Download 5.5 Basics IX : Lie groups and Lie algebras

5.5. BASICS IX : LIE GROUPS AND LIE ALGEBRAS
17
coincide.
5.5
5.5.1
Basics IX : Lie groups and Lie algebras
The finite dimensional case
A n-dimensional manifold M is a set which locally looks like an open subset of Rn . In this section
and in the following ones, we will focus only on smooth manifolds, but let us first give a more
precise definition of this.
Figure 5.11:
altas and charts for the sphere S 2 .
Definition 1 Let M be a set. We say that M is a n-dimensional smooth manifold if there exists
an atlas, that is a collection of pairs (Ui , ϕi ) (called “charts”) such that:
- Each Ui is a subset of M and the Ui cover M .
- Each ϕi is a bijection of Ui onto an open subset Vi of Rn . And for any i, j, ϕi (Ui ∩ Uj ) is open
in Rn .
- For any i, j, the map
: ϕi (Ui ∩ Uj ) → ϕj (Ui ∩ Uj )
ϕj ϕ−1
i
is a C ∞ isomorphism.
The most basic examples of finite dimensional manifolds are: any open subset of Rn (it only needs
one chart with the identity map), a smooth curve or a smooth surface in the space, the sphere S n
(like the sphere S 2 of Figure 5.11), the torus Tn , the set of real n × n matrices, etc.
Among finite dimensional manifolds, some of them have moreover the property of being groups:
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CHAPTER 5. FACES AND FLEXIBLE TEMPLATES
Definition 2 A group G is a Lie group if it is a smooth (C ∞ ) manifold such that the group
operations
(g, h) 7→ gh and g 7→ g −1
are smooth.
For all g ∈ G, we can define the left and right translations on G by the maps:
Lg : x 7→ gx and Rg : x 7→ xg
Let e denote the identity element of G. The left translations Lg give a way to map any tangent
space Tg G into the tangent space Te G via De Lg . This shows in particular that the tangent bundle
T G is a trivial vector bundle since it is isomorphic to G × Te G.
The tangent space Te G is also called the Lie algebra of the group G, and, as it is usually done, we
will denote it by g.
A vector field X on G is called left-invariant (resp. right-invariant) if
∀g, h ∈ G Dh Lg .Xh = Xgh
(resp. Dh Rg .Xh = Xhg ) .
Then, the main property of a left-invariant (resp. right-invariant) vector field X is that the value
of X at any point g can be recovered from the value Xe of X at e by De Lg .Xe = Xg (resp.
De Rg .Xe = Xg ). This shows that the vector space of all left- or right-invariant vector fields on G
is isomorphic to the tangent space Te G = g.
The exponential map. Let X be a vector of Te G. We are interested in the curve g : t 7→ g(t)
starting at e at time t = 0, with initial tangent vector X. Moreover, we want this curve to be a
1-parameter subgroup, which means that we want:
∀t, s,
g(t + s) = g(t)g(s).
This condition implies that the tangent vector g 0 (t) at each time t is the left-and right-translated
of the initial vector X = g 0 (0), that is:
∀t,
g 0 (t) = De Lg(t) X = De Rg(t) X.
Then, such a curve exists and is unique. It is called the exponential map and it is denoted by
g(t) = exp(tX).
The bracket product. Let X and Y be two vectors of the Lie algebra g of G. We want to
“measure” how much X and Y commute in the following sense. We have the two exponential
maps t 7→ exp(tX) and t 7→ exp(tY ), and we thus can compare t 7→ exp(tX) exp(tY ) and t 7→
exp(tY ) exp(tX). For this, we look at t 7→ exp(tX) exp(tY ) exp(−tX) exp(−tY ) when t goes to 0.
At the first order, we get have
d
|t=0 exp(tX) exp(tY ) exp(−tX) exp(−tY ) = 0.
dt
5.5. BASICS IX : LIE GROUPS AND LIE ALGEBRAS
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The bracket product [X, Y ] is then the element of g defined thanks to the second order derivative:
[X, Y ] =
1 d2
|t=0 exp(tX) exp(tY ) exp(−tX) exp(−tY ).
2 dt2
The adjoint representation. Let g be an element of G, it then defines an inner automorphism of
G by the map h 7→ ghg −1 = Lg Rg−1 h. Considering its tangent application at the identity element
e of G, we get a map, denoted by Adg in the Lie algebra g. For any vector X in g, it is given by
Adg X = Dg Rg−1 .De Lg .X
The map Ad : G → GL(g, g) is called the adjoint representation of G. Finally, the tangent
application De Ad of the map Ad at point e is denoted by ad : g → L(g, g), and it is called the
adjoint representation of the Lie algebra g.
One of the main properties of the adjoint representation ad is that it is related to the Lie bracket
by
∀X, Y ∈ g, ad(X)Y = [X, Y ].
Example 1: the linear group. Let G = GL(n, R) be the linear group of n × n real matrices
with non-zero determinant. It is a smooth manifold of dimension n2 , since it is an open subset of
2
the set M (n, R) ' Rn of all n × n matrices. Thus, its tangent space at e = In is simply M (n, R).
The Lie algebra g of G is the algebra of all real n × n matrices with the usual bracket product on
matrices given by
[A, B] = AB − BA.
And the exponential map is the usual exponential map for matrices:
exp(A) = In + A +
A2
Ak
+...+
+....
2!
k!
Example 2: the orthogonal group. Let G = O(n) be the orthogonal group, that is the set of
n × n real matrices H such that HH t = In . If we consider the map
F : M (n, R) → Sym(n, R) ' Rn(n+1)/2
M 7→ M M t
then O(n) = F −1 (In ). Moreover for all A ∈ M (n, R), the differential DA F is given by DA F.H =
AH t + HAt , and it is surjective for all A ∈ O(n) since for S ∈ Sym(n), one has DA F. SA
2 = S. This
shows that F is a submersion at every point M of O(n). Thus, O(n) is a manifold of dimension
n2 − n(n+1)
= n(n−1)
. The tangent space at a point H ∈ O(n) is
2
2
TH G = {M ∈ M (n, R); DH F.M = 0} = {M ∈ M (n, R); HM t + M H t = 0}.
In particular, the tangent space at e = In is the set of matrices M such that M + M t = 0, that
is the space of real n × n skew symmetric matrices. The Lie algebra of O(n) is the tangent space
at e = In with the same bracket product as in the case of GL(n) (Notice that if At = −A and
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CHAPTER 5. FACES AND FLEXIBLE TEMPLATES
B t = −B, then [A, B]t = B t At − At B t = BA − AB = −[A, B], so that the bracket [A, B] belongs
to the algebra. Notice also that if At = −A then for all t real, exp(tA)t = exp(−tA) = exp(tA)−1
and it thus belongs to the group).
Metrics. Let G be a Lie group. Given an inner product on Te G, denoted by h., .ie , one can then,
using left translations, define an inner product on any Tg G by setting :
∀u, v ∈ Tg G,
hu, vig = hDg Lg−1 u, Dg Lg−1 vie .
For a smooth path γ : [0, 1] → G, its length is defined by
Z 1q
hγ 0 (t), γ 0 (t)iγ(t) dt.
`(γ) =
0
The distance d(x, y) between two points x and y of G is then given by
d(x, y) = Inf{`(γ); γ smooth path such that γ(0) = x and γ(1) = y}.
The inner product on each Tg G was defined using left translations, and the metric thus obtained
on G is left-invariant, i.e. d(x, y) = d(gx, gy). Indeed, for a smooth path γ on G and for g ∈ G,
one has `(gγ) = `(γ). Usually this metric will not coincide with the one that can be constructed
thanks to right translations.
Example. Given two real n × n matrices A = (aij )i,j and B = (bij )i,j , one can define their inner
product by
X
hA, Bie =
aij bij = tr(AB t ).
i,j
Let G = O(n) and let h ∈ G. The inner product on Th G is then given by
hA, Bih = hh−1 A, h−1 Bie = tr(h−1 AB t h−t ) = tr(AB t )
because hht = I. The distance thus obtained on the group G = O(n) is then both left- and
right-invariant. If two matrices do not belong to the same connect component, their distance is
infinite.
5.5.2
The infinite dimensional setting, diffeomorphisms and curves
Ideally, the definition of infinite dimensional manifolds would be exactly like the one in the finite
dimensional case, except that we would have to replace Rn by a Banach space E (that is a topological vector space whose topology is given by a norm, and which is moreover complete). This
is the definition that one can usually find in books [Lan85]. But, unfortunately, things are not so
simple. When one works with infinite dimensional spaces, which is our case with function spaces,
we have to deal with spaces which are not Banach spaces. This is for instance the case of the
5.5. BASICS IX : LIE GROUPS AND LIE ALGEBRAS
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space C ∞ (Rn , Rn ) of smooth functions of Rn into itself: it is not a Banach space, and neither
is Cc∞ (Rn , Rn ) (the space of smooth functions with compact support). So, instead of modeling
a manifold on a Banach space E, we will use more general vector spaces (in particular function
spaces like Cc∞ (Rn , Rn )). Let us illustrate this with two important examples: the space of smooth
curves and the group of diffeomorphisms.
Example 1: the space of smooth curves. Let us describe with this example what charts and
atlas may look like in the infinite dimensional setting. Let us consider the space S of smooth plane
curves. Let C ∈ S be a fixed curve, parametrized by s 7→ C(s). Then we can define a local chart
near C by
s 7→ Ca (s) = C(s) + a(s).~n(s),
where ~n(s) is the unit normal to C and a is a smooth function. Let UC denote the set of functions
a which are such that the corresponding curve Ca is smooth. Then (UC , a 7→ Ca ) is a local chart
near C, and the a(s) are the local linear coordinates. The collection of the charts for all C ∈ S
gives the atlas.
Figure 5.12:
a local chart in the space of smooth curves.
Example 2: the Lie group Diff(Rn ). It is known [KM97], that ”there is no way to model the
diffeomorphism group on Banach spaces as a manifold”. Anyway, it can be seen as a Lie group
in the following sense. The group G = Diff(Rn ) of all smooth diffeomorphisms of Rn which are
the identity outside a compact subset of Rn is an open subset of the space C ∞ (Rn , Rn ) for a
convenient strong topology and the group operations of composition and inversion are smooth for
this topology (more details about this can be found in [KM97] or [Hir94]).
The group G = Diff(Rn ) can be seen as a manifold modeled on the space of functions with compact
support. The Lie algebra g of G is the tangent space Te G at point e = Id. It is given by the vector
space of smooth vector fields ~v with compact support on Rn via ~x −→ ~x + ε~v (~x). Then a tangent
vector at the point φ ∈ G is again equivalent to a vector field ~v with compact support which we
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CHAPTER 5. FACES AND FLEXIBLE TEMPLATES
can define either a) as the infinitesimal deformation φ(~x) + ε~v (φ(~x)) of φ (using right translation
by φ) or b) as the infinitesimal deformation φ(~x) + εD~x φ.~v (~x) (using left translation by φ). We
will use the first definition which is perhaps more natural. Notice that in this case, left and right
translations do not provide the same description of the tangent space at a point φ of the group
G. For the left translation, we need the derivative Dφ; which shows that we ”loose” one degree of
regularity. But in our case, we avoid all these questions by working directly in the C ∞ setting.
All the previous setting remains valid if we replace the space Rn by a domain D of Rn . In all the
following, we shall denote by G either the group Diff(Rn ) of smooth diffeomorphisms of Rn which
are the identity outside a compact subset of Rn or the group Diff(D) of smooth diffeomorphisms
of a domain D ⊂ Rn which are the identity ouside a compact subset of D.
MetricRon G. Assume that we have an inner product on the Lie algebra g given by k ~v k 2 =
L~v .~v = D k ~v (~y ) k2S d~y , where L is a self-adjoint operator. Let Φ : [0, 1] → G be a path on G, i.e.
x, t)
for each t, the map ~x 7→ Φ(~x, t) is a diffeomorphism of D. Then, for each t, the map ~x 7→ ∂φ
∂t (~
belongs to TΦ(·,t) G. Using right translations, we can define (as we did it for finite dimensional Lie
groups) an inner product on each TΦ(·,t) G and thus define the length of the path Φ by
`(Φ) =
5.5.3
Z
1
0
Z
D
k
∂φ −1
(Φ (~y , t), t) k2S d~y
∂t
1/2
dt.
Some gradient flows
Let f be a real-valued function defined on a smooth manifold. The tangent application Dx f of f at
a point x ∈ M is by definition a linear mapping from Tx M into R, i.e. it is a co-vector (its belongs
to Tx M ∗ the dual of Tx M ). If we want to convert it into a vector, we need an inner product on
Tx M . If h., .i denotes such an inner product, then one can define the gradient ∇f (x) of f at point
x as the vector of Tx M such that
∀h ∈ Tx M, Dx f.h = h∇f (x), hi.
Example: gradient of length in the space of plane curves. Let M be the space of plane
smooth curves. We already saw in the previous section that this set is a manifold and that if C
is a plane smooth curve, then the tangent space TC M can be identified with the set of smooth
functions. We want here to compute the gradient of the map C 7→ l(C) = length of C.
Let C be a smooth curve, and assume that we have a parametrization s 7→ C(s) by arc-length
(which means that k C 0 (s) k= 1). Let a be a smooth function on [0, l(C)], that is an element of
TC M . Let Cε be the curve
Cε : s 7→ C(s) + εa(s)~n(s),
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5.6. GEODESICS ON A LIE GROUP
where ~n(s) is the unit normal to C. Let κ(s) = h~n0 (s), C 0 (s)i be the curvature of the curve C.
Then
Z l(C)
l(Cε ) =
k Cε0 (s) k ds
0
=
=
Z
Z
l(C)
0
l(C)
0
k C 0 (s) + εa0 (s)~n(s) + εa(s)~n0 (s) k ds
(1 + εhC 0 (s), a0 (s)~n(s) + a(s)~n0 (s)i) ds + O(ε2 )
= l(C) + ε
Z
l(C)
0
a(s)κ(s)ds + O(ε2 ).
This computation shows that the gradient of the length on the space of smooth plane curves is the
curvature: ∇l(C) = s 7→ κ(s)
Geometric heat equation for plane shapes. Starting from of smooth closed curve C 0 , we can
make it evolve according to the following partial differential equation called the “geometric heat
equation”:
∂Ct (s)
= −κt (s)~nt (s).
∂t
This evolution t 7→ Ct will make the length of the curves decreases: it “pushes” the concave parts
outside while “pulling” the convex parts inside (see Figure 5.13 for an example of this).
Figure 5.13:
5.6
A cat silhouette evolving with the geometric heat equation.
Geodesics on a Lie group
A key question is whether there are paths minimizing path-length between any two diffeomorphisms, that is to say, are they connected by a geodesic? This is a difficult existence question and
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CHAPTER 5. FACES AND FLEXIBLE TEMPLATES
we will not be able to give here a general answer. Let us mention the results of Trouvé and Younes
[TY05] about this question. Their framework is not exactly the same as the one we have described
above, and this is necessary in order to prove the existence of geodesics as solutions of a variational
problem. One needs to work with complete spaces. They first extend the space of vector fields
into a complete space B. They then prove, for each v ∈ L1 (B, [0, 1]) the existence of the flow
∂Φ
(x, t) = v(Φ(x, t), t) and Φ(x, 0) = x.
∂t
And finally, if A is the set of all homeomorphisms that can be reached at time t = 1, they prove
that there is a path that minimizes path-length between any two homeomorphisms of A.
In this section, we will seek to characterize geodesics by a differential equation. On arbitrary
Lie groups, Arnold discovered a very general way to describe this equation but it can be derived
directly by variational calculus on the group G = Diff(D). In this case geodesics are described by
a variant of Euler’s fluid flow equation, called “EPDiff” (which stands for Euler-Poincaré equation
for diffeomorphisms).
5.6.1
The direct derivation of EPDiff
We have to calculate the first variation of the energy U (Φ) of a path Φ:
Z 1 Z
2
∂Φ −1
(Φ (~y , t), t) k d~y dt .
U (Φ) =
k
∂t
0
D
~ x, t), where f~ is a function with compact support in D × [0, 1], and let the
Let Φ −→ Φ + εf(~
2
−1
velocity be ~v (~y , t) = ∂Φ
(~y , t), t), then we can write k ~v k = L~v · ~v where L is a self-adjoint
∂t (Φ
~ −1 (~y , t), t).
operator. And finally let ~g be the function defined by ~g (~y , t) = f(Φ
1/ We first get some formula useful in what follows.
Since Φ(Φ−1 (~y , t), t) = ~y we get, by differentiation with respect to t:
∂(Φ−1 )
∂Φ −1
(~y , t) = −(D~x Φ)−1 (Φ−1 (~y , t), t) ·
(Φ (~y , t), t) .
∂t
∂t
We will need the first variation of Φ−1 . For this, we use the fact that
~ ◦ (Φ−1 + εδΦ−1 )(~y ) = ~y, and then get
(Φ + εf)
~ −1 (~y , t), t) .
δΦ−1 (~y ) = −(D~x Φ)−1 (Φ−1 (~y , t), t) · f(Φ
We will also need
∂~g
∂t
=
=
∂ ~ −1
f(Φ (~y , t), t)
∂t
∂ f~
∂Φ
− D~x f~ · (D~x Φ)−1 ·
.
∂t
∂t
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5.6. GEODESICS ON A LIE GROUP
2/ We now can calculate the first variation δ~v of ~v using the results of the first part.
∂ ∂
−1
~
~
δ~v =
(Φ + εf)((Φ + εf ) (~y , t), t)
∂ε ∂t
ε=0
∂Φ −1
∂ f~ −1
~ −1 (~y , t), t)
(Φ (~y , t), t) − D~x
(Φ (~y , t), t) · (D~x Φ)−1 (Φ−1 (~y , t), t) · f(Φ
=
∂t
∂t
∂~g
∂Φ
∂Φ
=
(~y , t) + D~x f~ · (D~x Φ)−1 ·
− D~x
· D~x Φ · ~g
∂t
∂t
∂t
And then since
D~x
and
∂Φ −1
(Φ (~y , t), t) · D~x Φ(Φ−1 (~y , t), t) = D~y~v (~y , t)
∂t
~ −1 (~y , t), t) · (D~x Φ)−1 (Φ−1 (~y , t), t) = D~y~g (~y , t),
D~x f(Φ
we finally have
δ~v =
∂~g
+ D~y~g · ~v − D~y~v · ~g.
∂t
3/ And so the first variation of U is
δU = −2
δU = −2
Z
0
1
Z X
D
j
Z
0
1
Z
D
(L~v · δ~v ) d~y dt .
X
∂
∂vi X ∂
(Lvj ) +
+
(Lvj .vi )
Lvi .
∂t
∂yj
∂yi
i
i
!
gj d~y dt .
And so if we want δU = 0 for all ~g , we get that ~v is solution of the following equation:
∂
~
(L~v ) + (~v · ∇)(L~
v ) + (div~v ).L~v + L~v · D~v = 0.
∂t
This equation is called EPDiff.
Example 1: the case of dimension 1.
When the operator L is simply the Identity, then the preceeding equation becomes:
∂
~ v ) + (div~v ).~v + 1 ∇(k
~ ~v k2 ) = 0
~v + (~v · ∇)(~
∂t
2
In dimension 1, then v satisfies Burger’s equation
vt + 3v.vy = 0
and Φ satisfies the equation:
Φtt Φx + 2Φtx Φt = 0.
This should not be nicely integrable and, in fact, it is the simplest example of a shock-wave
producing equation as shows the following proposition (see also Figure 5.14).
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CHAPTER 5. FACES AND FLEXIBLE TEMPLATES
Proposition 1 (Burger’s equation)
If (x, t) 7→ u(x, t) satisfies ut + C.u.ux = 0 and u(x0 , 0) = a then
u(x0 + C.a.t, t) ≡ a.
t
?
shock
x
u(x,0) > 0
u(x,0) < 0
Figure 5.14:
Burger’s equation ut + C.u.ux = 0 is the simplest example of a shock-wave producing equation. If
u(x0 , 0) = a then u(x0 + C.a.t, t) ≡ a (see Proposition 1).
Example 2: the group of volume preserving diffeomorphisms.
Theorem 2 (V.Arnold) If G is the group of volume preserving diffeomorphisms and
2
k ~v k =
Z
2
D
k ~v (~y ) k d~y ,
then the geodesics are given by Euler’s equation for inviscid, incompressible fluid.
If G is the group of volume preserving diffeomorphisms, then in order to define a variation in the
volume preserving group, ~v and ~g must be such that div(~v ) = 0 and div(~g) = 0. And so we must
have δU = 0 for all ~g such that div(~g ) = 0. Then using Lagrange multipliers, we get
∂
~ v ) + 1 ∇(k
~ ~v k2 ) = ∇p̃
~
~v + (~v · ∇)(~
∂t
2
for some p̃, which means
∂
~ v ) = ∇p
~
~v + (~v · ∇)(~
∂t
for some p. This equation is Euler’s equation for inviscid, incompressible flow (p is a pressure and
~ v ) is called the “transport term”).
(~v · ∇)(~
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5.6. GEODESICS ON A LIE GROUP
5.6.2
Momentum and Arnold’s general description of geodesics
In his paper [Arn66], Arnold gives a very general description of geodesics on a Lie group. He
characterizes them as curves along which momentum is preserved. Let us describe this more
precisely, using the notations and defintions introduced in Section 5.5. We will then see what this
momentum becomes in the case of the special orthogonal group SO(n) and in the case of the group
of diffeomorphisms G.
Let G be a Lie group and let g denotes its Lie algebra. Let L : g → g∗ be a self-adjoint operator
symmetric and positive definite. It defines an inner product on g by
∀u, v ∈ g, hu, vi = Lu.v = Lv.u.
Using left translations, we thus obtain an inner product on any tangent space Tg G by:
∀u, v ∈ Tg G, hu, vig = hdg Lg−1 u, dg Lg−1 vi.
Finally let β : g × g → g be the operator such that
∀u, v, w ∈ g, hu, [v, w]i = hβ(u, v), wi.
Let t 7→ g(t) be a curve on G and let ω(t) = Dg(t) Lg(t)−1 ġ(t) be the velocity. The energy of the
curve is defined by
Z
Z
1
1
hġ(t), ġ(t)ig(t) dt =
hω(t), ω(t)idt.
2
2
Then the curve t 7→ g(t) is said to be a geodesic on G if its an extremum of the energy.
In Section 5.5, we defined the adjoint representation Ad, which provides for all g ∈ G a map
Adg : g → g. If g∗ denotes the dual of g (that is the space of continuous linear forms on g), then
the dual map Ad∗g : g∗ → g∗ is defined by
∀ξ ∈ g∗ , ∀η ∈ g,
(Ad∗g ξ, η) = (ξ, Adg η).
Arnold proved in [Arn66], the following theorem about the conservation of the momentum:
Theorem 3 Let t 7→ g(t) be a geodesic on G. Then
∀t, ω̇(t) = β(ω(t), ω(t))
and
Ad∗g(t)−1 Lω(t) = constant.
(5.1)
(5.2)
This motivates calling ω(t) the momentum of the geodesic flow. Equation (5.2) is the conservation
of momentum for a Riemannian metric on a group. It is very strong and it is an integrated form
of Equation (5.1) which is the differential form of the geodesic equation.
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CHAPTER 5. FACES AND FLEXIBLE TEMPLATES
Example 1: Geodesics on the group SO(n).
Let G = SO(n) be the group of n × n orthogonal matrices with determinant 1. We already saw
in Section 5.5, that G is a Lie group and that its Lie algebra g is the space of skew symmetric
matrices. In order to define the length of a path on G, we first need an inner product on g. Let S
is an invertible symmetric matrix, we then define:
1
∀A, B ∈ g, hA, Big = tr(At SB) = −tr(ASB) = − tr(A(SB + BS)).
2
The corresponding operator is thus L : g → g, with LB = SB + BS. It is an isomorphism, and we
will denote its inverse by L−1 . We then need to compute the operator β : g × g → g. According to
its definition, it has to be such that for all A,B and C skew symmetric:
tr(β(A, B)SC) = tr(AS(BC − CB)) = tr(ASBC) − tr(BASC).
Since it holds for every skew symmetric C, it implies that the skew symmetric part of β(A, B)S −
ASB + BAS has to be 0. That is:
β(A, B)S + Sβ(A, B) = ASB − BSA − BAS + SAB.
And in particular, we get Lβ(A, A) = β(A, A)S + Sβ(A, A) = −A2 S + SA2 = [S, A2 ].
Let t 7→ g(t) be a path on G. And let Ω(t) = g −1 (t)ġ(t) be the velocity. The energy of this path
is then
Z
Z
hg(t)−1 ġ(t), g(t)−1 ġ(t)ig dt = − tr(Ω(t)SΩ(t))dt.
The equations for geodesics are thus in this case:
LΩ̇(t) = Lβ(Ω, Ω) = [S, Ω2 ].
(5.3)
And the momentum which is preserved is
M (t) = g(t)(Ω(t)S + SΩ(t))g(t)−1 = constant .
Now all this can be directly derived by taking infinitesimal deformations of the path t 7→ g(t) of the
form t 7→ g(t)(I + εa(t)) where for all t, the matrix a(t) is skew symmetric. Then, computing the
first variation of the energy of the path, one can directly obtain Equation (5.3) for the geodesics.
When the dimension is n = 3, and S is a diagonal matrix of the form



0
I2 + I 3
0
0
 and if we write Ω =  Ω3
0
I3 + I1
0
S=
−Ω2
0
0
I 1 + I2
Then the equation of geodesics becomes

 I1 Ω̇1 − Ω2 Ω3 (I2 − I3 ) = 0
I Ω̇ − Ω3 Ω1 (I3 − I1 ) = 0
 2 2
I3 Ω̇3 − Ω1 Ω2 (I1 − I2 ) = 0
−Ω3
0
Ω1

Ω2
−Ω1  ,
0
29
5.6. GEODESICS ON A LIE GROUP
These are exactly the usual Euler equations in mechanics. And in this case, S is the inertia matrix
and (Ω1 , Ω2 , Ω3 ) is the angular velocity.
Example 2: Geodesics on the group of diffeomorphisms G.
We consider the regular Lie group G = Diff(Rn ) of all smooth diffeomorphisms which are the
identity outside a compact subset of Rn . Its Lie algebra g is the space of smooth vector fields with
compact support. As we already saw, we can equip Diff(Rn ) with a right invariant metric given
by an inner product on g:
Z
hX, Y ig =
Rn
hLX, Y i dx,
where L is a self-adjoint operator (one can for example take a Sobolev inner-product: L = (1−∆) k ).
What is the form of the conservation of momentum for a geodesic Φ(t) in this metric, that is to
say, a flow x 7→ Φ(x, t) on Rn ? We need to work out Ad∗Φ first. Using the definition, we see:
Z
Z
Z
hLX, Ad∗Φ (Y )i :=
hLAdΦ (X), Y i =
h(DΦ.X) ◦ Φ−1 , LY i
n
n
n
R
R
Z
ZR
hX, det(DΦ).DΦt .(LY ◦ Φ)i
det(DΦ)hDΦ.X, LY ◦ Φi =
=
Rn
Rn
hence:
Ad∗Φ (Y ) = L−1 det(DΦ).DΦt .(LY ◦ Φ) .
−1
Let v(t) = ∂Φ
be the velocity of the flow Φ. Theorem 3 about the conservation of momentum
∂t ◦ Φ
for geodesics Φ(t) says that:
∂Φ
L−1 det(DΦ)(t).DΦ(t)t . L(
◦ Φ−1 ) ◦ Φ
is independent of t.
∂t
This can be put in a much more transparent form. First, L doesn’t depend on t, so we cross out
the outer L−1 . Now let u(t) = Lv(t), so that:
det(DΦ)(t).DΦ(t)t .(u(t) ◦ Φ(t))
is independent of t.
R
We should not think of u(t) as a vector field on Rn : this is because we want Rn u.v to make
invariant sense. This means we should think of u as expanding to the differential form:
ω(t) = (u1 .dx1 + u2 .dx2 + . . . + un .dxn ) ⊗ µ
where µ = dx1 ∧ dx2 ∧ . . . ∧ dxn , is the volume form on Rn . But then the pullback of the differential
form ω(t) is:
Φ(t)∗ (ω(t)) = det(DΦ)(t).hDΦt .(u ◦ Φ(t)), dxi ⊗ µ
so conservation of momentum says simply:
Φ(t)∗ ω(t)
is independent of t.
This is the integrated form of the geodesic equation. If we differentiate it with respect to t, we
find again EPDiff: the differential equation for the geodesics on the group of diffeomorphisms.