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