INVARIANT DIFFERENTIAL OPERATORS AND EXTERIOR
DIFFERENTIAL SYSTEMS.
LECTURE I: PROLONGATION AND INFINITESIMAL SYMMETRIES
JM LANDSBERG
1. Introduction
The purpose of these four lectures is to motivate and describe the machinery of Kostant
and Bernstein-Gelfand-Gelfand (henceforth BGG) respectively for for computing Lie algebra
cohomology and relating differential operators on sections of vector bundles to operators on
corresponding homology bundles. The exterior differential systems involved are of a very special
type - those where some finite prolongation is zero. In this lecture I first discuss how Spencer
cohomology groups arise in the study of the moduli space of local solutions through a point to a
system of partial differential equations. I will then take a natural example - that of finding the
infinitesimal symmetries (Killing fields) of a G-structure, and describe the familiar Riemannian
case in detail. In the second lecture Mike will revisit this example and discuss others where the
computation of Spencer cohomology reduces to a Lie algebra cohomology calculation, describe
how integral manifolds of systems of finite type can be viewed as covariant constant sections
of certain vector bundles (called tractor bundles) equipped with preferred connections, and
introduce the BGG perspective. In the third lecture Andreas will explain how Kostant’s Hodge
theory allows one to obtain the preferred connections, and in the fourth lecture he will present
new joint work with Soucek describing how the tractor bundles can be “pruned” to homology
bundles.
2. Prolongation
2.1. Notation. I will use the index ranges
1 ≤ i, j, k, l, m, u ≤ n
1 ≤ a, b, c ≤ s,
P
the summation convention: xi yi := i xi yi , and the Penrose type conventions: x(ij) := 12 (xij +
xji ) and x[ij] := 12 (xij − xji ) and similarly for symmetrizing or skew-symmetrizing over several
indices.
2.2. Linear Pfaffian systems. In exterior differential systems, one usually expresses systems
of PDE in terms of differential forms instead of equations.
Definition 2.2.1. Let Σ be a manifold, a linear Pfaffian system on Σ is a 2-step filtration of
the cotangent bundle
(1)
I ⊂ J ⊂ T ∗Σ
such that dθ ≡ 0 mod J ∀θ ∈ I. The bundle J/I is called the independence condition. An integral
manifold is an immersed submanifold i : M → Σ such that i∗ (I) = 0 and i∗ (J/I) = T ∗ M .
1
2
JM LANDSBERG
Example 2.2.2. Let J 1 (Rn , Rs ) denote the space of 1-jets of mappings from Rn → Rs , and
equip it with coordinates (xi , ua , pai ). There is the tautological system I = {θ a := dua − pai dxi },
J = {θ a , ω i := dxi } whose integral manifolds are exactly the 1-jets of mappings Rn → Rs , that
a
is, given f : Rn → Rs , its one-jet is j 1 (f ) = (xi , f a (x), ∂f
(x)).
∂xi
Example 2.2.3. Let
f r (ua , xi ,
(2)
∂ua
) = 0, 1 ≤ r ≤ R
∂xi
be a system of PDE for maps Rn → Rs . Consider the subset Σ ⊂ J 1 (Rn , Rs ) defined by the
equations f r (xi , ua , pai ) = 0, 1 ≤ r ≤ R, and let (IΣ , JΣ ) denote the pullback of (I, J) to Σ.
Then integral manifolds of (IΣ , JΣ ) are exactly the 1-jets of solutions to (2).
Example 2.2.4. Say J = T ∗ Σ. Then we are in the situation of the Frobenius theorem: If we
give Σ a coframing {θ a , ω i } where I = {θ a }, then dθ a ≡ Tija ω i ∧ ω j mod I and the system is
Frobenius (i.e. there exists a unique integral manifold through each point) if and only if the
functions Tija are zero.
In general, one can give T ∗ Σ a coframing (θ a , ω i , π ǫ ) adapted to the filtration (1), where
I = {θ a }, J = {θ a , ω i }. Then, since dθ a ≡ 0 mod I,
dθ a ≡ Aaǫi π ǫ ∧ ω i + Tija ω i ∧ ω j mod I.
for some functions Aaǫi , Tija . Here the functions Tija are apparent obstructions to having an
integral manifold as the ω i ∧ ω j must be linearly independent on an integral manifold. However
the functions Tija will change under a change in choice of complement to J in T ∗ Σ, so we must
quotient out by this ambiguity to obtain the actual obstruction.
Cartan’s test for the existence of integral manifolds begins as follows: Fix a general point
x ∈ Σ. (The theory works best in the analytic category where the notion of a general point
makes sense.) Let V ∗ = (J/I)x and W ∗ = Ix , and write v i = ωxi , wa = θxa . Let vi , wa be the
corresponding dual basis vectors.
Define the tableau at x by
A = Ax := {Aaǫi v i ⊗wa ⊆ V ∗ ⊗W | 1 ≤ ǫ ≤ r} ⊆ W ⊗V ∗ .
Let δ denote the natural skew-symmetrization map δ : W ⊗V ∗ ⊗V ∗ → W ⊗Λ2 V ∗ and let
(3)
(4)
H 0,2 (A) := (W ⊗Λ2 V ∗ )/δ(A⊗V ∗ )
A(1) := ker δ|A⊗V ∗ .
Exercise 2.2.5: Verify that A, A(1) and H 0,2 (A) depend only on (I, J) and the choice of x.
Show that in contrast the tensor Tija wa ⊗v i ∧ v j does depend on choices.
The torsion of (I, J) at x is
[T ]x := [Tija wa ⊗v i ∧ v j ] ∈ H 0,2 (A).
It is the obstruction to a solution through x. If it vanishes, A(1) measures the dimension of
the space
of admissible two-jets of a solution through a point. (In particular its dimension is
s n+1
for
the tautological system of Example 2.2.2.) When [T ]x = 0, Cartan’s test compares
2
a crude estimate for dim A(1) with its actual dimension. If equality holds, the Cartan-Kähler
theorem guarantees local existence of solutions in the analytic category (and existence of a a
formal solution in the C ∞ category). If equality fails, one must prolong the system. One starts
over on a new space which is locally a submanifold of the space of two-jets J 2 (Rn , Rs ). Formally
PROLONGATION AND INFINITESIMAL SYMMETRIES
3
one works on a subbundle (subsheaf) of the Grassmann bundle G(n, T Σ) → Σ. In practice on
simply adds new variables (differential forms) corresponding to a basis of A(1) .
Here is an overview of the Cartan algorithm - for precise definitions see [2, Chap.5]:
Rename Σ′ as Σ
-
Input:
linear Pfaffian system
(I, J) on Σ;
calculate dI mod I
6
N
Q
Q
′
Q
Q
Is
Σ
empty?
Q
Q
k
Q
Q
Q Q
Q
Y
Q
Q
?
Q
Done:
there are no
integral manifolds
?
Q
Q Y
Q
Is
[T
]
=
0? Q
Q
Q
Q
Q
N
?
Restrict to Σ′ ⊂ Σ
defined by [T ] = 0
and Ω |Σ′ 6= 0
“Prolong”, i.e., start over
on a larger space Σ̃;
rename Σ̃ as Σ
and new system as (I, J)
6
N
Q
Q
Q
- Is tableau Q
Q involutive? Q
Q
Q
Y
?
Done:
local existence of
integral manifolds
2.3. Exercises.
(1) Verify that if [T ]x 6= 0 then there are no integral manifolds through x ∈ Σ.
(2) Show that the Fundamental Lemma of Riemannian geometry is equivalent to the statements H 0,2 (so(V )) = 0 (existence of a torsion free connection) and so(V )(1) = 0 (uniqueness of a torsion free connection). Here so(V ) ⊂ V ∗ ⊗V and one takes W = V .
(3) Show that if the k-th prolongation of A is zero, then integral manifolds (if there are any)
are uniquely determined by at most the k-jet of an integral manifold at a single point.
In this lecture all the EDS we work with will be as in Exercise (3), which are called systems
of finite type. Their integral manifolds depend at most on constants. For these systems one does
not need the full force of the Cartan-Kähler theorem, but the perspective of prolongation will
be extremely useful.
2.4. Spencer Cohomology. Let A(k) be the kernel of the map A⊗S k V ∗ → W ⊗Λ2 V ∗ ⊗S k−1 V ∗ ,
called the k-th prolongation of A.
Remark 2.4.1. For those familiar with the representation theory of the general linear group, note
that W ⊗Λ2 V ∗ ⊗S k−1 V ∗ = W ⊗Sk−1,1,1 V ∗ ⊕ W ⊗Sk,1 V ∗ and the image must lie in the second
factor, so our map and the maps below can be pruned. When our system is invariant under
a group G ⊂ GL(V ), the GL(V )-module maps can be pruned further to take into account the
G-module structure.
Definition 2.4.2. The Spencer cohomology groups H i,j (A) of a tableau A ⊂ W ⊗V ∗ are defined
as follows: Let
(5)
δj : W ⊗S i V ∗ ⊗Λj V ∗ → W ⊗S i−1 V ∗ ⊗Λj+1 V ∗
be defined by
δj (f ⊗ξ) = df ∧ ξ,
where, for f ∈ W ⊗S i V ∗ , ξ ∈ Λj V ∗ , we define df by considering f as a W -valued function on
V , and extend δj by linearity. Note that δj (A(i) ⊗Λj V ∗ ) ⊆ A(i−1) ⊗Λj+1 V ∗ .
4
JM LANDSBERG
Define
H i,j (A) :=
ker δj (A(i−1) ⊗Λj V ∗ )
.
Image δj−1 (A(i) ⊗Λj−1 V ∗ )
Remark 2.4.3. For a linear Pfaffian system such that H 0,k (A) = 0 for all k ≥ 2, there exist
integral manifolds through a general point.
3. Infinitesimal symmetries
3.1. Killing vector fields. Let (M, g) be a Riemannian manifold. If the metric g admits any
continuous symmetries, there will be a one parameter group acting on M preserving the metric.
As is usual in mathematics, it is easier to work infinitesimally, so we look for a vector field
X ∈ Γ(T M ) such that LX g = 0, then the flow of X will provide the desired one parameter
group. Such a vector field is called a Killing vector field. We just consider
local question.
P i the
1
n
i
Say we have a local orthonormal coframing (η , . . . , η ), so g = i η ⊚ η . Then
X
(6)
LX g = 2
(X dη i ) ⊚ η i + d(η i (X)) ⊚ η i
i
Let’s set up an EDS for such a vector field. To avoid making choices, we work on the bundle
of orthonormal frames of M (equivalently orthonormal coframes) which we denote F → M .
The fundamental lemma of Riemannian geometry states that F has a canonical coframe (η i , ηji ),
where the η i span the semi-basic forms, ηji + ηij = 0, and
dη i = −ηji ∧ η j .
Moreover, we may write
i
dηji = −ηki ∧ ηjk + Rjkl
ηk ∧ ηl
i (e ⊗η j )⊗(η k ∧ η l ) is a basic tensor (the Riemann curvature tensor). Here the e are
where Rjkl
i
i
dual to the η i . A point of F may be described as (p, e1 , . . . , en ) where p ∈ M and the ej form
an orthonormal basis of Tp M .
We work on F, and solve for the horizontal lift X̃ of the desired X. Since the lift is horizontal,
we may write X̃ = xi ei , for some functions xi on F. The (lifted) right hand side of (6) becomes
X
(7)
2
(xj ηji + dxi ) ⊚ η i
i
The connection forms ηji define a covariant differential operator ∇ by ∇X = (xj ηji + dxi )⊗ei .
Let ♯ : Tx M → Tx∗ M denote the isomorphism induced by gx , and πS : V ∗ ⊗V ∗ → S 2 V ∗ the
symmetrization map, then
(8)
LX g = πS (Id⊗♯)∇X
where this expression is well defined on M . In indices
(9)
(LX g)ij = [(Id⊗♯)∇X](ij) .
In summary:
Proposition 3.1.1. LX g = 0 if and only if ∇X ∈ Γ(so(T M )).
Exercise 3.1.2: Generalize Proposition 3.1.1 by showing that if ∇ is an affine connection
preserving a tensor τ , then a necessary condition for X ∈ Γ(T M ) to preserve ∇ is that
∇X ∈ Γ(gτ (T M )), where (gτ )x is the Lie algebra of the group Gτ ⊂ GL(Tx M ) preserving
τ.
PROLONGATION AND INFINITESIMAL SYMMETRIES
5
Exercise 3.1.2 implies:
If M is equipped with a G-structure and compatible affine connection ∇, a vector field X ∈
Γ(T M ) is an infinitesimal symmetry of the G-structure only if ∇X ∈ Γ(g(T M )).
Now the question becomes how to determine such vector fields. We will set up the EDS in
the Riemannian case, and the generalization to an arbitrary G-structure should be clear.
3.2. The EDS. Write dxi = xij η j + 12 xi j ηkj , where xi j = −xi k , then (7) becomes
(k)
(k)
(j )
X
1
(10)
2
(δki xj + xi k )ηjk ⊚ η i + xik η k ⊚ η i .
2 (j )
i
Since the forms
(11)
(12)
ηjk , η i
are all linearly independent, for (10) to vanish we must have
xi k = −δki xj + δji xk
(j )
xik + xki = 0
So define the EDS on F × Rn × so(n), where the second and third factors respectively have
coordinates (xi ) and (xij = −xji ), by
(13)
I = {θ i := dxi − xij η j + xj ηji }.
Since ∇X = (dxi + xj ηji )⊗ei ≡ xij η j ⊗ei mod I, we see that we are indeed requiring that ∇X be
so(V )-valued.
Remark 3.2.1. Were we in a situation where g(1) 6= 0 (and H 0,2 (g) = 0), we would have started
on the larger space, (FG × g(1) ) × Rn × g to take into account the freedom in the choice of
connection form. Note that in this case, although there is not a unique choice of connection,
the algorithm one follows is unchanged.
i ηk ∧ ηl ,
We compute, recalling dηji = −ηki ∧ ηjk + 12 Rjkl
1 i k
dθ i ≡ −[dxil − xij ηlj + xjl ηji + xj Rjkl
η ] ∧ η l mod I
2
Inspired by the proof of the fundamental lemma of Riemannian geometry, we rewrite this as
1
i
η k ] ∧ η l mod I
(14)
dθ i ≡ −[dxil − xij ηlj + xjl ηji + xj Rljk
4
Since the term in the brackets is skew in (i, l), we conclude we must add it to the ideal, i.e.,
so(V )(1) = 0.
Thus we add the forms
1
i
ηk
θli := dxil − xij ηlj + xjl ηji + xj Rljk
4
to our ideal.
We now must differentiate the new forms in our ideal. We get
1
k
i
i
i
k
i
i
i
k
ηm
)∧η m + (xik Rjlm
ηlk −Rjkl
ηjk −Rjmk
ηki −Rkml
+Rjml
dθji = xl (dRjml
)η l ∧η m
+2xum Rjlu
−xkj Rklm
2
l
i
k
k
i
k η i − Ri
i
+ Rjml
and observe the first term is torsion because dRjml
kml ηj − Rjmk ηl − Rjkl ηm are
k
components of ∇R. Thus we are in the situation of the Frobenius theorem.
Setting z = xij ei ⊗η j ∈ so(V ), we rewrite the equation as
(15)
X ∇R + z.R = 0
where z. denotes the Lie algebra action.
6
JM LANDSBERG
Mike would like you to think of (15) as a condition on sections of the bundle T M ⊕Λ2 T ∗ M , or
since we are in the Riemannian case, T ∗ M ⊕ Λ2 T ∗ M . The punch line will be that the left hand
side of (15) may be interpreted as a covariant derivative with respect to a natural connection on
this bundle, which will be an example of a tractor bundle. (More precisely, the tractor bundle
is a filtered bundle whose associated graded bundle is T ∗ M ⊕ Λ2 T ∗ M .) Once you believe that,
Mike and Andi will try to convince you that this is a very special case of a much larger picture.
Regardless, we summarize:
Theorem 3.2.2. Let (M, g) be Riemannian, and let p ∈ M be a sufficiently general point. The
space of Killing vector fields is parametrized by the subspace of (X, z) ∈ Tp M ⊕ Λ2 Tp∗ M defined
by the equations
X ∇R + z.R = 0.
In particular, the dimension of the set of Killing vector fields on M is at most n + n2 .
Corollary 3.2.3. Let
(M, g) be Riemannian. The dimension of the set of Killing vector fields
n
on M equals n + 2 if and only if M has constant sectional curvature.
Proof of Corollary. If M has constant sectional curvature, then ∇R = 0 and R = sg ⊡ g, where
gp ⊡ gp ∈ S22 Tp∗ M is the Kulkarni-Nomizu product. Then z.R = s[(z.g) ⊡ g + g ⊡ (z.g)] = 0 as
so(V ) annihilates the metric. To see the other direction, if M does not have constant sectional
curvature, there will be some z ∈ so(V ) that does not annihilate R (because R does not take
value in a trivial so(V )-module), so the equations are non-trivial.
Example 3.2.4. Consider the special case of flat space. Fix constants xi , xij and coordinates
(y 1 , . . . , y n ) on En , such that (dy 1 , . . . , dy n ) is an orthonormal coframing. Then the general
Killing vector field is of the form
∂
X = (xi + y i xji ) j
∂y
which integrates to a translation plus a rotation.
Remark 3.2.5. In Mike’s lecture he will explain how the curvature of his natural connection on
the tractor bundle vanishes if and only if (M, g) is a space form.
Exercise 3.2.6: Let X ∈ K, show LX ◦ ∇ = 0.
3.3. The general case. One can similarly show that in general the dimension of the space of
Killing vector fields is finite dimensional if some prolongation of g ⊂ gl(V ) is zero (one says g is
(j)
of finite type), and when g is of finite type, it is of dimension at most dim(⊕∞
j=−1 g ) where the
sum is finite, and by convention g(−1) = V and g(0) = g.
3.4. Conformal Killing fields.
Proposition 3.4.1. If dim V ≥ 3, then co(V )(1) ≃ V and co(V )(2) = 0.
Proof. Fix g ∈ [g]. We have two so(V )-maps V → V ⊗V ∗ ⊗V ∗ , v 7→ v⊗g and v 7→ IdV ⊗v ♯ where
v ♯ (w) = g(v, w). The first clearly lands in V ⊗S 2 V ∗ and the second in so(V )⊗V ∗ .
Exercise 3.4.2: Prove that v 7→ v⊗g− 12 IdV ⊗v ♯ lies in co(V )(1) , independent of choice of g ∈ [g].
Prove moreover that V is the only intersection of the spaces, and prove that co(V )(2) = 0.
Proposition 3.4.3. On a conformally flat Rn , the general conformal Killing vector field is of
the form
1
X(x) = (si + mij xj + λxi + xi hr, xi − |x|2 r i )ei
2
PROLONGATION AND INFINITESIMAL SYMMETRIES
7
∂
where ei = ∂x
i is an orthonormal frame for some h , i ∈ [g], |x| is the norm with respect to that
inner-product, and si , mij = −mji , λ, r i are constants.
Proof. This is an immediate consequence of Proposition 3.4.1 (note that the term (xi hr, xi −
1
2 i
2 |x| r )ei is the image of x when v = r) and the fact that the EDS for such X is given by
∇X ∈ co(n).
The interesting part of these vector fields is the term coming from co(V )(1) , so consider a
vector field of the form
1
(16)
X(x) = (xi hr, xi − |x|2 r i )ei
2
for some r ∈ Rn .
Proposition 3.4.4. The flow of (16) φX (t) = (φ1 , . . . , φn ) is given by
φj =
(17)
xj − 2t r j |x|2
.
1 − thr, xi + 41 t2 |r|2 |x|2
∂
Exercise 3.4.5: Prove the identity es ∂t f (t) = f (t + s).
Proof. To prove 3.4.4 just differentiate at t = 0. However a direct proof is also instructive to
see what the flow is doing. The idea is to change coordinates via the the antipodal map of the
sphere, which on Rn \0 = S n \{p, q} is x 7→ y := |x|x 2 . Then
∂
1 ∂
2xj xk ∂
=
−
.
∂xj
|x|2 ∂y j
|x|4 ∂y k
so in these coordinates
∂
1
X(x) = (xi hr, xi − |x|2 r i ) i
2
∂x
1 ∂
2xj xk ∂
1
)
= r j (xi xj − |x|2 δij )( 2 j −
2
|x| ∂y
|x|4 ∂y k
xi xj ∂
xj xk ∂
1 ∂
xk xj ∂
−
2
−
+
]
|x|2 ∂y i
|x|2 ∂y k
2 ∂y j
|x|2 ∂y k
∂
1
= − rj j
2 ∂y
= rj [
Now we compute
x(t) = etX x
y
|y 2 |
y − 2t r
=
|y − 2t r|2
t j ∂
r
∂y j
= e2
=
So
xj (t) =
x
− 2t r
|x|2
.
| |x|x 2 − 2t r|2
xj − 2t r j |x|2
.
1 − thr, xi + 41 t2 |r|2 |x|2
8
JM LANDSBERG
Remark 3.4.6. From the proof we see that the map φ may be thought of as a composition of
the map x 7→ |x|x 2 , followed by a translation, followed by x 7→ |x|x 2 again. This suggests working
on the sphere S n instead of Rn , as there the map x 7→ |x|x 2 is simply the antipodal map. In
fact Cartan describes the general inversion this way, and a translation is just a special inversion
where the point at infinity is the fixed point of the inversion. In general, an inversion, viewed
from the conformal sphere, may be thought of as the flow with one fixed point, which, were we
to view that flow on Rn by stereographic projection from that fixed point, the flow would be a
translation.
When studying such maps we can set t = 1 and rescale r if necessary. From our discussion
one can recover the following theorem:
Theorem 3.4.7. (Liouville) Any diffeomorphism ψ : Rn → Rn with ψ(0) = 0 and dψ0 = IdT0 Rn
and such that ψ ∗ h , i = σ −2 h , i for some function σ is of the form (17) with t = 1.
4. Future work
In the next lectures we will see how to use Lie algebra cohomology for certain systems of finite
type. Recently there has been progress extending the use of Lie algebra cohomology to systems
were it was not naı̈vely applicable, e.g., [3, 1], and I expect there will be steady progress in this
direction. In another direction, if the system is not of finite type, but governed by Lie algebra
data (e.g. if the tableau is a Lie algebra g where g(k) 6= 0 for all k ≥ 0), similar methods should
be applicable.
References
1. Andreas Čap, Infinitesimal automorphisms and deformations of parabolic geometries, J. Eur. Math. Soc.
(JEMS) 10 (2008), no. 2, 415–437. MR 2390330 (2009e:32041)
2. Thomas A. Ivey and J. M. Landsberg, Cartan for beginners: differential geometry via moving frames and
exterior differential systems, Graduate Studies in Mathematics, vol. 61, American Mathematical Society, Providence, RI, 2003. MR 2 003 610
3. J. M. Landsberg and C. Robles, Fubini-griffiths-harris rigidity and lie algebra cohomology, Asian Math. J.
(2013).
E-mail address: jml@math.tamu.edu