Curvature of the Virasoro-Bott group

16 Pages·0.2 MB·English

by  Peter W. Michor

To appear in:
J. of Lie Theory
ON THE GEOMETRY OF THE VIRASORO-BOTT GROUP
8
9
9
1 Peter W. Michor
n Tudor S. Ratiu
a
J Erwin Schr¨odinger International Institute

6
of Mathematical Physics, Wien, Austria
2

]
G February 1, 2008
D
.
h
t Abstract. We consider a natural Riemannian metric on the infinite dimensional
a
manifold of all embeddings from a manifold into a Riemannian manifold, and derive
m
itsgeodesic equationinthe case Emb(R,R)whichturnsout tobe Burgers’equation.
[
Then we derive the geodesic equation, the curvature, and the Jacobi equation of a

1 right invariant Riemannian metric on an infinite dimensional Lie group, which we
v apply to Diff(R), Diff(S1), and the Virasoro-Bott group. Many of these results are
5
well known, the emphasis is on conciseness and clarity.
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1
1
0
8
9
/ Table of contents
h
t
a 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
m
2. The general setting and a motivating example . . . . . . . . . . . . . 2
:
v 3. Right invariant Riemannian metrics on Lie groups . . . . . . . . . . . 4
i
X 4. The diffeomorphism group of the circle revisited . . . . . . . . . . . . 9
r 5. The Virasoro-Bott group and the Korteweg-de Vries-equation . . . . . . 11
a
1. Introduction
We consider a natural Riemannian metric on the infinite dimensional manifold
of all embeddings from a manifold into a Riemannian manifold, derive its geodesic
equation in the case Emb(R,R) which turns out to be Burgers’ equation. For the
general case see [9]. Then we give a careful exposition of the derivation of the
1991 Mathematics Subject Classification. 58D05, 58F07, Secondary 35Q53.
Key words and phrases. diffeomorphism group, connection, Jacobi field, symplecticstructure,
KdV equation.
P.W.M. was supported by ‘Fonds zur Fo¨rderung der wissenschaftlichen Forschung, Projekt
P 10037 PHY’.
T.R. acknowledges the partial support of NSF Grant DMS-9503273 and DOE contract DE-
FG03-95ER25245-A000.
Typeset by AMS-TEX
1
2 PETER W. MICHOR, TUDOR RATIU
geodesic equation, the curvature, and the Jacobi equation of a right invariant Rie-
mannian metric on an infinite dimensional Lie group. This is a careful presentation
and extension of results in [1], [2], [3]. The formulas obtained in this way are then
applied to Diff(R), Diff(S1), and the Virasoro-Bottgroup, where the geodesic equa-
tion is the Korteweg-de Vries equations. This is due to [8], [10], [23], and also [22].
A fast overview on the geometry of the Virasoro-Bott group can also be found in
[24]. The emphasis of this paper is on a unified setting for these results, and on
conciseness and clarity. Thanks to Hermann Schichl and to the referee for detailed
checks of the computations.
2. The general setting and a motivating example
2.1. The principal bundle of embeddings. Let M and N be smooth finite
dimensional manifolds, connected and second countable without boundary, such
that dimM ≤ dimN. The space Emb(M,N) of all embeddings (immersions which
are homeomorphisms on their images) from M into N is an open submanifold of
∞
C (M,N)which isstableunder theright actionofthediffeomorphism groupofM.
∞
Here C (M,N) is a smooth manifold modeled on spaces of sections with compact
∗
supportΓ (f TN). Inparticularthetangentspaceatf iscanonicallyisomorphicto
c
the space of vector fields along f with compact support in M. If f and g differ on a
∞
non-compact set then they belong to different connected components ofC (M,N).
See [19] and [14]. Then Emb(M,N) is the total space of a smooth principal fiber
bundle with structure group the diffeomorphism group of M; the base is called
B(M,N), it is a Hausdorff 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. This result is based on an idea implicitly contained in [25], it
was fully proved in [5] for compact M and for general M in [18]. The clearest
presentation is in [19], section 13. If we take a Hilbert space H instead of N, then
B(M,H) is the classifying space for Diff(M) if M is compact, and the classifying
bundle Emb(M,H) carries also a universal connection. This is shown in [21].
2.2. If (N,g) is a Riemannian manifold then on the manifold Emb(M,N) there
∗
is a naturally induced weak Riemannian metric given, for s ,s ∈ Γ (f TN) and
1 2 c
ϕ ∈ Emb(M,N), by
∗
G (s ,s ) = g(s ,s )vol(φ g), φ ∈ Emb(M,N),
φ 1 2 1 2
ZM
where vol(g) denotes the volume form on N induced by the Riemannian metric
∗ ∗
g and vol(φ g) the volume form on M induced by the pull back metric φ g. The
covariant derivative and curvature of the Levi-Civita connection induced by G were
investigated in [4] if N = RdimM+1 (endowed with the standard inner product) and
in [9] for the general case. We shall not reproduce the general formulae here
ThisweakRiemannianmetricisinvariant undertheactionofthediffeomorphism
group Diff(M) by composition from the right and hence it induces a Riemannian
metric on the base manifold B(M,N).
2.3. Example. Let us consider the special case M = N = R, that is, the space
Emb(R,R) of all embeddings of the real line into itself, which contains the diffeo-
morphism group Diff(R) as an open subset. The case M = N = S1 is treated
ON THE GEOMETRY OF THE VIRASORO-BOTT GROUP 3
in a similar fashion and the results of this paper are also valid in this situation,
where Emb(S1,S1) = Diff(S1). For our purposes, we may restrict attention to the
space of orientation-preserving embeddings, denoted by Emb+(R,R). The weak
Riemannian metric has thus the expression
G (h,k) = h(x)k(x)|f′(x)|dx, f ∈ Emb(R,R), h,k ∈ C∞(R,R).
f c
R
Z
We shall compute the geodesic equation for this metric by variational calculus. The
energy of a curve f of embeddings is
b b
E(f) = 1 G (f ,f )dt = 1 f2f dxdt.
2 f t t 2 t x
Za ZaZR
If we assume that f(x,t,s) is a smooth function and that the variations are with
fixed endpoints, then the derivative with respect to s of the energy is
∂ ∂ b
E(f( ,s)) = 1 f2f dxdt
∂s ∂s 2 t x
(cid:12)0 (cid:12)0 ZaZR
(cid:12) (cid:12)b
(cid:12) (cid:12)
(cid:12) = 1 (cid:12) (2f f f +f2f )dxdt
2 t ts x t xs
ZaZR
b
= −1 (2f f f +2f f f +2f f f )dxdt
2 tt s x t s tx t tx s
ZaZR
b f f
t tx
= − f +2 f f dxdt,
tt s x
f
ZaZR(cid:18) x (cid:19)
so that the geodesic equation with its initial data is:
f f
(1) f = −2 t tx, f( ,0) ∈ Emb+(R,R), f ( ,0) ∈ C∞(R,R)
tt f t c
x
=: Γ (f ,f ),
f t t
where the Christoffel symbol Γ : Emb(R,R)×C∞(R,R)×C∞(R,R) → C∞(R,R)
c c c
is given by symmetrisation:
hk +h k (hk)
x x x
(2) Γ (h,k) := − = − .
f
f f
x x
For vector fields X,Y on Emb(R,R) the covariant derivative is given by the ex-
pression ∇EmbY = dY(X) − Γ(X,Y). The Riemannian curvature R(X,Y)Z =
X
(∇ ∇ −∇ ∇ −∇ )Z is then determined in terms of the Christoffel form
X Y Y X [X,Y]
by
R (h,k)ℓ = −dΓ(f)(h)(k,ℓ)+dΓ(f)(k)(h,ℓ)+Γ (h,Γ (k,ℓ))−Γ (k,Γ (h,ℓ))
f f f f f
h(kℓ)x k(hℓ)x
= −hx(kℓ)x + kx(hℓ)x + fx x − fx x
f2 f2 (cid:16) f (cid:17) (cid:16) f (cid:17)
x x x x
1
(3) = f h kℓ−f hk ℓ+f hk ℓ−f h kℓ+2f hk ℓ −2f h kℓ
xx x xx x x xx x xx x x x x x x
f3
x
(cid:16) (cid:17)
4 PETER W. MICHOR, TUDOR RATIU
The geodesic equation can be solved in the following way: if instead of the obvious
framing we change variables to T Emb = Emb×C∞ ∋ (f,h) 7→ (f,hf2) =: (f,F)
c x
then the geodesic equation becomes F = ∂ (f f2) = f2(f +2ftftx) = 0, so that
t ∂t t x x tt fx
F = f f2 is constant in t, or f (x,t)f (x,t)2 = f (x,0)f (x,0)2. From here, a
t x t x t x
standard separation of variables argument gives the solution; it blows up in finite
time for most initial conditions.
Now let us consider the trivialisation of T Emb(R,R) by right translation (this
is most useful for Diff(R)). The derivative of the inversion Inv : g 7→ g−1 is given
by
h◦g−1
T (Inv)h = −T(g−1)◦h◦g−1 = for g ∈ Emb(R,R), h ∈ C∞(R,R).
g g ◦g−1 c
x
Defining
u := f ◦f−1, or, in more detail, u(x,t) = f (f( ,t)−1(x),t),
t t
we have
1 f
u = (f ◦f−1) = (f ◦f−1) = tx ◦f−1,
x t x tx f ◦f−1 f
x x
u = (f ◦f−1) = f ◦f−1 +(f ◦f−1)(f−1)
t t t tt tx t
1
= f ◦f−1 +(f ◦f−1) (f f−1)
tt tx f f−1 t
x
which, by (1) and the first equation becomes
f f f f
u = f ◦f−1 − tx t ◦f−1 = −3 tx t ◦f−1 = −3u u.
t tt x
f f
x x
(cid:18) (cid:19) (cid:18) (cid:19)
The geodesic equation on Emb(R,R) in right trivialization, that is, in Eulerian
formulation, is hence
(4) u = −3u u,
t x
which is just Burgers’ equation.
3. Right invariant Riemannian metrics on Lie groups
3.1. Geodesics of a right invariant metric on a Lie group. Let G be a Lie
group which may be infinite dimensional, with Lie algebra g. Let µ : G×G → G
be the multiplication, let µ be left translation and µy be right translation, given
x
by µ (y) = µy(x) = xy = µ(x,y). We also need the right Maurer-Cartan form κ =
x
κr ∈ Ω1(G,g), given by κ (ξ) := T (µx−1)·ξ. It satisfies the right Maurer-Cartan
x x
equation dκ − 1[κ,κ]∧ = 0, where [ , ]∧ denotes the wedge product of g-valued
2
forms on G induced by the Lie bracket. Note that 1[κ,κ]∧(ξ,η) = [κ(ξ),κ(η)].
2
Let h , i : g×g → R be a positive definite bounded (weak) inner product. Then
(1) G (ξ,η) = hT(µx−1)·ξ,T(µx−1)·η)i = hκ(ξ),κ(η)i
x
ON THE GEOMETRY OF THE VIRASORO-BOTT GROUP 5
is a right invariant (weak) Riemannian metric on G, and any (weak) right invariant
bounded Riemannian metric is of this form, for suitable h , i.
Let g : [a,b] → G be a smooth curve. The velocity field of g, viewed in the
right trivializations, coincides with the right logarithmic derivative T(µg−1)·∂ g =
t
κ(∂ g) = (g∗κ)(∂ ), where ∂ = ∂ . The energy of the curve g(t) is given by
t t t ∂t
b b
E(g) = 1 G (g′,g′)dt = 1 h(g∗κ)(∂ ),(g∗κ)(∂ )idt.
2 g 2 t t
Za Za
For a variation g(t,s) with fixed endpoint we have then, using the right Maurer-
Cartan equation and integration by parts,
b
∂ E(g) = 1 2h∂ (g∗κ)(∂ ), (g∗κ)(∂ )idt
s 2 s t t
Za
b
∗ ∗ ∗
= h∂ (g κ)(∂ )−d(g κ)(∂ ,∂ ), (g κ)(∂ )idt
t s t s t
Za
b
∗ ∗ ∗ ∗ ∗
= (−h(g κ)(∂ ), ∂ (g κ)(∂ )i−h[(g κ)(∂ ),(g κ)(∂ )], (g κ)(∂ )i) dt
s t t t s t
Za
b
∗ ∗ ∗ ⊤ ∗
= − h(g κ)(∂ ), ∂ (g κ)(∂ )+ad((g κ)(∂ )) ((g κ)(∂ ))idt
s t t t t
Za
∗ ⊤ ∗
where ad((g κ)(∂ )) : g → g is the adjoint of ad((g κ)(∂ )) with respect to the
t t
inner product h , i. In infinite 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,
given by u(t) := g∗κ(∂ ) = T (µg(t)−1) · g′(t), the geodesic equation has the
t g(t)
expression
⊤
(2) u = −ad(u) u.
t
This is, of course, just the Euler-Poincar´e equation for right invariant systems using
the Lagrangian given by the kinetic energy (see [15], section 13) and the above
derivation is done directly without invoking this theorem.
3.2. The covariant derivative. Our next aim is to derive the Riemannian cur-
vature and for that we develop the basis-free version of Cartan’s method of moving
frames in this setting, which also works in infinite dimensions. The right trivializa-
∞
tion, or framing, (κ,π ) : TG → g×G induces the isomorphism R : C (G,g) →
G
X(G), given by R(X)(x) := R (x) := T (µx)·X(x), for X ∈ C∞(G,g) and x ∈ G.
X e
Here X(G) := Γ(TG) denote the Lie algebra of all vector fields. For the Lie bracket
and the Riemannian metric we have
(1) [R ,R ] = R(−[X,Y] +dY ·R −dX ·R ),
X Y g X Y
R−1[R ,R ] = −[X,Y] +R (Y)−R (X),
X Y g X Y
G (R (x),R (x)) = hX(x),Y(x)i, x ∈ G.
x X Y
∞
In the sequel we shall compute in C (G,g) instead of X(G). In particular, we shall
use the convention
∇ Y := R−1(∇ R ) for X,Y ∈ C∞(G,g).
X RX Y
to express the Levi-Civita covariant derivative.
6 PETER W. MICHOR, TUDOR RATIU
Lemma. Assume that for all ξ ∈ g the adjoint ad(ξ)⊤ with respect to the inner
⊤
product h , i exists and that ξ 7→ ad(ξ) is bounded. Then the Levi-Civita covariant
∞
derivative of the metric 2.1(1) exists and is given for any X,Y ∈ C (G,g) in terms
of the isomorphism R by
(2) ∇ Y = dY.R + 1 ad(X)⊤Y + 1 ad(Y)⊤X − 1 ad(X)Y.
X X 2 2 2
Proof. Easy computations show that this formula satisfies the axiomsof a covariant
derivative, that relative to it the Riemannian metric is covariantly constant, since
R hY,Zi = hdY.R ,Zi+hY,dZ.R i = h∇ Y,Zi+hY,∇ Zi,
X X X X X
and that it is torsion free, since
∇ Y −∇ X +[X,Y] −dY.R +dX.R = 0. (cid:3)
X Y g X Y
⊤
For ξ ∈ g define α(ξ) : g → g by α(ξ)η := ad(η) ξ. With this notation, the
∞
previous lemma states that for all X ∈ C (G,g) the covariant derivative of the
Levi-Civita connection has the expression
(3) ∇ = R + 1 ad(X)⊤ + 1α(X)− 1 ad(X).
X X 2 2 2
3.3. The curvature. First note that we have the following relations:
(1) [R ,ad(Y)] = ad(R (Y)), [R ,α(Y)] = α(R (Y)),
X X X X
⊤ ⊤ ⊤ ⊤ ⊤
[R ,ad(Y) ] = ad(R (Y)) , [ad(X) ,ad(Y) ] = −ad([X,Y] ) .
X X g
The Riemannian curvature is then computed by
(2)
R(X,Y) = [∇X,∇Y]−∇−[X,Y]g+RX(Y)−RY(X)
= [R + 1 ad(X)⊤ + 1α(X)− 1 ad(X),R + 1 ad(Y)⊤ + 1α(Y)− 1 ad(Y)]
X 2 2 2 Y 2 2 2
−R−[X,Y]g+RX(Y)−RY(X) − 21 ad(−[X,Y]g +RX(Y)−RY(X))⊤
− 1α(−[X,Y] +R (Y)−R (X))+ 1 ad(−[X,Y] +R (Y)−R (X))
2 g X Y 2 g X Y
= −1[ad(X)⊤ +ad(X),ad(Y)⊤ +ad(Y)]
4
+ 1[ad(X)⊤ −ad(X),α(Y)]+ 1[α(X),ad(Y)⊤ −ad(Y)]
4 4
+ 1[α(X),α(Y)]+ 1α([X,Y] ).
4 2 g
If we plug in all definitions and use 4 times the Jacobi identity we get the following
expression
h4R(X,Y)Z,Ui = +2h[X,Y],[Z,U]i−h[Y,Z],[X,U]i+h[X,Z],[Y,U]i
−hZ,[U,[X,Y]]i+hU,[Z,[X,Y]]i−hY,[X,[U,Z]]i−hX,[Y,[Z,U]]i
⊤ ⊤ ⊤ ⊤ ⊤ ⊤
+had(X) Z,ad(Y) Ui+had(X) Z,ad(U) Yi+had(Z) X,ad(Y) Ui
⊤ ⊤ ⊤ ⊤ ⊤ ⊤
−had(U) X,ad(Y) Zi−had(Y) Z,ad(X) Ui−had(Z) Y,ad(X) Ui
⊤ ⊤ ⊤ ⊤
−had(U) X,ad(Z) Yi+had(U) Y,ad(Z) Xi.
ON THE GEOMETRY OF THE VIRASORO-BOTT GROUP 7
3.4. Jacobi fields, I. Wecompute first the Jacobi equation directly viavariations
of geodesics. So let g : R2 → G be smooth, t 7→ g(t,s) a geodesic for each s. Let
∗
again u = κ(∂ g) = (g κ)(∂ ) be the velocity field along the geodesic in right
t t
⊤
trivialization which satisfies the geodesic equation u = −ad(u) u. Then y :=
t
∗
κ(∂ g) = (g κ)(∂ ) is the Jacobi field corresponding to this variation, written in
s s
the right trivialization. From the right Maurer-Cartan equation we then have:
∗ ∗ ∗
y = ∂ (g κ)(∂ ) = d(g κ)(∂ ,∂ )+∂ (g κ)(∂ )+0
t t s t s s t
∗ ∗
= [(g κ)(∂ ),(g κ)(∂ )] +u
t s g s
= [u,y]+u .
s
Using the geodesic equation, the definition of α, and the fourth relation in 3.3.(1),
this identity implies
⊤ ⊤ ⊤
u = u = ∂ u = −∂ (ad(u) u) = −ad(u ) u−ad(u) u
st ts s t s s s
⊤ ⊤
= −ad(y +[y,u]) u−ad(u) (y +[y,u])
t t
⊤ ⊤ ⊤
= −α(u)y −ad([y,u]) u−ad(u) y −ad(u) ([y,u])
t t
⊤ ⊤ ⊤ ⊤
= −ad(u) y −α(u)y +[ad(y) ,ad(u) ]u−ad(u) ad(y)u.
t t
Finally we get the Jacobi equation as
y = [u ,y]+[u,y ]+u
tt t t st
⊤ ⊤
= ad(y)ad(u) u+ad(u)y −ad(u) y
t t
⊤ ⊤ ⊤
−α(u)y +[ad(y) ,ad(u) ]u−ad(u) ad(y)u,
t
⊤ ⊤ ⊤
(1) y = [ad(y) +ad(y),ad(u) ]u−ad(u) y −α(u)y +ad(u)y .
tt t t t
3.5. Jacobi fields, II. Let y be a Jacobi field along a geodesic g with right trivi-
alized velocity field u. Then y should satisfy the analogue of the finite dimensional
Jacobi equation
∇ ∇ y +R(y,u)u = 0
∂t ∂t
We want to show that this leads to same equation as 3.4.(1). First note that from
3.2.(2) we have
∇ y = y + 1 ad(u)⊤y + 1α(u)y − 1 ad(u)y
∂t t 2 2 2
⊤
so that, using u = −ad(u) u, we get:
t
∇ ∇ y = ∇ y + 1 ad(u)⊤y + 1α(u)y − 1 ad(u)y
∂t ∂t ∂t t 2 2 2
= y +(cid:16) 1 ad(u )⊤y + 1 ad(u)⊤y + 1α(u )y (cid:17)
tt 2 t 2 t 2 t
+ 1α(u)y − 1 ad(u )y − 1 ad(u)y
2 t 2 t 2 t
+ 1 ad(u)⊤ y + 1 ad(u)⊤y + 1α(u)y − 1 ad(u)y
2 t 2 2 2
(cid:16) (cid:17)
+ 1α(u) y + 1 ad(u)⊤y + 1α(u)y − 1 ad(u)y
2 t 2 2 2
− 1 ad(u(cid:16)) y + 1 ad(u)⊤y + 1α(u)y − 1 ad(u)(cid:17)y
2 t 2 2 2
(cid:16) (cid:17)
8 PETER W. MICHOR, TUDOR RATIU
⊤
= y +ad(u) y +α(u)y −ad(u)y
tt t t t
− 1α(y)ad(u)⊤u− 1 ad(y)⊤ad(u)⊤u− 1 ad(y)ad(u)⊤u
2 2 2
+ 1 ad(u)⊤ 1α(y)u+ 1 ad(y)⊤u+ 1 ad(y)u
2 2 2 2
(cid:16) (cid:17)
+ 1α(u) 1α(y)u+ 1 ad(y)⊤u+ 1 ad(y)u
2 2 2 2
(cid:16) (cid:17)
− 1 ad(u) 1α(y)u+ 1 ad(y)⊤u+ 1 ad(y)u .
2 2 2 2
(cid:16) (cid:17)
In the second line of the last expression we use
−1α(y)ad(u)⊤u = −1α(y)ad(u)⊤u− 1α(y)α(u)u
2 4 4
and similar forms for the other two terms to get:
⊤
∇ ∇ y = y +ad(u) y +α(u)y −ad(u)y
∂t ∂t tt t t t
+ 1[ad(u)⊤,α(y)]u+ 1[ad(u)⊤,ad(y)⊤]u+ 1[ad(u)⊤,ad(y)]u
4 4 4
+ 1[α(u),α(y)]u+ 1[α(u),ad(y)⊤]u+ 1[α(u),ad(y)]u
4 4 4
− 1[ad(u),α(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 using 3.3.(2):
R(y,u)u = −1[ad(y)⊤ +ad(y),ad(u)⊤ +ad(u)]u
4
+ 1[ad(y)⊤ −ad(y),α(u)]u+ 1[α(y),ad(u)⊤ −ad(u)]u
4 4
+ 1[α(y),α(u)]+ 1α([y,u])u
4 2
= −1[ad(y)⊤ +ad(y),ad(u)⊤]u− 1[ad(y)⊤ +ad(y),ad(u)]u
4 4
+ 1[ad(y)⊤,α(u)]u− 1[ad(y),α(u)]u+ 1[α(y),ad(u)⊤ −ad(u)]u
4 4 4
+ 1[α(y),α(u)]u+ 1 ad(u)⊤ad(y)u.
4 2
Summing up we get
⊤
∇ ∇ y +R(y,u)u = y +ad(u) y +α(u)y −ad(u)y
∂t ∂t tt t t t
− 1[ad(y)⊤ +ad(y),ad(u)⊤]u
2
+ 1[α(u),ad(y)]u+ 1 ad(u)⊤ad(y)u.
2 2
Finally we need the following computation using 3.3.(1):
1[α(u),ad(y)]u = 1α(u)[y,u]− 1 ad(y)α(u)u
2 2 2
= 1 ad([y,u])⊤u− 1 ad(y)ad(u)⊤u
2 2
= −1[ad(y)⊤,ad(u)⊤]u− 1 ad(y)ad(u)⊤u.
2 2
Inserting we get the desired result:
⊤
∇ ∇ y +R(y,u)u = y +ad(u) y +α(u)y −ad(u)y
∂t ∂t tt t t t
⊤ ⊤
−[ad(y) +ad(y),ad(u) ]u.
ON THE GEOMETRY OF THE VIRASORO-BOTT GROUP 9
3.6. The weak symplectic structure on the space of Jacobi fields. Let us
assume now that the geodesic equation in g
⊤
u = −ad(u) u
t
admitsauniquesolutionforsometimeinterval,dependingsmoothlyonthechoiceof
theinitialvalueu(0). FurthermoreweassumethatGisaregularLiegroup(see[13],
5.3) so that each smooth curve u in g is the right logarithmic derivative of a smooth
∗
curve g in G which depends smoothly on u, so that u = (g κ)(∂ ). Furthermore
t
we have to assume that the Jacobi equation along u admits a unique solution for
some time, depending smoothly on the initial values y(0) and y (0). These are non-
t
trivial assumptions: in [13], 2.4 there are examples of ordinary linear differential
equations ‘with constant coefficients’ which violate existence or uniqueness. These
assumptions have to be checked in the special situations. Then the space J of all
u
Jacobi fields along the geodesic g described by u is isomorphic to the space g×g
of all initial data.
There is the well known symplectic structure on the space J of all Jacobi fields
u
along a fixed geodesic with velocity field u, see e.g. [11], II, p.70. It is given by the
following expression which is constant in time t:
σ(y,z) : = hy,∇ zi−h∇ y,zi
∂t ∂t
= hy,z + 1 ad(u)⊤z + 1α(u)z − 1 ad(u)zi
t 2 2 2
−hy + 1 ad(u)⊤y + 1α(u)y − 1 ad(u)y,zi
t 2 2 2
= hy,z i−hy ,zi+h[u,y],zi−hy,[u,z]i−h[y,z],ui
t t
= hy,z −ad(u)z + 1α(u)zi−hy −ad(u)y + 1α(u)y,zi.
t 2 t 2
It is worth while to check directly from the Jacobi field equation 3.4.(1) that σ(y,z)
is indeed constant in t. Clearly σ is a weak symplectic structure on the relevant
vector space J ∼= g × g, i.e., σ gives an injective (but in general not surjective)
u
∗
linear mapping J → J . This is seen most easily by writing
u u
σ(y,z) = hy,z −Γ (u,z)i| −hy −Γ (u,y),zi|
t g t=0 t g t=0
∗
which is induced from the standard symplectic structure on g×g by applying first
the automorphism (a,b) 7→ (a,b−Γ (u,a))to g×g and then by injecting the second
g
∗
factor g into its dual g .
For regular (infinite dimensional) Lie groups variations of geodesics exist, but
there is no general theorem stating that they are uniquely determined by y(0) and
y (0). For concrete regular Lie groups, this needs to be shown directly.
t
4. The diffeomorphism group of the circle revisited
4.1. Geodesics and curvature. We consider again the Lie groups Diff(R) and
Diff(S1) with Lie algebras X (R) and X(S1) where the Lie bracket [X,Y] = X′Y −
c
′
XY isthenegativeoftheusualone. FortheinnerproducthX,Yi = X(x)Y(x)dx
integration by parts gives
R
′ ′ ′ ′ ⊤
h[X,Y],Zi = (X YZ −XY Z)dx = (2X YZ +XYZ )dx = hY,ad(X) Zi,
R R
Z Z
10 PETER W. MICHOR, TUDOR RATIU
which in turn gives rise to
⊤ ′ ′
(1) ad(X) Z = 2X Z +XZ ,
′ ′
(2) α(X)Z = 2Z X +ZX ,
⊤ ′
(3) (ad(X) +ad(X))Z = 3X Z,
⊤ ′ ′
(4) (ad(X) −ad(X))Z = X Z +2XZ = α(X)Z.
Equation(4) states that −1α(X)is the skew-symmetrization of ad(X) with respect
2
to to the inner product h , i. From the theory of symmetric spaces one then
expects that −1α is a Lie algebra homomorphism and indeed one can check that
2
−1α([X,Y]) = −1α(X),−1α(Y)
2 2 2
(cid:2) (cid:3)
holds for any vector fields X,Y. From (1) we get the same geodesic equation as in
2.3(4), namely Burgers’ equation:
⊤
u = −ad(u) u = −3u u.
t x
Using the above relations and the general curvature formula 3.3.(2), we get
′′ ′′ ′ ′ ′ ′ ′ ′
R(X,Y)Z = −X YZ +XY Z −2X YZ +2XY Z = −2[X,Y]Z −[X,Y] Z
(5) = −α([X,Y])Z.
If we change the framing of the tangent bundle:
h h f −h f
X = h◦f−1, X′ = x ◦f−1, X′′ = xx x x xx ◦f−1,
f f3
(cid:18) x(cid:19) (cid:18) x (cid:19)
and similarly for Y = k◦f−1 and Z = ℓ◦f−1, for h,k,ℓ ∈ C∞(R,R) or C∞(S1,R),
c
then (R(X,Y)Z)◦f given by (5) coincides with formula 2.3.(3) for the curvature.
4.2. Jacobi fields. A Jacobi field y along a geodesic g with velocity field u is a
solution of the partial differential equation 3.4.(1), which in our case becomes:
⊤ ⊤ ⊤
(1) y = [ad(y) +ad(y),ad(u) ]u−ad(u) y −α(u)y +ad(u)y
tt t t t
= −3u2y −4uy −2u y
xx tx x t
u = −3u u.
t x
Since the geodesic equation has solutions, locally in time (see the argument in 2.3)
it is to be expected that the space of all Jacobi fields exists and is isomorphic to
the space of all initial data (y(0),y (0)) ∈ C∞(S1,R)2 or C∞(R,R)2, respectively.
t c
The weak symplectic structure on it is given by 3.6:
σ(y,z) = hy,z − 1u z +2uz i−hy − 1u y +2uy ,zi
t 2 x x t 2 x x
(2) = (yz −y z +2u(yz −y z))dx.
t t x x
ZS1or R