds exists and is isomorphic to C ∞ (S 1 , R)2 or Cc (R, R)2 , respectively.

∞

The weak symplectic structure on it is given by (46.9):

σ(y, z) = y, zt ’ 2 ux z + 2uzx ’ yt ’ 1 ux y + 2uyx , z

1

2

(yzt ’ yt z + 2u(yzx ’ yx z)) dx.

(2) =

S 1 or R

46.12. Geodesics on the Virasoro-Bott group. For • ∈ Di¬ + (S 1 ) let • :

S 1 ’ R+ be the mapping given by Tx •.‚x = • (x)‚x . Then

c : Di¬ + (S 1 ) — Di¬ + (S 1 ) ’ R

log(• —¦ ψ) d log ψ = log(• —¦ ψ)d log ψ

c(•, ψ) :=

S1 S1

is a smooth group cocycle with c(•, •’1 ) = 0, and S 1 — Di¬ + (S 1 ) becomes a Lie

group S 1 —c Di¬(S 1 ) with the operations

’1

•’1

•—¦ψ

• ψ •

. = , = .

a’1

ab e2πic(•,ψ)

a b a

46.12

506 Chapter IX. Manifolds of mappings 46.12

The Lie algebra of this Lie group turns out to be R —ω X(S 1 ) with the bracket

h k ’ hk

h k

, = ,

a b ω(h, k)

where ω : X(S 1 ) — X(S 1 ) ’ R is the Lie algebra cocycle

h k

1

ω(h, k) = ω(h)k = h dk = h k dx = det dx,

2 h k

S1 S1 S1

a generator of the bounded Chevalley cohomology H 2 (X(S 1 ), R). Note that the

Lie algebra cocycle makes sense on the Lie algebra Xc (R) of all vector ¬elds with

compact support on R, but it does not integrate to a group cocycle on Di¬(R). The

following considerations also make sense on Xc (R). Note also that H 2 (Xc (M ), R) =

0 for each ¬nite dimensional manifold of dimension ≥ 2 (see [Fuks, 1984]), which

blocks the way to ¬nd a higher dimensional analogue of the Korteweg “ de Vries

equation in a way similar to that sketched below. We shall use the following inner

product on X(S 1 ):

h k

, := hk dx + a.b.

a b 1

S

Integrating by parts we get

h k ’ hk

h k

ad , = ,

a b c ω(h, k) c

(h k ’ hk + ch k ) dx =

= (2h + h + ch )k dx

S1 S1

k h

= , ad ,

b a c

h 2h + h + ch

ad = ,

a c 0

so that in matrix notation we have (where ‚ := ‚x )

h h ’ h‚ 0

ad = ,

ω(h) 0

a

h 2h + h‚ h

ad = ,

0 0

a

h + 2h‚ + a‚ 3

h h 0

± = ad = ,

0 0

a a

h h 3h h

ad + ad = ,

ω(h) 0

a a

h h h + 2h‚ h

’ ad

ad = .

’ω(h) 0

a a

46.12

46.13 46. The Korteweg “ De Vries equation as a geodesic equation 507

From (46.4.2) we see that the geodesic equation on the Virasoro-Bott group is

’3u u ’ au

ut u u

= ’ad = ,

at a a 0

so that c is a constant in time, and ¬nally the geodesic equation is the periodic

Korteweg-De Vries equation

ut + 3ux u + auxxx = 0.

If we use Xc (R) we get the usual Korteweg-De Vries equation.

46.13. The curvature. Now we compute the curvature. Recall from (46.12) the

matrices ad h , ± h , and ad h whose entries are integro-di¬erential operators,

a a a

and insert them into formula (46.6.2). For the computation recall that the matrix is

applied to vectors of the form c where c a constant. Then we see that 4R h , k

a b

is the following 2 — 2-matrix whose entries are integro-di¬erential operators:

(4) (4)

«

4(h1 h2 ’ h1 h2 ) + 2(a1 h2 ’ a2 h1 )

2(h1 h2 ’ h1 h2 ) ·

+(8(h1 h2 ’ h1 h2 ) + 10(a1 h2 ’ a2 h1 ))‚

¬

¬ ·

(4) (4)

¬ ·

+18(a1 h2 ’ a2 h1 )‚ 2 +2(h1 h2 ’ h1 h2 ) ·

¬

¬ ·

¬ ·

(6) (6)

+(12(a1 h2 ’ a2 h1 ) + 2ω(h1 , h2 ))‚ 3 +(a1 h2 ’ a2 h1 ) · .

¬

¬ ·

¬ ·

’h1 ω(h2 ) + h2 ω(h1 )

¬ ·

¬ ·

¬ ·

¬ ·

ω(h2 )(4h1 + 2h1 ‚ + a1 ‚ 3 )

¬ ·

0

’ω(h1 )(4h2 + 2h2 ‚ + a2 ‚ 3 )

This leads to the following expression for the sectional curvature:

h1 h2 h1 h2

4R , , =

a1 a2 a1 a2

4(h1 h2 ’ h1 h2 )h1 h2 + 8(h1 h2 ’ h1 h2 )h1 h2

=

S1

(4) (4)

+ 2(a1 h2 ’ a2 h1 )h1 h2 + 10(a1 h2 ’ a2 h1 )h1 h2

+ 18(a1 h2 ’ a2 h1 )h1 h2

+ 12(a1 h2 ’ a2 h1 )h1 h2 + 2ω(h1 , h2 )h1 h2

’ h1 ω(h2 , h1 )h2 + h2 ω(h1 , h1 )h2

+ 2(h1 h2 ’ h1 h2 )a1 h2

(4) (4)

+ 2(h1 h2 ’ h1 h2 )a1 h2

(6) (6)

+ (a1 h2 ’ a2 h1 )a1 h2

+ (4h1 h1 h2 + 2h1 h1 h2 + a1 h1 h2

’ 4h2 h1 h1 ’ 2h2 h1 h1 ’ a2 h1 h1 )a2 dx

46.13

508 Chapter IX. Manifolds of mappings 46.14

(4) (4)

’ 4[h1 , h2 ]2 + 4(a1 h2 ’ a2 h1 )(h1 h2 ’ h1 h2 + h1 h2 ’ h1 h2 )

=

S1

’ (h2 )2 a2 + 2h1 h2 a1 a2 ’ (h1 )2 a2 dx

1 2

+3ω(h1 , h2 )2 .

46.14. Jacobi ¬elds. A Jacobi ¬eld y = y along a geodesic with velocity ¬eld

b

u

a is a solution of the partial di¬erential equation (46.7.1) which in our case looks

as follows.

ytt y y u u

= ad + ad , ad

btt b b a a

u yt u yt u yt

’ ad ’± + ad

a bt a bt a bt

u

3yx yxxx 2ux + u‚x uxxx

= ,

ω(y) 0 0 0 a

3 yt

’2ux ’ 4u‚x ’ a‚x ’uxxx

+ ,

ω(u) 0 bt

which leads to

ytt = ’u(4ytx + 3uyxx + ayxxxx ) ’ ux (2yt + 2ayxxx )

(1)

’ uxxx (bt + ω(y, u) ’ 3ayx ) ’ aytxxx ,

(2) btt = ω(u, yt ) + ω(y, 3ux u) + ω(y, auxxx ).

Let us consider ¬rst equation (2):

(2™) btt = (’ytxxx u + yxxx (3ux u + auxxx ))dx

S1

Next we consider the disturbing integral term in equation (1), and using the geodesic

equation for u we check that its derivative with respect to t equals equation (2™),

so it is a constant:

(3) bt + ω(y, u) = bt + yxxx u dx =: B1 since

S1

(ytxxx u + yxxx (’3ux u ’ auxxx )) dx = 0.

btt + (ytxxx u + yxxx ut ) dx = btt +

S1 S1

Note that b(t) can be explicitly solved as

t

b(t) = B0 + B1 t ’

(4) yxxx u dx dt.

S1

a

The ¬rst line of the Jacobi equation on the Virasoro-Bott group is a genuine partial

di¬erential equation and we get the following system of equations:

ytt = ’u(4ytx + 3uyxx + ayxxxx ) ’ ux (2yt + 2ayxxx )

+ (3ayx ’ B1 )uxxx ’ aytxxx ,

(5)

ut = ’3ux u ’ auxxx ,

a = constant,

46.14

46.15 46. The Korteweg “ De Vries equation as a geodesic equation 509

where u(t, x), y(t, x) are either smooth functions in (t, x) ∈ I —S 1 or in (t, x) ∈ I —R,

where I is an interval or R, and where in the latter case u, y, yt have compact

support with respect to x.

46.15. The weak symplectic structure on the space of Jacobi ¬elds on the

Virasoro Lie algebra. Since the Korteweg - de Vries equation has local solutions

depending smoothly on the initial conditions (and global solutions if a = 0), the

space of all Jacobi ¬elds exists and is isomorphic to (R —ω X(S 1 )) — (R —ω X(S 1 )).

The weak symplectic structure is given by (46.9):

y z y zt yt z u y z

’

σ , = , , + , ,

b c b ct bt c a b c

y u z y z u

’ ’

, , , ,

b a b b c a

(yzt ’ yt z + 2u(yzx ’ yx z)) dx

=

S1 or R

+ b(ct + ω(z, u)) ’ (bt + ω(y, u)) ’ aω(y, z)

(yzt ’ yt z + 2u(yzx ’ yx z)) dx

(1) =

S1 or R

+ bC1 ’ B1 c ’ ay z a dx.

S1 or R

46.15

510 Chapter IX. Manifolds of mappings

Complements to Manifolds of Mappings

For a compact smooth ¬nite dimensional manifold M , some results on the topolog-

ical type of the di¬eomorphism group Di¬(M ) are available in the literature. In

[Smale, 1959] it is shown that Di¬(S 2 ) is homotopy equivalent to O(3, R). This

result has been extended in [Hatcher, 1983] where it is shown that Di¬(S 3 ) is ho-

motopy equivalent to O(4, R). The component group π0 (Di¬ + (S n )) of the group of

orientation preserving di¬eomorphisms on the sphere S n is isomorphic to the group

of homotopy spheres of dimension n + 1 for n > 4, see [Kervaire, Milnor, 1963].

If M is a product of spheres then π0 (Di¬(M )) has been computed by [Browder,

1967] and [Turner, 1969]. For a simply connected orientable compact manifold M

in [Sullivan, 1978] it is shown that π0 (Di¬(M )) is commensurable to an arithmetic

group, where two groups are said to be commensurable if there is a ¬nite sequence

of homomorphisms of groups between them with ¬nite kernel and cokernel. By

[Borel, Harish-Chandra, 1962], any arithmetic group is ¬nitely presented, and by

[Borel, Serre, 1973] it is even of ¬nite type, which means that its classifying space is

homotopy equivalent to a CW-complex with ¬nitely many cells in each dimension.

Two di¬eomorphisms f, g of M are called pseudo-isotopic if there is a di¬eomor-

phism F : M — I ’ M — I restricting to f and g at the two ends of M — I, respec-

tively. Let D(M ) be the group of pseudo-isotopy classes of di¬eomorphisms of M ,

a quotient of Di¬(M )/ Di¬ 0 (M ), where Di¬ 0 (M ) is the connected component. By

[Cerf, 1970], if M is simply connected then π0 (Di¬(M )) = D(M ), whereas for non

simply connected M there is in general an abelian kernel A in an exact sequence

0 ’ A ’ π0 (Di¬(M )) ’ D(M ) ’ 0,

and A has been computed by [Hatcher, Wagoner, 1973] and [Igusa, 1984]. In

particular, A is ¬nitely generated if π0 (M ) is ¬nite.

In [Trianta¬llou, 1994] the following result is announced: If M is a smooth compact

orientable manifold of dimension ≥ 5 with ¬nite fundamental group, then D(M ) is

commensurable to an arithmetic group. Moreover, π0 (M ) is of ¬nite type.

511

Chapter X

Further Applications

47. Manifolds for Algebraic Topology . . . . . . . .... . . . . . . 512

48. Weak Symplectic Manifolds ......... .... . . . . . . 522

49. Applications to Representations of Lie Groups . .... . . . . . . 528

50. Applications to Perturbation Theory of Operators ... . . . . . . 536

51. The Nash-Moser Inverse Function Theorem .. .... . . . . . . 553

In section (47) we show how to treat direct limit manifolds like S ∞ = ’ S n or

lim

’

SO(∞, R) = lim SO(n, R) as real analytic manifolds modeled on R∞ = R(N) =

’’

N R. As topological spaces these are often used in algebraic topology, in partic-

ular the Grassmannians as classifying spaces of the groups SO(k). The di¬erential

calculus is very well applicable, and the groups GL(∞, R), SO(∞, R) turn out to

be regular Lie groups, where the exponential mapping is even locally a di¬eomor-

phism onto a neighborhood of the identity, since it factors over ¬nite dimensional

exponential mappings.

In section (48) we consider a manifold with a closed 2-form σ inducing an injective

(but in general not surjective) mapping σ : T M ’ T — M . This is called a weak

symplectic manifold, and there are di¬culties in de¬ning the Poisson bracket for

general smooth functions. We describe a natural subspace of functions for which

the Poisson bracket makes sense, and which admit Hamiltonian vector ¬elds.

For a (unitary) representation of a (¬nite dimensional) Lie group G in a Hilbert

space H one wishes to have the in¬nitesimal representation of the Lie algebra at

disposal. Classically this is given by unbounded operators and o¬ers analytical

di¬culties. We show in (49.5) and (49.10) that the dense subspaces H∞ and Hω of

smooth and real analytic vectors are invariant convenient vector spaces on which

the action G — H∞ ’ H∞ is smooth (resp. real analytic). These are well known

results. Our proofs are transparent and surprisingly simple; they use, however, the

uniform boundedness principles (5.18) and (11.12). Using this and the results from

section (48), we construct the moment mapping of any unitary representation.

Section (50) on perturbation theory of operators is devoted to the background and

proof of theorem (50.16) which says that a smooth curve of unbounded selfadjoint

operators on Hilbert space with compact resolvent admits smooth parameteriza-

tions of its eigenvalues and eigenvectors, under some condition. The real analytic

version of this theorem is due to [Rellich, 1940], see also [Kato, 1976, VII, 3.9], with

formally stronger notions of real analyticity which are quite di¬cult to handle.

512 Chapter X. Further Applications 47.2

Again the power of convenient calculus shows in the ease with which this result is

derived.

In section (51) we present a version of one of the hard implicit function theorems,

which is applicable to some non-linear partial di¬erential equations. Its origins

are the result of John Nash about the existence of isometric embeddings of Rie-

mannian manifolds into Rn ™s, see [Nash, 1956]. It was then identi¬ed by [Moser,

1961], [Moser, 1966] as an abstract implicit function theorem, and found the most

elaborate presentations in [Hamilton, 1982], and [Gromov, 1986]. But the original

application about the existence of isometric embeddings was ¬nally reproved in a

very simple way by [G¨nther, 1989, 1990], who composed the nonlinear perturba-

u

tion problem with the inverse of a Laplace operator and then applied the Banach

¬xed point theorem. This is characteristic for applications of hard implicit function

theorems: Each serious application is incredibly complicated, and ¬nally a simple ad

hoc method solves the problem. To our knowledge of the original applications only

two have not yet found direct simpler proofs: The result by Hamilton [Hamilton,

1982], that a compact Riemannian 3-manifold with positive Ricci curvature also

admits a metric with constant scalar curvature; and the application on the small

divisor problem in celestial mechanics. We include here the hard implicit function

theorem of Nash and Moser in the form of [Hamilton, 1982], in full generality and

without any loss, in condensed form but with all details.

47. Manifolds for Algebraic Topology

47.1. Convention. In this section the space R(N) of all ¬nite sequences with the

direct sum topology plays a an important role. It is also denoted by R∞ , mainly in

in algebraic topology. It is a convenient vector space. We consider it equipped with

the weak inner product x, y := xi yi , which is bilinear and bounded, therefore

smooth. It is called weak, since it is non degenerate in the following sense: the

associated linear mapping R(N) ’ (R(N) ) = RN is injective but far from being

surjective. We will also use the weak Euclidean distance |x| := x, x , whose

square is a smooth function.

47.2. Example: The sphere S ∞ . This is the set {x ∈ R(N) : x, x = 1},

the usual in¬nite dimensional sphere used in algebraic topology, the topological

inductive limit of S n ‚ S n+1 ‚ . . . . The inductive limit topology coincides with

the subspace topology since clearly lim S n ’ S ∞ ‚ R(N) is continuous, S ∞ as

’’

∞

(N)

closed subset of R with the c -topology is compactly generated, and since each

compact set is contained in a step of the inductive limit.

We show that S ∞ is a smooth manifold by describing an explicit smooth atlas, the

stereographic atlas. Choose a ∈ S ∞ (”south pole”). Let

x’ x,a a

U+ := S ∞ \ {a}, u+ : U+ ’ {a}⊥ , u+ (x) = 1’ x,a ,

x’ x,a a

U’ := S ∞ \ {’a}, u’ : U’ ’ {a}⊥ , u’ (x) = 1+ x,a .

47.2

47.2 47. Manifolds for algebraic topology 513

From an obvious drawing in the 2-plane through 0, x, and a it is easily seen that

u+ is the usual stereographic projection. We also get

|y|2 ’1

u’1 (y) = for y ∈ {a}⊥ \ {0}

2

|y|2 +1 a + |y|2 +1 y

+

and (u’ —¦u’1 )(y) = |y|2 . The latter equation can directly be seen from the drawing

y

+

using the intersection theorem.

The two stereographic charts above can be extended to charts on open sets in R(N)

in such a way that S ∞ becomes a splitting submanifold of R(N) :

u+ : R(N) \ [0, +∞)a ’ a⊥ + (’1, +∞)a

˜

z

u+ (z) := u+ ( ) + (|z| ’ 1)a

˜

|z|

= (1 + z, a )u’1 (z ’ z, a a)

+

Since the model space R(N) of S ∞ has the bornological approximation property by

(28.6), and is re¬‚exive, by (28.7) the operational tangent bundle of S ∞ equals the

kinematic one: DS ∞ = T S ∞ .

We claim that T S ∞ is di¬eomorphic to {(x, v) ∈ S ∞ — R(N) : x, v = 0}.

The Xx ∈ Tx S ∞ are exactly of the form c (0) for a smooth curve c : R ’ S ∞

with c(0) = x by (28.13). Then 0 = dt |0 c(t), c(t) = 2 x, Xx . For v ∈ x⊥ we use

d

v

c(t) = cos(|v|t)x + sin(|v|t) |v| .

The construction of S ∞ works for any positive de¬nite bounded bilinear form on

any convenient vector space.

The sphere is smoothly contractible, by the following argument: We consider the

homotopy A : R(N) — [0, 1] ’ R(N) through isometries which is given by A0 = Id

and by (44.22)

At (a0 , a1 , a2 , . . . ) = (a0 , . . . , an’2 , an’1 cos θn (t), an’1 sin θn (t),

an cos θn (t), an sin θn (t), an+1 cos θn (t), an+1 sin θn (t), . . . )

for n+1 ¤ t ¤ n , where θn (t) = •(n((n + 1)t ’ 1)) π for a ¬xed smooth function

1 1

2

• : R ’ R which is 0 on (’∞, 0], grows monotonely to 1 in [0, 1], and equals 1

on [1, ∞). The mapping A is smooth since it maps smooth curves (which locally

map into some RN ) to smooth curves (which then locally have values in R2N ).

(N)

Then A1/2 (a0 , a1 , a2 , . . . ) = (a0 , 0, a1 , 0, a2 , 0, . . . ) is in Reven , and on the other

(N)

hand A1 (a0 , a1 , a2 , . . . ) = (0, a0 , 0, a1 , 0, a2 , 0, . . . ) is in Rodd . This is a variant of

a homotopy constructed by [Ramadas, 1982]. Now At |S ∞ for 0 ¤ t ¤ 1/2 is a

(N)

smooth isotopy on S ∞ between the identity and A1/2 (S ∞ ) ‚ Reven . The latter set

is contractible in a chart.

One may prove in a simpler way that S ∞ is contractible with a real analytic homo-

topy with one corner: roll all coordinates one step to the right and then contract

in the stereographic chart opposite to (1, 0, . . . ).

47.2

514 Chapter X. Further Applications 47.4

47.3. Example. The Grassmannians and the Stiefel manifolds. The

Grassmann manifold G(k, ∞; R) = G(k, ∞) is the set of all k-dimensional lin-

ear subspaces of the space of all ¬nite sequences R(N) . The Stiefel manifold of

orthonormal k-frames O(k, ∞; R) = O(k, ∞) is the set of all linear isometries

Rk ’ R(N) , where the latter space is again equipped with the standard weak inner

product described at the beginning of (47.2). The Stiefel manifold of all k-frames

GL(k, ∞; R) = GL(k, ∞; R) is the set of all injective linear mappings Rk ’ R(N) .

)t : L(Rk , R(N) ) ’ L(R(N) , Rk ) which

There is a canonical transposition mapping (

is given by

incl A

At : R(N) ’ ’ RN = R(N) ’ (Rk ) = Rk

’ ’

and satis¬es At (x), y = x, A(y) . The transposition mapping is bounded and

linear, so it is real analytic. Then we have

GL(k, ∞) = {A ∈ L(Rk , R(N) ) : At —¦ A ∈ GL(k)},

since At —¦ A ∈ GL(k) if and only if Ax, Ay = At Ax, y = 0 for all y implies

x = 0, which is equivalent to A injective. So in particular GL(k, ∞) is open in

L(Rk , R(N) ). The Lie group GL(k) acts freely from the right on the space GL(k, ∞).

Two elements of GL(k, ∞) lie in the same orbit if and only if they have the same

image in R(N) . We have a surjective mapping π : GL(k, ∞) ’ G(k, ∞), given by

π(A) = A(Rk ), where the inverse images of points are exactly the GL(k)-orbits.

Similarly, we have

O(k, ∞) = {A ∈ L(Rk , R(N) ) : At —¦ A = Idk }.

The Lie group O(k) of all isometries of Rk acts freely from the right on the space

O(k, ∞). Two elements of O(k, ∞) lie in the same orbit if and only if they have

the same image in R(N) . The projection π : GL(k, ∞) ’ G(k, ∞) restricts to a

surjective mapping π : O(k, ∞) ’ G(k, ∞), and the inverse images of points are

now exactly the O(k)-orbits.

47.4. Lemma. Iwasawa decomposition. Let T (k; R) = T (k) be the group

of all upper triangular k — k-matrices with positive entries on the main diagonal.

Then each B ∈ GL(k, ∞) can be written in the form B = p(B) —¦ q(B), with unique

p(B) ∈ O(k, ∞) and q(B) ∈ T (k). The mapping q : GL(k, ∞) ’ T (k) is real

analytic, and p : GL(k, ∞) ’ O(k, ∞) ’ GL(k, ∞) is real analytic, too.

Proof. We apply the Gram Schmidt orthonormalization procedure to the vectors

B(e1 ), . . . , B(ek ) ∈ R(N) . The coe¬cients of this procedure form an upper trian-

gular k — k-matrix q(B) whose entries are rational functions of the inner products

B(ei ), B(ej ) and are positive on the main diagonal. So (B —¦ q(B)’1 )(e1 ), . . . , (B —¦

q(B)’1 )(ek ) is the orthonormalized frame p(B)(e1 ), . . . , p(B)(ek ).

47.4

47.5 47. Manifolds for algebraic topology 515

47.5. Theorem. The following are real analytic principal ¬ber bundles:

(π : O(k, ∞; R) ’ G(k, ∞; R), O(k, R)),

(π : GL(k, ∞; R) ’ G(k, ∞; R), GL(k, R)),

(p : GL(k, ∞; R) ’ O(k, ∞; R), T (k; R)).

The last one is trivial. The embeddings Rn ’ R(N) induce real analytic embeddings,

which respect the principal right actions of all the structure groups

O(k, n) ’ O(k, ∞),

GL(k, n) ’ GL(k, ∞),

G(k, n) ’ G(k, ∞).

All these cones are inductive limits in the category of real analytic (and smooth)

manifolds. All manifolds are smoothly paracompact.

Proof. Step 1. G(k, ∞) is a real analytic manifold.

For A ∈ O(k, ∞) we consider the open subset WA := {B ∈ GL(k, ∞) : At —¦ B ∈

GL(k)} of L(Rk , R(N) ), and we let VA := WA © O(k, ∞) = {B ∈ O(k, ∞) : At —¦ B ∈

GL(k)}. Obviously, VA is invariant under the action of O(k) and VAU = VA for

U ∈ O(k). So we may denote Uπ(A) := π(VA ). Let P := π(A) = A(Rk ) ∈ G(k, ∞).

We de¬ne the mapping

vA : VA ’ L(P, P ⊥ ),

vA (B) := B(At B)’1 At ’ AAt |P

= (IdR(N) ’AAt )B(At B)’1 At |P.

In order to visualize this de¬nition note that A —¦ At is the orthonormal projection

R(N) ’ P , and that the image of B in R(N) = P • P ⊥ is the graph of vA (B). It is

easily checked that vA (B) ∈ L(P, P ⊥ ) and that vA (BU ) = vA (B) = vAU (B) for all

U ∈ O(k). So we may de¬ne

uP : UP ’ L(P, P ⊥ ),

uP (π(B)) := vA (B).

For C ∈ L(P, P ⊥ ) the mapping A+C—¦A is a parameterization of the graph of C, it is

in GL(k, ∞), and we have (using p from lemma (47.3)) that u’1 (C) = π(p(A+CA)),

P

since for B ∈ VA the image of B equals the graph of C = uP (π(B)), which in turn

is equal to (A + CA)(Rk ) = (A + CA) q(A + CA)’1 (Rk ) = p(A + CA)(Rk ).

⊥

Now we check the chart changes: Let P1 = π(A1 ), P2 = π(A2 ), and C ∈ L(P1 , P1 ),

then we have

’1

uP2 —¦ u’1 (C) = (IdR(N) ’A2 At ) p(A1 + C A1 ) At p(A1 + C A1 ) At |P2 ,

2 2 2

P1

47.5

516 Chapter X. Further Applications 47.6

⊥

which is de¬ned on the open set of all C ∈ L(P1 , P1 ) for which At p(A1 + C A1 ) is

2

in GL(k) and which is real analytic there.

Step 2. The principal bundles.

We ¬x A ∈ O(k, ∞) and consider the section

sA : Uπ(A) ’ VA ,

sA (Q) := p(A + uπ(A) (Q) A)

and the principal ¬ber bundle chart

ψA : VA ’ Uπ(A) — O(k),

ψA (B) := π(B), sA (π(B))t B ,

’1

ψA (Q, U ) = sA (Q) U.

Clearly, these charts give a principal ¬ber bundle atlas with cocycle of transition

functions Q ’ sA2 (Q)t sA1 (Q) ∈ O(k).

The same formulas (for A still in O(k, ∞)) give ¬ber bundle charts ψA : WA ’

Uπ(A) — GL(k) for GL(k, ∞) ’ G(k, ∞).

The injection O(k, ∞) ’ GL(k, ∞) is a real analytic section of the real analytic

projection p : GL(k, ∞) ’ O(k, ∞), which by lemma (47.4) gives a trivial principal

¬ber bundle with structure group T (k). This fact implies that O(k, ∞) is a splitting

real analytic submanifold of the convenient vector space L(Rk , R(N) ).

Since R(N) is the inductive limit of the direct summands Rn in the category of

convenient vector spaces and real analytic (smooth) mappings, and since the chart

constructions above restrict to the usual ones on the ¬nite dimensional Grassman-

nians and bundles, the assertion on the inductive limits follows.

All these manifolds are smoothly paracompact. For R(N) this is in (16.10), so it holds

for L(Rk , R(N) ) and for the closed subspace O(k, ∞), see (47.3). Then it follows

for G(k, ∞) since O(k, ∞) ’ G(k, ∞) is a principal ¬ber bundle with compact

structure group O(k), by integrating the members of the partition over the ¬ber.

Then we get the result for GL(k, ∞) by bundle argumentation on GL(k, ∞) ’

G(k, ∞), since the ¬ber GL(k) is ¬nite dimensional, so the product is well behaved

by (4.16).

47.6. Theorem. The principal bundle (O(k, ∞), π, G(k, ∞)) is classifying for

¬nite dimensional principal O(k)-bundles and carries a universal real analytic O(k)-

connection ω ∈ „¦1 (O(k, ∞), o(k)).

This means: For each ¬nite dimensional smooth or real analytic principal O(k)-

bundle P ’ M with principal connection ωP there is a smooth or real analytic

mapping f : M ’ G(k, ∞) such that the pullback O(k)-bundle f — O(k, ∞) is iso-

morphic to P and the pullback connection f — ω equals ωP via this isomorphism.

For ∞ replaced by a large N and bundles where the dimension of the base is

bounded this is due to [Schla¬‚i, 1980].

47.6

47.6 47. Manifolds for algebraic topology 517

Proof. Step 1. The tangent bundle of O(k, ∞) is given by

T O(k, ∞) = {(A, X) ∈ O(k, ∞) — L(Rk , R(N) ) : X t A ∈ o(k)}.

We have O(k, ∞) = {A : At A = Idk }, thus TA O(k, ∞) ⊆ {X : X t A + At X = 0}.

Since At A = Idk is an equation of constant rank when restricted to GL(k, n) for

¬nite n, we have equality by the implicit function theorem.

Another argument which avoids the implicit function theorem is the following.

By theorem (47.5) the vertical tangent space {AZ : Z ∈ t(k)} at A ∈ O(k, ∞)

of the bundle GL(k, ∞) ’ O(k, ∞) is transversal to TA O(k, ∞), where t(k) is

the Lie algebra of T (k). We have equality since an easy computation shows that

{AZ : Z ∈ t(k)} © {X : X t A + At X = 0} = 0.

Step 2. The inner product on Rk and the weak inner product on R(N) induce a

bounded weak inner product on the space L(Rk , R(N) ) by X, Y = Trace(X t Y ) =

Trace(XY t ), where the second trace makes sense since XY t has ¬nite dimensional

range. With respect to this inner product we consider the orthonormal projection

¦A : TA O(k, ∞) ’ VA O(k, ∞) onto the vertical tangent space VA O(k, ∞) = {AY :

Y ∈ o(k)} of O(k, ∞) ’ G(k, ∞). Its kernel, the horizontal space, turns out to be

{Z ∈TA O(k, ∞) : Trace(Z t AY ) = 0 for all Y ∈ o(k)} =

= {Z : Z t A both skew and symmetric}

= {Z ∈ L(Rk , R(N) ) : Z t A = 0}

= {Z : Z(Rk )⊥A(Rk )}.

So ¦A : TA O(k, ∞) ’ VA O(k, ∞) turns out to be ¦A (Z) = AAt Z = AωA (Z),

where ωA (Z) := At Z = ’Z t A. Then ω ∈ „¦1 (O(k, ∞), o(k)) is an O(k)-equivariant

form which reproduces the generators of fundamental vector ¬elds of the principal

right action, so it is a principal O(k)-connection (37.19):

((rU )— ω)A (Z) = U t At ZU = Ad(U ’1 )ωA (Z),

ωA (AY ) = Y for Y ∈ o(k).

Step 3. The classifying process.

Let (p : P ’ M, O(k)) be a principal bundle with a principal connection form

ωP ∈ „¦1 (P, o(k)). We consider the obvious representation of O(k) on Rk and the

associated vector bundle E = P [Rk ] = P —O(k) Rk with its induced ¬ber Riemannian

metric gv and induced linear connection .

Now we choose a Riemannian metric gM on the base manifold M , which we pull

back to the horizontal bundle Hor E (with respect to ) in T E via the ¬berwise

isomorphism T p| Hor E : Hor E ’ T M . Then we use the vertical lift vlE :

E —M E ’ V E ‚ T E to heave the ¬ber metric gv to the vertical bundle. Finally,

we declare the horizontal and the vertical bundle to be orthogonal, and thus we

get a Riemannian metric gE := (T p| Hor E)— gM • (vprE )— gv on the total space

47.6

518 Chapter X. Further Applications 47.6

E. By the theorem of [Nash, 1956] (see also [G¨nther, 1989] and [Gromov, 1986]),

u

there is an isometric embedding (which can be chosen real analytic, if all data

are real analytic) i : (E, gE ) ’ RN into some high dimensional Euclidean space

which in turn is contained in R(N) . Let j := dv i(0E ) : E ’ R(N) be the vertical

d

derivative along the zero section of E which is given by dv i(0x )(ux ) = dt |0 i(tux ).

Then j : E ’ R(N) is a ¬ber linear smooth mapping which is isometric on each

¬ber.

Let us now identify the principal bundle P with the orthonormal frame bundle

O(Rk , (E, gv )) of its canonically associated Riemannian vector bundle. Then j— :

P u ’ j —¦ u ∈ O(k, ∞) de¬nes a smooth mapping which is O(k)-equivariant and

therefore ¬ts into the following pullback diagram

w O(k, ∞)

j—

P

p

u u

π

w G(k, ∞).

f

M

The factored smooth mapping f : M ’ G(k, ∞) is therefore classifying for the

bundle P , so that f — O(k, ∞) ∼ P .

=

In order to show that the canonical connection is pulled back to the given one we

consider again the associated Riemannian vector bundle E ’ M from above. Note

that T i(Hor E) is orthogonal to T i(V E) in R(N) , and we have to check that this

is still true for T j, see (37.26). This is a local question on M , so let E = U — Rn ,

then we have as in (29.9)

T (U — Rn ) = U — Rn — Rm — Rn (x, v; ξ, ω) ’

Ti

’ (i(x, v), d1 i(x, v).ξ + d2 i(x, v)ω) ∈ R(N) — R(N)

’

Tj

’’ (d2 i(x, 0)v, d1 d2 i(x, 0)(ξ, v) + d2 i(x, 0)ω),

Ti

V (U — Rn ) (x, v; 0, ω) ’ (i(x, v), d2 i(x, v)ω),

’

Tj

’’ (d2 i(x, 0)v, d2 i(x, 0)ω),

Hor (U — Rn ) (x, v; ξ, “x (ξ, v)) ’ (i(x, v), d1 i(x, v).ξ + d2 i(x, v)“x (ξ, v)),

Tj

’’ (d2 i(x, 0)v, d1 d2 i(x, 0)(ξ, v) + d2 i(x, 0)“x (ξ, v)),

0 = d2 i(x, v)ω, d1 i(x, v).ξ + d2 i(x, v)“x (ξ, v) for all ξ, ω.

In the last equation we replace v and ω both by tv and apply ‚t |0 to get the required

result

0 = d2 i(x, tv)tv, d1 i(x, tv).ξ + d2 i(x, tv)“x (ξ, tv) for all ξ, ω.

0 = d2 i(x, 0)v, d2 d1 i(x, 0).(v, ξ) + d2 i(x, 0)“x (ξ, v) for all ξ, ω.

47.6

47.7 47. Manifolds for algebraic topology 519

47.7. The Lie group GL(∞; R). The canonical embeddings Rn ’ Rn+1 onto

the ¬rst n coordinates induce injections GL(n) ’ GL(n + 1). The inductive limit

is given by

GL(∞; R) = GL(∞) := ’ GL(n)

lim

’ n’∞

in the category of sets or groups. Since each GL(n) also injects into L(R(N) , R(N) )

we can visualize GL(∞) as the set of all N — N-matrices which are invertible and

di¬er from the identity in ¬nitely many entries only.

We also consider the Lie algebra gl(∞) of all N — N-matrices with only ¬nitely

many nonzero entries, which is isomorphic to R(N—N) , and we equip it with this

convenient vector space structure. Then

gl(∞) = lim gl(n)

’’

n’∞

in the category of real analytic mappings, since it is a regular inductive limit in the

category of bounded linear mappings.

Claim. gl(∞) = L(RN , R(N) ) as convenient vector spaces. Composition is a boun-

ded bilinear mapping on gl(∞). The transposition

A ’ At = A —¦ i : RN ’ (RN ) ’ (RN ) = R(N)

on the space L(RN , R(N) ) induces a bibounded linear isomorphism of gl(∞) which

resembles the usual transposition of matrices.

N

Proof. Let T ∈ L(RN , R(N) ). Then T ∈ L(RN , RN ) = R(N) and hence is a

matrix with ¬nitely many non zero entries in every line. Since T has values in R(N) ,

there are also only ¬nitely many non zero entries in each column, since T (ej ) ∈ R(N) .

Suppose that T is not in gl(∞). Then the matrix of T has in¬nitely many nonzero

entries, so there are Tjik = 0 for ik ∞ and jk ∞ and such that jk is the last

k

index with nonzero entry in the line ik . Now one can choose inductively an element

(xi ) ∈ RN with T (x) ∈ R(N) , a contradiction. For both spaces the evaluations

/

evi,j generate the convenient vector space structure by (5.18), so the convenient

structures coincide.

Another argument leading to this conclusion is the following: Since both spaces are

nuclear we have for the injective tensor product

L(RN , R(N) ) ∼ R(N) —µ R(N) .

ˆ

=

By the same reason the injective and the projective tensor product coincide. Since

both spaces are (DF), separately continuous bilinear functionals are jointly continu-

ous, so the latter space coincides with the bornological tensor product R(N) —β R(N) ,

˜

which commutes with direct sums, since it is a left adjoint functor, so ¬nally we

get R(N—N) .

47.7

520 Chapter X. Further Applications 47.8

Composition is bounded since it can be written as

comp

L(RN , R(N) ) — L(RN , R(N) ) ’ L(RN , R(N) ) — L(RN , RN ) ’ ’ L(RN , R(N) ).

’’

The assertion about transposition is obvious, using L(RN , (RN ) ) ∼ L2 (RN ; R).

=

Then the (convenient) a¬ne space

Id +gl(∞) = lim (Id +gl(n))

’’

n’∞

is closed under composition, which is real analytic on it. The determinant is a real

analytic function there, too.

Now obviously GL(∞) = {A ∈ Id +gl(∞) : det(A) = 0}, so GL(∞) is an open

subset in Id +gl(∞) and is thus a real analytic manifold, in fact, it is the inductive

limit of all the groups GL(n) = {A ∈ Id∞ +gl(n) : det(A) = 0} in the category of

real analytic manifolds.

We consider the Killing form on gl(∞), which is given by the trace

k(X, Y ) := tr(XY ) for X, Y ∈ gl(∞).

This is the right concept, since for each n and X, Y ∈ gl(n) ‚ gl(∞) we have

1

trR(N) (XY ) = trRn (XY ) = trgl(n) (ad(X) ad(Y )) + 2 trR(N) (X) trR(N) (Y ) ,

2n

but ad(X)ad(Y ) ∈ L(R(N) , R(N) ) is not of trace class. We have the following short

exact sequence of Lie algebras and Lie algebra homomorphisms

tr

0 ’ sl(∞) ’ gl(∞) ’ R ’ 0.

’

t

It splits, using t ’ n · IdRn for an arbitrary n, but gl(∞) has no nontrivial abelian

ideal a, since we would have a © gl(n) ‚ R · Idn for every n. So gl(∞) is only the

semidirect product of R with the ideal sl(∞) and not the direct product.

47.8. Theorem. GL(∞) is a real analytic regular Lie group modeled on R(N)

with Lie algebra gl(∞) and is the inductive limit of the Lie groups GL(n) in

the category of real analytic manifolds. The exponential mapping is well de¬ned,

real analytic, and a local real analytic di¬eomorphism onto a neighborhood of the

identity. The Campbell-Baker-Hausdor¬ formula gives a real analytic mapping

near 0 and expresses the multiplication on GL(∞) via exp. The determinant

det : GL(∞) ’ R \ 0 is a real analytic homomorphism. We have a real ana-

lytic left action GL(∞) — R(N) ’ R(N) , such that R(N) \ 0 is one orbit, but the

injection GL(∞) ’ L(R(N) , R(N) ) does not generate the topology.

Proof. Since the exponential mappings are compatible with the inductive limits

and are di¬eomorphisms on open balls with radius π in norms in which the Lie

brackets are submultiplicative, all these assertions follow from the inductive limit

property. One may use the double of the operator norms.

Regularity is proved as follows: A smooth curve X : R ’ gl(∞) factors locally in

R into some gl(n), and we may integrate this piece of the resulting right invariant

time dependent vector ¬eld on GL(n).