‚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