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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

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