Prolongation of (8, 15)-Distribution of Type ₄ by Singular Curves
Cartan gives the model of (8, 15)-distribution with the exceptional simple Lie algebra ₄ as its symmetry algebra in his paper (1893), which was published one year before his thesis. In the present paper, we study abnormal extremals (singular curves) of Cartan's model from the viewpoints of sub-...
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| description | Cartan gives the model of (8, 15)-distribution with the exceptional simple Lie algebra ₄ as its symmetry algebra in his paper (1893), which was published one year before his thesis. In the present paper, we study abnormal extremals (singular curves) of Cartan's model from the viewpoints of sub-Riemannian geometry and geometric control theory. Then we construct the prolongation of Cartan's model based on the data related to its singular curves, and obtain the nilpotent graded Lie algebra which is isomorphic to the negative part of the graded Lie algebra ₄.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 076, 20 pages
Prolongation of (8, 15)-Distribution of Type F4
by Singular Curves
Goo ISHIKAWA a and Yoshinori MACHIDA b
a) Department of Mathematics, Hokkaido University, Kita 10 Nishi 8, Kita-ku,
Sapporo 060-0810, Japan
E-mail: ishikawa@math.sci.hokudai.ac.jp
b) Department of Mathematics, Faculty of Science, Shizuoka University,
836, Ohya, Suruga-ku, Shizuoka 422-8529, Japan
E-mail: machida.yoshinori@shizuoka.ac.jp, yomachi212@gmail.com
Received January 30, 2025, in final form September 12, 2025; Published online September 18, 2025
https://doi.org/10.3842/SIGMA.2025.076
Abstract. Cartan gives the model of (8, 15)-distribution with the exceptional simple Lie
algebra F4 as its symmetry algebra in his paper (1893), which is published one year before
his thesis. In the present paper, we study abnormal extremals (singular curves) of Cartan’s
model from viewpoints of sub-Riemannian geometry and geometric control theory. Then
we construct the prolongation of Cartan’s model based on the data related to its singular
curves, and obtain the nilpotent graded Lie algebra which is isomorphic to the negative part
of the graded Lie algebra F4.
Key words: exceptional Lie algebra; singular curve; constrained Hamiltonian equation
2020 Mathematics Subject Classification: 53C17; 58A30; 17B25; 34H05; 37J37; 49K15;
53D25
1 Introduction
Let M be a manifold of dimension 15 and D ⊂ TM a distribution, i.e., a vector subbundle of the
tangent bundle TM of rank 8. Then D is called an (8, 15)-distribution if D + [D,D] = TM for
the sheave D (resp. TM) of local sections to D (resp. TM). In this paper, we study a special
class of (8, 15)-distributions related to the simple Lie group F4.
Distributions are important subjects in manifold theory and global analysis. They are studied
also related to the theory of Lie groups, Lie algebras and their representations. Then the theory
of prolongations and equivalence problems of distributions are established by many authors
(see, for instance, [10, 34, 35, 38]). On symmetries for distributions, there are well-known several
powerful and beautiful methods to investigate, based on differential geometry and representation
theory; Cartan’s prolongation, Tanaka’s prolongation, and Kostant’s theorem on Bott–Borel–
Weil theory and so on [14, 21, 23, 31, 30, 38, 39, 40].
We provide, in this paper, a way of prolongations of (8, 15)-distributions of type F4 via the
notion of abnormal extremals or singular curves and related objects from viewpoints of sub-
Riemannian geometry and geometric control theory [3, 32, 33, 34] which recovers several results
explicitly. The relations of our constructions with those by the method of representation theory
are presented in Remark 4.4 of Section 4 in our paper.
The singular curves or abnormal extremals are extensively used to study distributions by
many authors (see, for instance, [4, 11, 16, 17]). In the previous papers (see [24, 25, 27]),
we study (2, 3, 5)-distributions or Cartan distributions [9, 14] using singular curves. Here
a (2, 3, 5)-distribution means a distribution D of rank 2 on a 5-dimensional manifold M such
that D(2) := D + [D,D] becomes the sheaf of local sections of a distribution D(2) of rank 3
mailto:ishikawa@math.sci.hokudai.ac.jp
mailto:machida.yoshinori@shizuoka.ac.jp
mailto:yomachi212@gmail.com
https://doi.org/10.3842/SIGMA.2025.076
2 G. Ishikawa and Y. Machida
and that TM = D(3) := D(2) +
[
D,D(2)
]
. Then we show the prolongation using the cone of
singular curves of any (2, 3, 5)-distribution has the nilpotent gradation algebra which is isomor-
phic to the negative part of the graded simple Lie algebra G2. Note that the prolongation
procedure is a partial case of twistor construction in the general framework of parabolic geom-
etry [6, 12].
In his book [34] on sub-Riemannian geometry, Montgomery gives expositions on (4, 7)-
distributions. In particular, Montgomery classifies (4, 7)-distributions into elliptic, hyperbolic
and parabolic (4, 7)-distributions and shows the non-existence of non-trivial singular curves
for elliptic (4, 7)-distribution. Moreover, he develops Cartan’s approach for (4, 7)-distributions
and studies their symmetry groups. In the previous paper [26], we study hyperbolic (4, 7)-
distributions and their prolongations via the cone of singular curves. Then we observe, contrary
to the case of (2, 3, 5)-distributions, the isomorphism classes of the nilpotent graded Lie algebra
of prolongations are never unique and then we specifies the class of C3-(4, 7)-distributions by
the condition that the graded algebra associated to the (4, 7)-distribution after prolongation is
isomorphic to the negative part of the simple Lie algebra C3.
Cartan, in his paper [14] which is published one year before his thesis [13], gives the model
of (8, 15)-distribution whose infinitesimal symmetry algebra is the simple Lie algebra F4. The
purpose of the present paper is to study Cartan’s model of (8, 15)-distribution from viewpoints
of sub-Riemannian geometry and geometric control theory. We construct its prolongation us-
ing the data related to abnormal or singular curves, and verify that the prolonged nilpotent
graded algebra obtained by our method is isomorphic to the negative part of the simple Lie
algebra F4.
Note that the complex simple Lie algebra F4 has three real forms; one compact type and
two non-compact types denoted as F4(4) and as F4(−20) (see [14, 13, 15, 18, 29]). In [21], F4(4)
(resp. F4(−20)) is denoted by F4I (resp. F4II), and in [20], by F̃4 (resp. F ′
4). Cartan’s model,
which we treat in the present paper, gives the (8, 15)-distribution corresponding to F4(4), which
maybe called the “hyperbolic” F4-(8, 15)-distribution. Nurowski [36] has given the explicit
models of (8, 15)-distributions of type F4 and (16, 24)-distributions of type E6. Though we
do not touch the details here, it can be observed that the real (8, 15)-distribution of type F4(−20)
in Nurowski’s normal form has the canonical definite conformal metric and it has no nontrivial
singular curves (cf. Sections 3 and 4 of this paper). Thus Nurowski’s (8, 15)-distribution of
type F4(−20) can be called “elliptic” F4-(8, 15)-distribution. Refer [36] also for related references
and historical remarks. Note also that both (8, 15)-distributions of type F4(4) and F4(−20) appear,
as two cases of real simple Lie algebras, in the classification of certain sub-Riemannian structures
in [5, 19].
In Section 2, we recall Cartan’s model
(
K15, D
)
of (8, 15)-distribution associated to the simple
Lie algebra F4. The basics on sub-Riemannian geometry and geometric control theory which
we need in this paper are given in Section 3. We study the singular curves of Cartan’s model
and show that there exist canonically the conformal metrics on D ⊂ TK15 and on D⊥ ⊂ T ∗K15
in Section 4. In Section 5, we construct null-flag manifold of dimension 9 which prolongs
(
K15, D
)
to (W,E) so that dim(W ) = 24 and E is of rank 4. In Section 6, we show that E has the small
growth vector (4, 7, 10, 13, 16, 18, 20, 21, 22, 23, 24) and the gradation algebra of E is isomorphic
to the negative part of the simple graded Lie algebra with respect to filtration defined by the
set of all roots of F4. In Section 7, we introduce the class of (8, 15)-distributions of type F4
regarding the arguments of previous sections and show that also the gradation algebra of the
prolongation of any (8, 15)-distributions of type F4 is isomorphic to the negative part of the
simple Lie algebra F4 with respect to the filtration defined by the set of all roots of F4 (Theo-
rem 7.3).
In this paper, all manifolds and maps are supposed to be of class C∞ unless otherwise
stated.
Prolongation of (8, 15)-Distribution of Type F4 by Singular Curves 3
2 Cartan’s model of (8, 15)-distributions of type F4
We recall Cartan’s model of (8, 15)-distribution [14, 40] which has, as the infinitesimal symme-
tries, the simple Lie algebra F4:
The Dynkin diagram of F4.
As for the exceptional Lie algebra F4, see, for instance, also [1, 2, 8, 22, 37].
The model of (8, 15)-distributions found by Cartan is derived from the homogeneous space
by the parabolic subgroup of the simple Lie group F4 which corresponds to {α4} for the simple
roots α1, α2, α3, α4 [14, 40]:
Here we have simply marked the corresponding root in black to the parabolic subgroup, which
not meant, say, the Satake diagram. Note that, as the standard way, a cross under the node can
be used to indicate a parabolic subgroup as in [6].
Let K = R or C. On K15 with the system of coordinates z, x1, x2, x3, x4, y1, y2, y3, y4, xij ,
1 ≤ i < j ≤ 4, and consider the C∞ (resp. holomorphic) 1-forms
ω = dz − y1dx1 − y2dx2 − y3dx3 − y4dx4,
ωij = dxij − (xidxj − xjdxi + yhdyk − ykdyh), 1 ≤ i < j ≤ 4,
where (i, j, h, k) is an even permutation of (1, 2, 3, 4). Let
Z, X12, X13, X14, X23, X24, X34, X1, X2, X3, X4, Y1, Y2, Y3, Y4
be the dual frame of TK15 to the frame
ω, ω12, ω13, ω14, ω23, ω24, ω34, dx1, dx2, dx3, dx4, dy1, dy2, dy3, dy4
of T ∗K15. Then D ⊂ TK15 is defined as the distribution generated by X1, X2, X3, X4, Y1, Y2,
Y3, Y4. Explicitly the distribution D ⊂ TK15 has the system of generators
X1 =
∂
∂x1
+ y1
∂
∂z
− x2
∂
∂x12
− x3
∂
∂x13
− x4
∂
∂x14
,
X2 =
∂
∂x2
+ y2
∂
∂z
+ x1
∂
∂x12
− x3
∂
∂x23
− x4
∂
∂x24
,
X3 =
∂
∂x3
+ y3
∂
∂z
+ x1
∂
∂x13
+ x2
∂
∂x23
− x4
∂
∂x34
,
X4 =
∂
∂x4
+ y4
∂
∂z
+ x1
∂
∂x14
+ x2
∂
∂x24
+ x3
∂
∂x34
,
Y1 =
∂
∂y1
− y4
∂
∂x23
+ y3
∂
∂x24
− y2
∂
∂x34
,
Y2 =
∂
∂y2
+ y4
∂
∂x13
− y3
∂
∂x14
+ y1
∂
∂x34
,
Y3 =
∂
∂y3
− y4
∂
∂x12
+ y2
∂
∂x14
− y1
∂
∂x24
,
Y4 =
∂
∂y4
+ y3
∂
∂x12
− y2
∂
∂x13
+ y1
∂
∂x23
.
4 G. Ishikawa and Y. Machida
Moreover, we have that Z = ∂
∂z and Xij =
∂
∂xij
, 1 ≤ i < j ≤ 4. Then we get the following
bracket relations:
[X1, X2] = 2X12, [X1, X3] = 2X13, [X1, X4] = 2X14,
[X2, X3] = 2X23, [X2, X4] = 2X24,
[X3, X4] = 2X34,
[Y1, Y2] = 2X34, [Y1, Y3] = −2X24, [Y1, Y4] = 2X23,
[Y2, Y3] = 2X14, [Y2, Y4] = −2X13,
[Y3, Y4] = 2X12,
[Y1, X1] = [Y2, X2] = [Y3, X3] = [Y4, X4] = Z, [Yi, Xj ] = 0, i ̸= j.
Moreover, we have
[Xi, Xjk] = 0, [Yi, Xjk] = 0, [Xi, Z] = 0, [Yi, Z] = 0 for any i, j, k.
Remark 2.1. We set, for 1 ≤ i < j ≤ 4, a sub-distribution Dij = ⟨Xi, Xj , Yh, Yk, Xij⟩ of D(2),
where (i, j, h, k) is a permutation of (1, 2, 3, 4). Then we see each Dij is completely integrable
and each leaf of the foliation induced by Dij of K15 has a contact structure. Thus we have six
contact foliations in K15. For example, for i = 1, j = 2, then the contact foliation is given by
the Pfaff system
dz − y1dx1 − y2dx2 = 0, dx3 = 0, dx4 = 0, dy1 = 0, dy2 = 0,
dx13 + x3dx1 + y2dy4 = 0, dx14 + x4dx1 − y2dy3 = 0, dx23 + x3dx2 + y1dy4 = 0,
dx24 + x4dx2 + y1dy3 = 0, dx34 = 0,
and with the 1-form
dx12 + x2dx1 − x1dx2 + y4dy3 − y3dy4,
which gives a contact structure on each leaf of the foliation defined by D12.
3 Abnormal bi-extremals and singular curves of distributions
Here we recall several notions in geometric control theory and sub-Riemannian geometry. For
details, consult, for instance, the references [3, 7, 33, 34].
Let M be a real C∞ manifold, D ⊂ TM a distribution endowed with a positive definite
metric g : D ⊗D → R on a manifold M , and γ : [a, b]→M an absolutely continuous curve sat-
isfying γ̇(t) ∈ Dγ(t) for almost all t ∈ I, which is called a D-integral curve. Then the arc-length
of γ is defined by L(γ) :=
∫ b
a
√
g(γ̇(t), γ̇(t))dt. A curve γ is called a D-geodesic if it minimises
the arc-length locally.
Let rank(D) = r and, just for simplicity, ξ1, . . . , ξr be an orthonormal frame of (D, g) over M .
Then we define F : D ∼= M × Rr → TM by F (x, u) =
∑r
i=1 uiξi(x).
Consider the optimal control problem for the energy function on D defined by
e =
1
2
g
(
r∑
i=1
uiξi(x),
r∑
i=1
uiξi(x)
)
=
1
2
r∑
i=1
ui(t)
2.
Note that the problem of minimising arc-length and that of minimising energy function are
known to be equivalent up to re-parametrisations [3, 34]. Then the Hamiltonian function
on (D ×M T ∗M)× R is given by
H(x, p, u, p0) =
〈
p,
r∑
i=1
uiξi(x)
〉
+ p0
(
1
2
r∑
i=1
u2i
)
.
Prolongation of (8, 15)-Distribution of Type F4 by Singular Curves 5
Here D ×M T ∗M = {(x, u), (x′, p) ∈ D × T ∗M | x = x′} ∼= T ∗M × Rr and p0 is an additional
parameter.
Regarding the optimal control problem for minimising the energy function of D-integrable
curves, we have, by Pontryagin’s maximum principle, if γ is a D-geodesic, then, for γ̇(t) =
(x(t), u(t)), there exists a Lipschitz curve (x(t), p(t)) ∈ T ∗M and non-positive constant p0 ≤ 0
such that the following constrained Hamiltonian equation in terms of H = H
(
x, p, u, p0
)
is
satisfied:
ẋi(t) =
∂H
∂pi
(
x(t), p(t), u(t), p0
)
, 1 ≤ i ≤ m,
ṗi(t) = −
∂H
∂xi
(
x(t), p(t), u(t), p0
)
, 1 ≤ i ≤ m,
with constraints ∂H
∂uj
(
x(t), p(t), u(t), p0
)
= 0, 1 ≤ j ≤ r,
(
p(t), p0
)
̸= 0.
If p0 < 0, then the curve (x(t), p(t)) (resp. x(t)) of a solution of the above constrained Hamilto-
nian equation is called a normal bi-extremal (resp. normal extremal) respectively. If p0 = 0, then
bi-extremals and extremals are called abnormal. Note that the notion of abnormal (bi-)extremals
is independent of the metric g on D and depends only on the distribution D.
The constraint ∂H
∂uj
= 0 is equivalent to that p0uj = −⟨p, ξj(x)⟩. In the normal case, i.e.,
p0 < 0, we have uj = − 1
p0
⟨p, ξj(x)⟩. Because the Hamiltonian is linear on
(
p, p0
)
, by normalising
as p0 = −1, we have H = 1
2
∑r
i=1⟨p, ξi(x)⟩2.
For abnormal extremals, the constrained Hamiltonian equation reads as
ẋ = u1ξ1(x) + u2ξ2(x) + · · ·+ urξi(x),
ṗ = −
(
u1
∂Hξ1
∂x
+ u2
∂Hξ2
∂x
+ · · ·+ ur
∂Hξr
∂x
)
,
with constraints Hξ1 = 0,Hξ2 = 0, . . . , Hξr = 0 and p ̸= 0, where Hξi(x, p) := ⟨p, ξi(x)⟩.
Given a distribution D ⊂ TM , for any x ∈M , we define the subbundle D⊥ ⊂ T ∗M by
D⊥
x := {α ∈ T ∗
xM | ⟨α, v⟩ = 0, for any v ∈ Dx}.
Then the above constraints mean that p(t) ∈ D⊥
x(t).
The notion of abnormal extremals coincides with that of singular curves, i.e., critical points
of the end-point mapping [33, 34]. Let x0 ∈ M and I = [a, b] an interval. Let Ω be the
set of Lipschitz continuous curves γ : I → M with γ̇(t) ∈ Dγ(t) for almost all t ∈ I, which is
called a D-integral curve, and γ(a) = x0. Then the endpoint mapping End: Ω→M is defined by
End(γ) := γ(b). A curve γ ∈ Ω is called a D-singular curve if γ is a critical point of the endpoint
mapping, i.e., the differential map dγEnd: TγΩ → Tγ(b)M is not surjective, for an appropriate
manifold structure of Ω (and M).
We introduce the key notion of the present paper.
Definition 3.1. We define the singular velocity cone SVC(D) ⊂ TM of a distribution D ⊂ TM
by the set of tangent vectors v ∈ TxM , x ∈ M such that there exists a D-singular curve
γ : (R, 0)→ (M,x) with γ′(0) = v.
Note that SVC(D) is a cone field over M , i.e., SVC(D) is invariant under the fibrewise
R×-multiplication on TM .
The following lemma is used in the following sections. We have given a proof using coordinates
to make sure.
Lemma 3.2 ([3] and [7, Section 4.2]). For a distribution D generated by ξ1, . . . , ξr, we have,
along abnormal bi-extremals (x(t), p(t)) and corresponding u(t), that
d
dt
Hξi(t) =
r∑
j=1
uj(t)H[ξi,ξj ](t), 1 ≤ i ≤ r.
6 G. Ishikawa and Y. Machida
Proof. We put p =
∑r
j=1 pjdxj and ξi =
∑r
k=1 ξik
∂
∂xk
. Then H(x, p, u) =
∑
1≤i,j≤r uipjξij(x)
and Hξi =
∑r
j=1 pjξij(x). By the Hamiltonian equation, for 1 ≤ i ≤ r, we have
d
dt
Hξi(t) =
r∑
j=1
(
p′jξij + pjξ
′
ij
)
=
r∑
j=1
(
p′jξij +
r∑
ℓ=1
pj
∂ξij
∂xℓ
x′ℓ
)
=
r∑
j=1
(
−∂H
∂xj
ξij +
r∑
ℓ=1
pj
∂ξij
∂xℓ
∂H
∂pℓ
)
= −
∑
kℓj
ukpℓ
∂ξkℓ
∂xj
ξij +
∑
jℓk
pj
∂ξij
∂xℓ
ukξkℓ
= −
∑
kℓj
ukpℓ
∂ξkℓ
∂xj
ξij +
∑
ℓjk
pℓ
∂ξiℓ
∂xj
ukξkj =
∑
kℓ
ukpℓ
(
r∑
j=1
(
ξij
∂ξkℓ
∂xj
− ξkj
∂ξiℓ
∂xj
))
=
r∑
k=1
uk⟨p, [ξi, ξk]⟩ =
r∑
j=1
ujH[ξi,ξj ]. ■
Remark 3.3. We have defined the notion of abnormal (bi-)extremals and singular curves over
the real. In the complex analytic case K = C, we can (and do) define abnormal (bi-)extremals
and singular curves, forgetting about end-point mapping, just by the complex analytic con-
strained Hamiltonian equation for a complex analytic distribution D ⊂ TM over a complex
analytic manifold M , which is defined similarly as explained in this section.
4 Conformal metric on Cartan’s (8, 15)-distribution
and singular velocity cone
Let us determine the singular curves of Cartan’s model
(
K15, D
)
explained in Section 2. On the
cotangent bundle T ∗K15 with base coordinates z, x1, x2, x3, x4, y1, y2, y3, y4, xij , 1 ≤ i < j ≤ 4
and fiber coordinates s, p1, p2, p3, p4, q1, q2, q3, q4, rij , 1 ≤ i < j ≤ 4, we have the Hamiltonian
of the distribution D ⊂ TX,
H = u1HX1 + u2HX2 + u3HX3 + u4HX4 + v1HY1 + v2HY2 + v3HY3 + v4HY4 ,
where
HX1 = p1 + y1s− x2r12 − x3r13 − x4r14, HX2 = p2 + y2s+ x1r12 − x3r23 − x4r24,
HX3 = p3 + y3s+ x1r13 + x2r23 − x4r34, HX4 = p4 + y4s+ x1r14 + x2r24 + x3r34,
HY1 = q1 − y4r23 + y3r24 − y2r34, HY2 = q2 + y4r13 − y3r14 − y1r34,
HY3 = q3 − y4r12 + y2r14 − y1r24, HY4 = q4 + y3r12 − y2r13 + y1r23.
The constrained Hamiltonian equation is given by
ż = u1y1 + u2y2 + u3y3 + u4u4,
ẋ1 = u1, ẋ2 = u2, ẋ3 = u3, ẋ4 = u4, ẏ1 = v1, ẏ2 = v2, ẏ3 = v3, ẏ4 = v4,
ẋ12 = −x2u1 + x1u2 − y4v3 + y3v4, ẋ13 = −x3u1 + x1u3 + y4v2 − y2v4,
ẋ14 = −x4u1 + x1u4 − y3v2 + y2v3, ẋ23 = −x3u2 + x2u3 − y4v1 + y1v4,
ẋ24 = −x4u2 + x2u4 + y3v1 − y1v3, ẋ34 = −x4u3 + x3u4 − y2v1 + y1v2,
ṡ = 0, . . . . . .
ṗ1 = −u2r12 − u3r13 − u4r14, ṗ2 = u1r12 − u3r23 − u4r24,
ṗ3 = u1r13 + u2r23 − u4r34, ṗ4 = u1r14 + u2r24 + u3r34,
q̇1 = −u1s− v2r34 + v3r24 − v4r23, q̇2 = −u2s+ v1r34 − v3r14 + v4r13,
q̇3 = −u3s− v1r24 + v2r14 − v4r12, q̇2 = −u4s+ v1r23 − v2r13 + v3r12,
ṙ12 = 0, ṙ13 = 0, ṙ14 = 0, ṙ23 = 0, ṙ24 = 0, ṙ34 = 0,
(4.1)
Prolongation of (8, 15)-Distribution of Type F4 by Singular Curves 7
with constraints
HX1 = 0, HX2 = 0, HX3 = 0, HX4 = 0,
HY1 = 0, HY2 = 0, HY3 = 0, HY4 = 0,
and s(t), p1(t), p2(t), p3(t), p4(t), q1(t), q2(t), q3(t), q4(t), rij(t) are not all zero for any t.
By the constraints, if s, rij are all zero, then pi, qj , 1 ≤ i, j ≤ 4 are also zero. So s, rij ,
1 ≤ i < j ≤ 4 must be not all zero.
Remark 4.1. In Cartan’s model, we have that s and rij are locally constant by the Hamiltonian
equation. However, we do not use this property in the following arguments.
For instance, from the constraint HX1 = 0, we have, along any solution curve by Lemma 3.2,
that
0 =
d
dt
HX1 =
4∑
i=1
uiH[X1,Xi] +
4∑
j=1
vjH[X1,Yj ].
Then similarly from the constraint, we have the following equality in a general form:
0 H[X1,X2] H[X1,X3] H[X1,X4] H[X1,Y1] H[X1,Y2] H[X1,Y3] H[X1,Y4]
H[X2,X1] 0 H[X2,X3] H[X2,X4] H[X2,Y1] H[X2,Y2] H[X2,Y3] H[X2,Y4]
H[X3,X1] H[X3,X2] 0 H[X3X4] H[X3,Y1] H[X3,Y2] H[X3,Y3] H[X3,Y4]
H[X4,X1] H[X4,X2] H[X4X3] 0 H[X4,Y1] H[X4,Y2] H[X4,Y3] H[X4,Y4]
H[Y1,X1] H[Y1,X2] H[Y1,X3] H[Y1,X4] 0 H[Y1,Y2] H[Y1,Y3] H[Y1,Y4]
H[Y2,X1] H[Y2,X2] H[Y2,X3] H[Y2,X4] H[Y2,Y1] 0 H[Y2,Y3] H[Y2,Y4]
H[Y3,X1] H[Y3,X2] H[Y3,X3] H[Y3,X4] H[Y3,Y1] H[Y3,Y2] 0 H[Y3,Y4]
H[Y4,X1] H[Y4,X2] H[Y4,X3] H[Y4,X4] H[Y4,Y1] H[Y4,Y2] H[Y4,Y3] 0
u1
u2
u3
u4
v1
v2
v3
v4
=
0
0
0
0
0
0
0
0
.
Explicitly, we have in fact
0 2r12 2r13 2r14 −s 0 0 0
−2r12 0 2r23 2r24 0 −s 0 0
−2r13 −2r23 0 2r34 0 0 −s 0
−2r14 −2r24 −2r34 0 0 0 0 −s
s 0 0 0 0 2r34 −2r24 2r23
0 s 0 0 −2r34 0 2r14 −2r13
0 0 s 0 2r24 −2r14 0 2r12
0 0 0 s −2r23 2r13 −2r12 0
u1
u2
u3
u4
v1
v2
v3
v4
=
0
0
0
0
0
0
0
0
. (4.2)
Equivalently, we have
−v1 2u2 2u3 2u4 0 0 0
−v2 −2u1 0 0 2u3 2u4 0
−v3 0 −2u1 0 −2u2 0 2u4
−v4 0 0 −2u1 0 −2u2 −2u3
u1 0 0 0 2v4 −2v3 2v2
u2 0 −2v4 2v3 0 0 −2v1
u3 2v4 0 −2v2 0 2v1 0
u4 −2v3 2v2 0 −2v1 0 0
s
r12
r13
r14
r23
r24
r34
=
0
0
0
0
0
0
0
0
. (4.3)
Write (4.2) as(
A11 −sI
sI A22
)(
u
v
)
=
(
0
0
)
,
8 G. Ishikawa and Y. Machida
where u = t(u1, u2, u3, u4), v = t(v1, v2, v3, v4) and I is the 4 × 4 unit matrix. We denote by A
the skew-symmetric 8× 8 matrix
(
A11 −sI
sI A22
)
and by U the 8× 7 matrix which appeared in (4.2)
and (4.3), respectively.
Then the condition (4.2) is equivalent to that A11u = sv, A22v = −su. Note that det(A11) =
det(A22) = {4(r12r34 − r13r24 + r14r23)}2 and that A11A22 = A22A11 = −4(r12r34 − r13r24 +
r14r23)I. Then the condition (4.2) implies that{
s2 − 4(r12r34 − r13r24 + r14r23)
}
u = 0,
{
s2 − 4(r12r34 − r13r24 + r14r23)
}
v = 0.
Therefore, if (u, v) ̸= (0, 0), then we have
s2 − 4(r12r34 − r13r24 + r14r23) = 0.
Suppose s ̸= 0. Then, since A11 is skew-symmetric, we have that tu · v = 1
s
tu · (A11u) =
1
s (
tuA11) · u = 1
s
t(tA11u)u = −1
s
t(A11u)u = −tv · u = −tu · v. Therefore, we have that
tu · v = u1v1 + u2v2 + u3v3 + u4v4 = 0.
Suppose s = 0. Then A11u = 0 and A22v = 0. Note that A11A22 = A22A11 = 0. Since A11
and A22 are non-zero and skew-symmetric, we have rank(A11) = 2, rank(A22) = 2, and therefore
Ker(A11) = Im(A22) and Im(A11) = Ker(A22). Then we have u = A22ũ and v = A11ṽ for
some ũ, ṽ, and thus tu · v = t(A22ũ) ·A11ṽ = tũ tA22A11ṽ = −tũA22A11ṽ = 0.
Proposition 4.2. The singular velocity cone SVC(D) of Cartan’s model D is given by
SVC(D) =
{
4∑
i=1
uiXi +
∑
j=1
vjYj
∣∣∣∣∣u1v1 + u2v2 + u3v3 + u4v4 = 0
}
.
Proof. That SVC(D) is contained in the right hand side is already shown. Let us show the
converse inclusion. All columns of the 8 × 7 matrix U which appeared in (4.3) are null and
orthogonal to each other with respect to the metric tu · v = u1v1 + u2v2 + u3v3 + u4v4 on K8.
Note that the metric is non-degenerate for K = C and is of signature (4, 4) if K = R. In any case
we have that rank(U) ≤ 4 < 7, because the subspace generated by all columns of U is a null
space in K8 with respect to the metric tu · v. Hence, for any non-zero constant vector (u, v)
with tu · v = 0, there exists (s, rij) ̸= 0 such that (4.3) holds, and therefore that (4.2) holds.
Thus we see that, given non-zero (u, v) with tu · v = 0, there exist constants s, pi, 1 ≤ i ≤ 4, qj ,
1 ≤ j ≤ 4, rij , 1 ≤ i < j ≤ 4, which are not all zero, and functions xi, 1 ≤ i ≤ 4, yj , 1 ≤ j ≤ 4,
xij , 1 ≤ i < j ≤ 4 such that the linear ordinary differential equation (4.1) for singular curves is
satisfied. Thus we see the required equality. ■
We define a quadratic form Q on K8 and R on K7, respectively, by
Q(u, v) := u1v1 + u2v2 + u3v3 + u4v4, R(s, rij) := s2 − 4(r12r34 − r13r24 + r14r23).
The quadratic form Q induces the bi-linear form(
(u, v),
(
u′, v′
))
=
1
2
(
u1v
′
1 + v1u
′
1 + u2v
′
2 + v2u
′
2 + u3v
′
3 + v3u
′
3 + u4v
′
4 + v4u
′
4
)
on K8 ×K8. Moreover, the quadratic form R induces the bilinear form((
s, rij
)
|
(
s′, r′ij
))
= ss′ − 2
(
r12r
′
34 + r34r
′
12 − r13r
′
24 + r24r
′
13 + r14r
′
23 + r23r
′
14
)
on K7 ×K7.
Prolongation of (8, 15)-Distribution of Type F4 by Singular Curves 9
Corollary 4.3. The distribution D ⊂ TK15 has the canonical non-degenerate metric ( , ) for
K = C and the canonical conformal (4, 4)-metric ( , ) for K = R. The distribution D⊥ ⊂ T ∗K15
has the canonical non-degenerate metric ( | ) for K = C and the canonical conformal (4, 3)-
metric ( | ) for K = R.
Remark 4.4. Let G = F4(4), P = Pα4 , the parabolic subgroup of F4(4) corresponding to the
root α4, X = G/Pα4 = O′P 2
0 , that is the hyperplane section of the split Cayley projective
space O′P 2 and H = Spin(4, 3). Then we have the decomposition TG = T1 ⊕ T2 into H-
modules, where T1 (resp. T2) is regarded as the 8-dimensional spin representation of Spin(4, 3);
T1
∼= O′, (resp. the 7-dimensional vector representation; T2
∼= ImO′). Moreover, the closed H-
orbit Y1 ⊂ P(T1) (resp. Y2 ⊂ P(T2)) is a 6-dimensional quadric (resp. is a 5-dimensional quadric)
with a conformal structure of type (3, 3) (resp. of type (3, 2)) (see [31, Section 6.3]). See also [31,
Section 2] and [6, 28] for general constructions in simple Lie algebras.
Consider the Clifford algebra Cl(4, 3) ⊃ T1. Let N be the totality of 3-dimensional null
subspaces in T1. We set Ns := {z ∈ T2 | z(s) = 0} for s ∈ T1. If Ns ∈ N , s is called a pure
spinor. Denote by PS(4, 3) the set of pure spinors and by P(PS(4, 3)) its projectivisation. Then
the correspondence [s] ∈ P(PS(4, 3)) 7→ Ns ∈ N turns to be an isomorphism. See, for instance,
[20, p. 241 and p. 283].
Now in the left hand side of the equality (4.2) in our argument in this section, the action
u = t(u, v) 7→ Au corresponds to the spinor representation of T2 ⊂ Cl(4, 3) to T1. Moreover, we
see that the set D of solutions u to the equation Au = 0 is exactly equal to the set PS(4, 3) of
pure spinors. Thus we see that D = T1 and that SVC(D) ∼= Ŷ1 ∼= PS(4, 3). Therefore, invariant
cone Ŷ1 is constructed from D = T1 algebraically from the viewpoint of representation theory.
Further D⊥ = (TX/T1)
∗ = T ∗
2 (⊂ T ∗X) has the H-invariant (4, 3)-metric. In this paper, we
have characterised these objects known in representation theory by using singular curves from
the viewpoint of geometric control theory.
5 Null flags associated to abnormal bi-extremals
We continue to analyse the equation (4.3) appeared in the previous section. Recall the 8 × 7
matrix U which appeared in (4.3). Write U =
(
U ′
U ′′
)
using 4× 7 matrices U ′, U ′′. Then we have
that
tUU =
(
tU ′′ tU ′)(U ′
U ′′
)
=
−2Q 0 0 0 0 0 0
0 0 0 0 0 0 4Q
0 0 0 0 0 −4Q 0
0 0 0 0 4Q 0 0
0 0 0 4Q 0 0 0
0 0 −4Q 0 0 0 0
0 4Q 0 0 0 0 0
,
where Q = u1v1 + u2v2 + u3v3 + u4v4. Note that det
(
tUU
)
= 213Q8.
If Q ̸= 0, then rank(U) = 7. If Q = 0, then, since tUU = O, regarding U : K7 → K8 and
tU : K8 → K7, we have that Im(U) ⊆ Ker
(
tU
)
, so that rank(U) ≤ 8 − rank(U). Thus we have
rank(U) ≤ 4 again. Moreover, if we set R = s2 − 4(r12r34 − r13r24 + r14r23), then we have that
if (u, v) ̸= (0, 0) and Q = 0, then Ker(U) ⊆ R−1(0). So we have Ker(U) is a null subspace for
the non-degenerate metric R′ induced by the quadratic form R and that dimKer(U) ≤ 3. Thus
we have, in fact, rank(U) = 4 and dimKer(U) = 3, if (u, v) ̸= (0, 0) and Q = 0. Therefore, we
observe that, for any (null) line in Q−1(0), there corresponds a null 3-pace in R−1(0). Conversely,
for any null 3-space in R−1(0), there corresponds a null line in Q−1(0). However, for any null
10 G. Ishikawa and Y. Machida
line in R−1(0), naturally there corresponds, not a null 3-space, but a null 4-space in Q−1(0) by
the equation (4.2), since, on R−1(0) \ {0}, we see det(A11) ̸= 0 and the matrix A is of rank 4.
In fact we have
Lemma 5.1. To any null-flag Λ1 ⊂ Λ2 ⊂ Λ3 ⊂ R−1(0) for R = s2− 4(r12r34− r13r24 + r14r23),
where dim(Λi) = i, i = 1, 2, 3, there corresponds uniquely, by the equation (4.2), a null-flag
V1 ⊂ V2 ⊂ V4 ⊂ Q−1(0) for Q = u1v1 + u2v2 + u3v3 + u4v4, where dim(Vk) = k, k = 1, 2, 4.
Proof. The conformal orthogonal group CO(R) of the quadratic form R acts transitively on the
null Grassmannian {(Λ1,Λ2,Λ3)} on the metric space D⊥
m
∼= R4,3, m ∈ K15 defined by R. We
take the basis of D⊥
m: ε1 =
∂
∂s , ε2 =
∂
∂r12
, ε3 =
∂
∂r13
, ε4 =
∂
∂r14
, ε5 =
∂
∂r23
, ε6 =
∂
∂r24
, ε7 =
∂
∂r34
.
Then the representation matrix of the (4, 3)-metric on R becomes
1 0 0 0 0 0 0
0 0 0 0 0 0 −2
0 0 0 0 0 2 0
0 0 0 0 −2 0 0
0 0 0 −2 0 0 0
0 0 2 0 0 0 0
0 −2 0 0 0 0 0
Then we set the base point
(
Λ0
1,Λ
0
2,Λ
0
3
)
of the null flag manifold F ′, where
Λ0
1 = ⟨ε2⟩, Λ0
2 = ⟨ε2, ε3⟩, Λ0
3 = ⟨ε2, ε3, ε4⟩,
We take the frame
f1 = z11ε1 + ε2 + z13ε3 + z14ε4 + z15ε5 + z16ε6 + z17ε7,
f2 = z21ε1 + ε3 + z24ε4 + z25ε5 + z26ε6 + z27ε7,
f3 = z31ε1 + ε4 + z35ε5 + z36ε6 + z37ε7,
associated to a (not necessarily null) flag (Λ1,Λ2,Λ3) with Λ1 = ⟨f1⟩, Λ2 = ⟨f1, f2⟩ and Λ3 =
⟨f1, f2, f3⟩ in a neighbourhood of the base point
(
Λ0
1,Λ
0
2,Λ
0
3
)
.
Then the condition that (Λ1,Λ2,Λ3) is a null flag is equivalent to that
(f1|f1) = z11 − 4z17 + 4z13z16 − 4z14z15 = 0,
(f1|f2) = z11z21 − 2z27 + 2z13z26 − 2z14z25 − 2z15z24 + 2z16 = 0,
(f1|f3) = z11z31 − 2z37 + 2z13z36 − 2z14z35 − 2z15 = 0,
(f2|f2) = z221 + 4z26 − 4z24z25 = 0,
(f2|f3) = z21z31 + 2z36 − 2z24z35 − 2z25 = 0,
(f3|f3) = z231 − 4z35 = 0.
Thus the null flag manifold F ′ has a system of local coordinates (z11, z13, z14, z15, z16, z21, z24,
z25, z31) and dimF ′ = 9. For Λ1 = ⟨f1⟩, the equation (4.2) is equivalent to that
2u2 + 2z13u3 + 2z14u4 − z11v1 = 0, − 2u1 + 2z15u3 + 2z16u4 − z11v2 = 0,
− 2z13u1 − 2z15u2 + 2z17u4 − z11v3 = 0, − 2z14u1 − 2z16u2 − 2z17u3 − z11v4 = 0,
z11u1 + 2z17v2 − 2z16v3 + 2z15v4 = 0, z11u2 − 2z17v1 + 2z14v3 − 2z13v4 = 0,
z11u3 + 2z16v1 − 2z14v2 + 2v4 = 0, z11u4 − 2z15v1 + 2z13v2 − 2v3 = 0,
Prolongation of (8, 15)-Distribution of Type F4 by Singular Curves 11
and, in fact, to that
u1 = z15u3 + z16u4 −
1
2
z11v2, u2 = −z13u3 − z14u4 +
1
2
z11v1,
v3 =
1
2
z11u4 − z15v2 + z13v2, v4 = −
1
2
z11u3 − z16v1 + z14v2.
Thus we see that the solutions form a null 4-space V4 in Dm
∼= K8
V4 =
〈
t
(
z15,−z13, 1, 0, 0, 0, 0,−1
2z11
)
, t
(
z16,−z14, 0, 1, 0, 0, 12z11, 0
)
t
(
0, 12z11, 0, 0, 1, 0,−z15,−z16
)
, t
(
−1
2z11, 0, 0, 0, 0, 1, z13, z14
)〉
via the frame (X1, X2, X3, X4, Y1, Y2, Y3, Y4). For Λ2 = ⟨f1, f2⟩, we get the equation (4.2) for f1
as above and, in addition, the equation (4.2) applied to f2,
2u3 + 2z24u4 − z21v1 = 0, − 2z25u3 + 2z26u4 − z21v2 = 0,
− 2u1 − 2z25u2 + 2z27u4 − z21v3 = 0, − 2z24u1 − 2z26u2 − 2z27u3 − z21v4 = 0,
z21u1 + 2z27v2 − 2z26v3 + 2z25v4 = 0, z21u2 − 2z27v1 + 2z24v3 − 2v4 = 0,
z21u3 + 2z26v1 − 2z24v2 = 0, z21u4 − 2z25v1 + 2v2 = 0.
Then, by the two systems of linear equations for f1 and f2, we have
u1 =
(
−z15z24 + z16 +
1
4
z11z21
)
u4 +
(
1
2
z15z21 −
1
2
z11z25
)
v1
u2 = (z13z24 − z14)u4 +
(
−1
2
z13z21 +
1
2
z11
)
v1,
u3 = −z24u4 +
1
2
z21v1,
v2 = −
1
2
z21u4 +
1
2
z21v1
v3 =
(
1
2
z11 −
1
2
z13z21
)
u4 + (−z15 + z13z25)v1,
v4 =
(
1
2
z11z24 −
1
2
z14z21
)
u4 +
(
−1
4
z11z21 − z16 + z14z25
)
v1.
Thus we see Λ2 corresponds to the null 2-plane V2 in Dm
∼= K8, by (4.2), generated by two
vectors
t
(
−z15z24+ z16+
1
4z11z21, z13z24− z14,−z24, 1, 0,−1
2z21,
1
2z11−
1
2z13z21,
1
2z11z24−
1
2z14z21
)
,
t
(
1
2z15z21−
1
2z11z25,−
1
2z13z21+
1
2z11,
1
2z21, 0, 1, z25,−z15+ z13z25,−1
4z11z21− z16+ z14z25
)
.
For Λ3 = ⟨f1, f2, f3⟩, we obtain the additional condition (4.2) applied to f3, which is given by
2u4 − z31v1 = 0, 2z35u3 + 2z36u4 − z31v2 = 0,
− 2z35u2 + 2z37u4 − z31v3 = 0, − 2u1 − 2z36u2 − 2z37u3 − z31v4 = 0,
z31u1 + 2z37v2 − 2z36v3 + 2z35v4 = 0, z31u2 − 2z37v1 + 2v3 = 0,
z31u3 + 2z36v1 − 2v2 = 0, z31u4 − 2z35v1 = 0.
Then, from the conditions (4.2) for f1, f2 and f3, we have
u1 =
(
−1
2
z11z25 +
1
2
z16z31 +
1
8
z11z21z31 −
1
2
z15z24z31 +
1
2
z15z21
)
v1,
12 G. Ishikawa and Y. Machida
u2 =
(
1
2
z11 −
1
2
z13z21 −
1
2
z14z31 +
1
2
z13z24z31
)
v1,
u3 =
(
1
2
z21 −
1
2
z24z31
)
v1,
u4 =
1
2
z31v1,
v2 =
(
z25 −
1
4
z21z31
)
v1,
v3 =
(
−z15 +
1
4
z11z31 + z13z25 −
1
4
z13z21z31
)
v1,
v4 =
(
−z16 −
1
4
z11z21 + z14z25 +
1
4
z11z24z31 −
1
4
z14z21z31
)
v1.
Therefore, if we set, by taking v1 = 1,
η1 =
−1
2z11z25 +
1
2z16z31 +
1
8z11z21z31 −
1
2z15z24z31 +
1
2z15z21
1
2z11 −
1
2z13z21 −
1
2z14z31 +
1
2z13z24z31
1
2z21 −
1
2z24z31
1
2z31
1
z25 − 1
4z21z31
−z15 + 1
4z11z31 + z13z25 − 1
4z13z21z31
−z16 − 1
4z11z21 + z14z25 +
1
4z11z24z31 −
1
4z14z21z31
,
then we see that Λ3 corresponds to the null line V1 generated by η1. Moreover, we set η2 by
t
(
z16 + 1
4z11z21 − z15z24, −z14 + z13z24, −z24, 1, 0, −1
2z21,
1
2z11 −
1
2z13z21,
1
2z11z24 −
1
2z14z21
)
,
η3 by t(z15,−z13, 1, 0, 0, 0, 0,−1
2z11), and η4 by t
(
−1
2z11, 0, 0, 0, 0, 1, z13, z14
)
. Then we have
that (η1, η2, η3, η4) is a frame of V4 satisfying V1 = ⟨η1⟩ ⊂ V2 = ⟨η1, η2⟩ ⊂ V4 = ⟨η1, η2, η3, η4⟩. ■
Remark 5.1. The total null flag bundle F̃ constructed from D which consists of all null flags
V1 ⊂ V2 ⊂ V4 ⊂ Dm
∼= R4,4, m ∈ K15 is of dimension 15+11 = 26. The total null flag bundle F̃ ′
constructed from D⊥ which consists of Λ1 ⊂ Λ2 ⊂ Λ3 ⊂ D⊥
m
∼= R4,3, m ∈ K15, is of dimension
15 + 9 = 24. Then we have obtained, as above, the embedding F̃ ′ → F̃ of codimension 2.
6 Prolongation of Cartan’s model
The theory of prolongations and equivalence problems of distributions are established by many
authors (see, for instance, [10, 34, 35, 38]). Here we provide, related to the notion of singular
curves of distributions, a way of prolongations from viewpoints of sub-Riemannian geometry
and geometric control theory.
We set, as the prolonged space, W = F̃ ′ ∼= K15 × F ′ in F̃ ∼= K15 × F by the null flag
manifold F ′. Note that dim(F ′) = 9 and that dim(W ) = 24: W has a local coordinate system
(z, x1, x2, x3, x4, y1, y2, y3, y4, x12, x13, x14, x23, x24, x34; z11, z13, z14, z15, z16, z21, z24, z25, z31).
We are going to define and study the canonical distribution E on W = F̃ ′.
Take any point w0 = (m0, (V1)0, (V2)0, (V4)0) of W . Then we define Ew0 ⊂ Tw0W as the set of
initial vectors
(
m′(0), V ′
1(0), V
′
2(0), V
′
4(0)
)
of curves (m(t), V1(t), V2(t), V4(t)) : (R, 0)→ W in W
which satisfy the condition m′(t) ∈ V1(t), η
′
1(t) ∈ V2(t), η
′
2(t) ∈ V4(t) for some (so equivalently
for any) framing V1(t) = ⟨η1(t)⟩, V2(t) = ⟨η1(t), η2(t)⟩.
Prolongation of (8, 15)-Distribution of Type F4 by Singular Curves 13
Now we calculate the canonical distribution E explicitly. The above condition for E ⊂ TW
reads, at t = 0, that
m′(0) = pη1(m(0)), η′1(0) = qη1(m(0)) + rη2(m(0)),
f ′
2(0) = sη1(m(0)) + uη2(m(0)) + vη3(m(0)) + wη4(m(0)),
for some p, q, r, s, u, v, w ∈ R.
By the above second condition η′1 = q η1 + r η2 at t = 0, we see q = 0 and 1
2z
′
31 = r at t = 0.
Moreover, after some straightforward calculations, we have
z′11 − z21z
′
13 − z13z
′
21 − z31z
′
14 + z24z31z
′
13 + z13z31z
′
24 = 0,
z′21 − z31z
′
24 = 0, z′25 −
1
4
z231z
′
24 = 0,
z′15 −
1
4
z31z
′
11 − z25z
′
13 − z13z
′
25 +
1
4
z21z31z
′
13 +
1
4
z13z31z
′
21 = 0,
z′16 +
1
4
z21z
′
11 +
1
4
z11z
′
21 − z25z
′
14 − z14z
′
25 −
1
4
z24z31z
′
11 −
1
4
z11z31z
′
24
+
1
4
z21z31z
′
14 +
1
4
z14z31z
′
21 = 0,
at t = 0, for the coordinate functions of the curve η1 on F ′. By the above third condition
f ′
2 = s η1 + u η2 + v η3 + w η4 at t = 0, we have that s = u = 0 and that −z′24 = v, −1
2z
′
21 = w
at t = 0. Moreover, we have that
z′16 − z24z
′
15 +
1
4
z21z
′
11 = 0, z′14 − z24z
′
130, z
′
11 − z21z
′
13 = 0,
at t = 0. In term of differential 1-forms, the above conditions are reduced to that
dz11 − z21dz13 = 0, dz21 − z31dz24 = 0,
dz14 − z24dz13 = 0, dz25 −
1
4
z231dz24 = 0,
dz15 − z25dz13 = 0, dz16 −
(
z24z25 −
1
4
z221
)
dz13 = 0,
at t = 0. To get the frame of E, we set
ζ = A
∂
∂z11
+B
∂
∂z13
+ C
∂
∂z14
+D
∂
∂z15
+ F
∂
∂z16
+G
∂
∂z21
+H
∂
∂z24
+ I
∂
∂z25
+ J
∂
∂z31
.
The condition that ζ belongs to E is given by
A− z21B = 0, G− z31H = 0, C − z25B = 0,
I − 1
4
z231H = 0, D − z25B = 0, F −
(
z24z25 −
1
4
z221
)
B = 0,
and thus we have, for some B,H, J ∈ R,
ζ = B
{
∂
∂z13
+ z21
∂
∂z11
+ z24
∂
∂z14
+ z25
∂
∂z15
+
(
z24z25 −
1
4
z221
)
∂
∂z16
}
+H
(
∂
∂z24
+ z31
∂
∂z21
+
1
4
z231
∂
∂z25
)
+ J
∂
∂z31
,
at t = 0.
Thus, adding the generator which comes from the condition m′(0) = p η1(m(0)), we have the
following lemma.
14 G. Ishikawa and Y. Machida
Lemma 6.1. We have on the 24-dimensional space W = R15 ×F ′ with local coordinates z, xi,
yj, xij, zkℓ, the prolonged distribution E with the system of generators
ζ1 =
∂
∂z13
+ z21
∂
∂z11
+ z24
∂
∂z14
+ z25
∂
∂z15
+
(
z24z25 −
1
4
z221
)
∂
∂z16
,
ζ2 =
∂
∂z24
+ z31
∂
∂z21
+
1
4
z231
∂
∂z25
,
ζ3 =
∂
∂z31
,
ζ4 =
(
−1
2
z11z25 +
1
2
z15z21 +
1
2
z16z31 +
1
8
z11z21z31 −
1
2
z15z24z31
)
X1
+
(
1
2
z11 −
1
2
z13z21 −
1
2
z14z31 +
1
2
z13z24z31
)
X2 +
(
1
2
z21 −
1
2
z24z31
)
X3 +
1
2
z31X4
+ Y1 +
(
z25 −
1
4
z21z31
)
Y2 +
(
−z15 +
1
4
z11z31 + z13z25 −
1
4
z13z21z31
)
Y3
+
(
−z16 −
1
4
z11z21 + z14z25 +
1
4
z11z24z31 −
1
4
z14z21z31
)
Y4.
Note that the vector field ζ4 in Lemma 6.1 is induced from η1 obtained in the previous
section. We have chosen the above system of generators regarding the F4-Dynkin diagram (see
Remark 6.3). Now we have the following.
Lemma 6.2. The growth vector of the distribution E defined in the previous Lemma 6.1 is given
by (4, 7, 10, 13, 16, 18, 20, 21, 22, 23, 24) and the following bracket relations for the generators ζ1,
ζ2, ζ3, ζ4 of E given in Lemma 6.1:
[ζ1, ζ2] = ζ5, [ζ1, ζ3] = 0, [ζ1, ζ4] = 0, [ζ2, ζ3] = ζ6,
[ζ2, ζ4] = 0, [ζ3, ζ4] = ζ7 in E(2);
[ζ1, ζ5] = 0, [ζ1, ζ6] = ζ8, [ζ1, ζ7] = 0, [ζ2, ζ5] = 0,
[ζ2, ζ6] = 0, [ζ2, ζ7] = ζ9, [ζ3, ζ5] = −ζ8, [ζ3, ζ6] = ζ10,
[ζ3, ζ7] = 0, [ζ4, ζ5] = 0, [ζ4, ζ6] = −ζ9, [ζ4, ζ7] = 0 in E(3);
[ζ1, ζ8] = 0, [ζ1, ζ9] = ζ11, [ζ1, ζ10] = ζ12, [ζ2, ζ8] = 0,
[ζ2, ζ9] = 0, [ζ2, ζ10] = 0, [ζ3, ζ8] = ζ12, [ζ3, ζ9] = ζ13,
[ζ3, ζ10] = 0, [ζ4, ζ8] = −ζ11, [ζ4, ζ9] = 0, [ζ4, ζ10] = −2ζ13 in E(4);
[ζ1, ζ11] = 0, [ζ1, ζ12] = 0, [ζ1, ζ13] = ζ14, [ζ2, ζ11] = 0,
[ζ2, ζ12] = ζ15, [ζ2, ζ13] = 0, [ζ3, ζ11] = ζ14, [ζ3, ζ12] = 0,
[ζ3, ζ13] = 0, [ζ4, ζ11] = 0, [ζ4, ζ12] = −2ζ14, [ζ4, ζ13] = ζ16 in E(5);
[ζ1, ζ14] = 0, [ζ1, ζ15] = 0, [ζ1, ζ16] = 0, [ζ2, ζ14] = ζ17,
[ζ2, ζ15] = 0, [ζ2, ζ16] = 0, [ζ3, ζ14] = 0, [ζ3, ζ15] = 0,
[ζ3, ζ16] = 0, [ζ4, ζ14] = ζ18, [ζ4, ζ15] = −2ζ17, [ζ4, ζ16] = 0 in E(6);
[ζ1, ζ17] = 0, [ζ1, ζ18] = 0, [ζ2, ζ17] = 0, [ζ2, ζ18] = ζ19,
[ζ3, ζ17] = ζ20, [ζ3, ζ18] = 0, [ζ4, ζ17] = ζ19, [ζ4, ζ18] = 0 in E(7);
[ζ1, ζ19] = 0, [ζ1, ζ20] = 0, [ζ2, ζ19] = 0, [ζ2, ζ20] = 0,
[ζ3, ζ19] = ζ21, [ζ3, ζ20] = 0, [ζ4, ζ19] = 0, [ζ4, ζ20] =
1
2
ζ21 in E(8);
[ζ1, ζ19] = 0, [ζ1, ζ20] = 0, [ζ2, ζ19] = 0, [ζ2, ζ20] = 0,
Prolongation of (8, 15)-Distribution of Type F4 by Singular Curves 15
[ζ3, ζ19] = ζ21, [ζ3, ζ20] = 0, [ζ4, ζ19] = 0, [ζ4, ζ20] =
1
2
ζ21 in E(8);
[ζ1, ζ21] = 0, [ζ2, ζ21] = 0, [ζ3, ζ21] = ζ22, [ζ4, ζ21] = 0 in E(9);
[ζ1, ζ22] = 0, [ζ2, ζ22] = ζ23, [ζ3, ζ22] = 0, [ζ4, ζ22] = 0 in E(10);
[ζ1, ζ23] = ζ24, [ζ2, ζ23] = 0, [ζ3, ζ23] = 0, [ζ4, ζ23] = 0 in E(11) = TW ,
which are calculated explicitly in the proof. In particular, the distribution E is isomorphic to
the (8, 15)-distribution on the quotient space by the parabolic subgroup associated to the root α4
of F4 in the complex case (resp. of F4(4) in the real case).
Remark 6.3. Between the simple roots α1, α2, α3, α4 of F4 (see, for instance, [8]) and the
generators ζ1, ζ2, ζ3, ζ4 of E, there exists the correspondence ζi ←→ −αi, i = 1, 2, 3, 4,
ζ5 ←→ −(α1 + α2), ζ6 ←→ −(α2 + α3),
ζ7 ←→ −(α3 + α4), ζ8 ←→ −(α1 + α2 + α3),
ζ9 ←→ −(α2 + α3 + α4), ζ10 ←→ −(α2 + 2α3),
ζ11 ←→ −(α1 + α2 + α3 + α4), ζ12 ←→ −(α1 + α2 + 2α3),
ζ13 ←→ −(α2 + 2α3 + α4), ζ14 ←→ −(α1 + α2 + 2α3 + α4),
ζ15 ←→ −(α1 + 2α2 + 2α3), ζ16 ←→ −(α2 + 2α3 + 2α4),
ζ17 ←→ −(α1 + α2 + 2α3 + α4), ζ18 ←→ −(α1 + α2 + 2α3 + 2α4),
ζ19 ←→ −(α1 + 2α2 + 2α3 + 2α4), ζ20 ←→ −(α1 + 2α2 + 3α3 + α4),
ζ21 ←→ −(α1 + 2α2 + 3α3 + 2α4), ζ22 ←→ −(α1 + 2α2 + 4α3 + 2α4),
ζ23 ←→ −(α1 + 3α2 + 4α3 + 2α4), ζ24 ←→ −(2α1 + 3α2 + 4α3 + 2α4).
Proof of Lemma 6.2. In fact, we have for the vector fields ζ1, ζ2, ζ3, ζ4 in Lemma 6.1:
[ζ1, ζ2] = −
∂
∂z14
− z31
∂
∂z11
− 1
4
z231
∂
∂z15
+
(
−z25 +
1
2
z21z31 −
1
4
z24z
2
31
)
∂
∂z16
=: ζ5,
[ζ1, ζ3] = 0, [ζ1, ζ4] = 0,
[ζ2, ζ3] = −
∂
∂z21
− 1
2
z31
∂
∂z25
=: ζ6, [ζ2, ζ4] = 0,
[ζ3, ζ4] =
(
1
2
z16 +
1
8
z11z21 −
1
2
z15z24
)
X1 +
(
−1
2
z14 +
1
2
z13z24
)
X2 −
1
2
z24X3 +
1
2
X4
− 1
4
z21Y2 +
(
1
4
z11 −
1
4
z13z21
)
Y3 +
(
1
4
z11z24 −
1
4
z14z21
)
Y4 =: ζ7.
So far, we have rankE(2) = 7.
Moreover, we have
[ζ1, ζ5] = 0, [ζ1, ζ6] =
∂
∂z11
+
1
2
z31
∂
∂z15
+
(
−1
2
z21 +
1
2
z24z31
)
∂
∂z16
=: ζ8,
[ζ1, ζ7] = 0,
[ζ2, ζ5] = 0, [ζ2, ζ6] = 0,
[ζ2, ζ7] =
(
−1
2
z15 +
1
8
z11z31
)
X1 +
1
2
z13X2 −
1
2
X3 −
1
4
z31Y2 −
1
4
z13z31Y3
+
(
1
4
z11 −
1
4
z14z31
)
Y4 =: ζ9
[ζ3, ζ5] = −
∂
∂z11
− 1
2
z31
∂
∂z15
+
(
1
2
z21 −
1
2
z24z31
)
∂
∂z16
= −[ζ1, ζ6] = −ζ8,
16 G. Ishikawa and Y. Machida
[ζ3, ζ6] = −
1
2
∂
∂z25
=: ζ10, [ζ3, ζ7] = 0,
[ζ4, ζ5] = 0, [ζ4, ζ6] = −[ζ2, ζ7] = −ζ9, [ζ4, ζ7] = 0.
Then we have rankE(3) = 10.
Further we have
[ζ1, ζ8] = 0,
[ζ1, ζ9] =
(
−1
2
z25 +
1
8
z21z31
)
X1 +
1
2
X2 −
1
4
z31Y3 +
(
1
4
z21 −
1
4
z24z31
)
Y4 =: ζ11,
[ζ1, ζ10] =
1
2
∂
∂z15
+
1
2
z24
∂
∂z16
=: ζ12,
[ζ2, ζ8] = 0, [ζ2, ζ9] = 0, [ζ2, ζ10] = 0,
[ζ3, ζ8] = [ζ1, ζ10] = ζ12, [ζ3, ζ9] =
1
8
z11X1 −
1
4
Y2 −
1
4
z13Y3 −
1
4
z14Y4 =: ζ13,
[ζ3, ζ10] = 0,
[ζ4, ζ8] = −[ζ1, ζ9] = −ζ11, [ζ4, ζ9] = 0, [ζ4, ζ10] = −2[ζ3, ζ9] = −2ζ13.
We get that rankE(4) = 13.
Further we have
[ζ1, ζ11] = 0, [ζ1, ζ12] = 0, [ζ1, ζ13] =
1
8
z21X1 −
1
4
Y3 −
1
4
z24Y4 =: ζ14,
[ζ2, ζ11] = 0, [ζ2, ζ12] =
1
2
∂
∂z16
=: ζ15, [ζ2, ζ13] = 0,
[ζ3, ζ11] = [ζ1, ζ13] = ζ14, [ζ3, ζ12] = 0, [ζ3, ζ13] = 0,
[ζ4, ζ11] = 0, [ζ4, ζ12] = −
1
4
z21X1 +
1
2
Y3 +
1
2
z24Y4 = −2[ζ1, ζ13] = −2ζ14,
[ζ4, ζ13] =
(
−1
8
z211 −
1
2
z13z16 +
1
2
z14z15
)
X12 +
1
2
z16X13 −
1
2
z15X14 −
1
2
z14X23
+
1
2
z13X24 −
1
2
X34 +
1
4
z11Z =: ζ16.
Therefore, we have rankE(5) = 16.
Furthermore,
[ζ1, ζ14] = 0, [ζ1, ζ15] = 0, [ζ1, ζ16] = 0,
[ζ2, ζ14] =
1
8
z31X1 −
1
4
Y4 =: ζ17, [ζ2, ζ15] = 0, [ζ2, ζ16] = 0,
[ζ3, ζ14] = 0, [ζ3, ζ15] = 0, [ζ3, ζ16] = 0,
[ζ4, ζ14] =
(
−1
2
z16 −
1
4
z11z21 +
1
2
z14z25 +
1
2
z15z24 +
1
8
z13z
2
21 −
1
2
z13z24z25
)
X2
+
(
−1
8
z221 +
1
2
z24z25
)
X13 −
1
2
z25X14 −
1
2
z24X23 +
1
2
X24 +
1
4
z12Z =: ζ18,
[ζ4, ζ15] = −2[ζ2, ζ14] = −2ζ17, [ζ4, ζ16] = 0.
Thus we see rankE(6) = 18.
Furthermore, we have
[ζ1, ζ17] = 0, [ζ1, ζ18] = 0, [ζ2, ζ17] = 0,
Prolongation of (8, 15)-Distribution of Type F4 by Singular Curves 17
[ζ2, ζ18] =
(
1
2
z15 −
1
4
z11z31 −
1
2
z13z25 +
1
8
z14z
2
31 +
1
4
z13z21z31 −
1
8
z13z24z
2
31
)
X12
+
(
1
2
z25 −
1
4
z21z31 +
1
8
z24z
2
31
)
X13 −
1
8
z231X14 −
1
2
X23 +
1
4
z31Z =: ζ19,
[ζ3, ζ17] =
1
8
X1 =: ζ20, [ζ3, ζ18] = 0,
[ζ4, ζ17] = [ζ2, ζ18] = ζ19, [ζ4, ζ18] = 0.
Thus we have rankE(7) = 20.
Moreover,
[ζ1, ζ19] = 0, [ζ1, ζ20] = 0,
[ζ2, ζ19] = 0, [ζ2, ζ20] = 0,
[ζ3, ζ19] =
(
−1
4
z11 +
1
4
z14z31 +
1
4
z13z21 −
1
4
z13z24z31
)
X12 +
(
−1
4
z21 +
1
4
z24z31
)
X13
− 1
4
z31X14 +
1
4
Z =: ζ21,
[ζ3, ζ20] = 0,
[ζ4, ζ19] = 0, [ζ4, ζ20] =
1
2
[ζ3, ζ19] =
1
2
ζ21.
We obtain that rankE(8) = 21.
We have
[ζ1, ζ21] = 0, [ζ2, ζ21] = 0,
[ζ3, ζ21] =
(
1
4
z14 −
1
4
z13z24
)
X12 +
1
4
z24X13 −
1
4
X14 =: ζ22
and we have [ζ4, ζ21] = 0. So we get that rankE(9) = 22.
Also we have
[ζ1, ζ22] = 0, [ζ2, ζ22] = −
1
4
z13X12 +
1
4
X13 =: ζ23,
[ζ3, ζ22] = 0, [ζ4, ζ22] = 0.
Then we have rankE(10) = 23.
Lastly, we have
[ζ1, ζ23] = −
1
4
X12 =: ζ24, [ζ2, ζ23] = 0,
[ζ3, ζ23] = 0, [ζ4, ζ23] = 0.
We have that rankE(11) = 24. This shows the claim. ■
Remark 6.4. By the calculations in the proof of Lemma 6.2, we observe that π−1
∗ (D) ⊂ E(7)
for the projection π : W →M , π(m, (V1, V2, V4)) = m.
7 (8, 15)-distributions of type F4
Inspired by our study on singular curves for Cartan model performed in the previous sections,
it would be natural to introduce the class of (8, 15)-distributions of type F4 including Cartan’s
model.
18 G. Ishikawa and Y. Machida
Definition 7.1. Let D ⊂ TM be a complex (resp. a real) (8, 15)-distribution. Then we
call D of type F4 (resp. of type F4(4)) if, for each point x0 ∈ M , there exists a local frame
{X1, X2, X3, X4, Y1, Y2, Y3, Y4} of D over an open neighbourhood of x0 such that, modulo D,
[X1, X2] ≡ [Y3, Y4], [X1, X3] ≡ −[Y2, Y4], [X1, X4] ≡ [Y2, Y3],
[X2, X3] ≡ [Y1, Y4], [X2, X4] ≡ −[Y1, Y3], [X3, X4] ≡ [Y1, Y2],
[X1, Y1] ≡ [X2, Y2] ≡ [X3, Y3] ≡ [X4, Y4], and [Xi, Yj ] ≡ 0 i ̸= j, 1 ≤ i, j ≤ 4,
and, if we set
X12 =
1
2
[X1, X2], X13 =
1
2
[X1, X3], X14 =
1
2
[X1, X4],
X23 =
1
2
[X2, X3], X24 =
1
2
[X2, X4], X34 =
1
2
[X3, X4],
and Z = [Y1, X1], then the vector fields
X1, X2, X3, X4, Y1, Y2, Y3, Y4X12, X13, X14, X23, X24, X34, Z,
form a local frame of TM .
Remark 7.2. Comparing with the relations on generators of Cartan’s model in Section 2,
the relations in Definition 7.1 are given modulo D. The class of (8, 15)-distributions of type F4
in Definition 7.1 coincides with the class of regular differential system of type mF in the sense
of Tanaka [38, 39, 40].
Then we have the following theorem.
Theorem 7.3. Let (M,D) be a complex (resp. real) (8, 15)-distribution of type F4 (resp. F4(4)).
Then there exist uniquely the conformal non-degenerate bilinear form (resp. (4, 4)-metric) on D
and the conformal non-degenerate bilinear form (resp. (4, 3)-metric) on D⊥ obtained from the
abnormal bi-extremals of D such that the null-cone C ⊂ D coincides with the singular velocity
cone SVC(D). Moreover, the flag manifold of null-subspaces
{
Λ1 ⊂ Λ2 ⊂ Λ3 ⊂ D⊥ ⊂ T ∗M
}
corresponds to a subclass of flags by null-subspaces {V1 ⊂ V2 ⊂ V4 ⊂ C ⊂ D ⊂ TM} in D. The
prolongation (W,E) of (M,D) by the above null-flags of D turns out to be a (4, 7, 10, 13, 16, 18, 20,
21, 22, 23, 24)-distribution such that its symbol algebra is isomorphic to the negative part of the
nilpotent algebra for the gradation by the full set {α1, α2, α3, α4} of simple roots of simple Lie
algebra F4 (resp. F4(4)).
Proof of Theorem 7.3. We re-examine the arguments on Cartan’s model of (8, 15)-distri-
bution defined in Section 2 and performed in Sections 4–6 for general (8, 15)-distributions of
type F4.
Let D ⊂ TM be an (8, 15)-distribution of type F4. Reversing the correspondence in Section 2,
we take the local frame
β1, β2, β3, β4, γ1, γ2, γ3, γ4, ω12, ω13, ω14, ω23, ω24, ω34, σ
of T ∗M which is dual to the local frame
X1, X2, X3, X4, Y1, Y2, Y3, Y4, X12, X13, X14, X23, X24, X34, Z
of TM in Definition 7.1. ThenD⊥ is generated by ω12, ω13, ω14, ω23, ω24, ω34 and σ. Any α ∈ D⊥
is expressed uniquely as α =
∑
1≤i<j≤4 rijωij + sσ. Then we have ⟨α,Xij⟩ = rij and ⟨α, σ⟩ = s.
The functions rij and s with local coordinates of the base manifold M form a system of local
Prolongation of (8, 15)-Distribution of Type F4 by Singular Curves 19
coordinates of the submanifold D⊥ ⊂ T ∗M . Then the equations (4.2) and (4.3) are obtained
other (linear) algebraic arguments in Section 4 work as well also for general (8, 15)-distributions
of type F4. Thus we have the same conclusion of Corollary 4.3 and moreover our discussions
on the correspondence of null-flags in D and D⊥ performed in Section 5 and the same proofs of
the results such as Lemma 6.2 which concern on the prolongations of D in Section 6 works well
also for any (8, 15)-distribution of type F4. This shows Theorem 7.3. ■
Remark 7.4. The above statement on (8, 15)-distribution of type F4 (resp. F4(4)) means that
the gradation sheaf, i.e., the sheaf of nilpotent graded Lie algebras m :=
⊕11
i=1
(
D(i)/D(i−1)
)
is
isomorphic to that for the model derived from the simple Lie algebra F4, which is described in
Section 2. It is stated in [40] (see Proposition 5.5 and the arguments in pp. 482–483) that any
(8, 15)-distribution of type F4 (resp. F4(4)) is isomorphic to Cartan’s model over C (resp. R) in
fact by Tanaka theory on simple graded Lie algebras. Note that we have proved our Theorem 7.3
without using this fact.
Acknowledgements
The authors would like to thank anonymous referees for valuable and helpful comments to
improve the paper. The first author is partially supported by JSPS KAKENHI Grant Num-
ber 24K06700, by JST CREST Geometrical Understanding of Spatial Orientation and by the
Research Institute for Mathematical Sciences in Kyoto University.
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1 Introduction
2 Cartan's model of (8, 15)-distributions of type F_4
3 Abnormal bi-extremals and singular curves of distributions
4 Conformal metric on Cartan's (8, 15)-distribution and singular velocity cone
5 Null flags associated to abnormal bi-extremals
6 Prolongation of Cartan's model
7 (8,15)-distributions of type F_4
References
|
| id | nasplib_isofts_kiev_ua-123456789-214092 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T15:00:09Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ishikawa, Goo Machida, Yoshinori 2026-02-19T11:10:55Z 2025 Prolongation of (8, 15)-Distribution of Type ₄ by Singular Curves. Goo Ishikawa and Yoshinori Machida. SIGMA 21 (2025), 076, 20 pages 1815-0659 2020 Mathematics Subject Classification: 53C17; 58A30; 17B25; 34H05; 37J37; 49K15; 53D25 arXiv:2501.02789 https://nasplib.isofts.kiev.ua/handle/123456789/214092 https://doi.org/10.3842/SIGMA.2025.076 Cartan gives the model of (8, 15)-distribution with the exceptional simple Lie algebra ₄ as its symmetry algebra in his paper (1893), which was published one year before his thesis. In the present paper, we study abnormal extremals (singular curves) of Cartan's model from the viewpoints of sub-Riemannian geometry and geometric control theory. Then we construct the prolongation of Cartan's model based on the data related to its singular curves, and obtain the nilpotent graded Lie algebra which is isomorphic to the negative part of the graded Lie algebra ₄. The authors would like to thank anonymous referees for their valuable and helpful comments to improve the paper. The first author is partially supported by JSPS KAKENHI Grant Number 24K06700, by JST CREST Geometrical Understanding of Spatial Orientation, and by the Research Institute for Mathematical Sciences in Kyoto University. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Prolongation of (8, 15)-Distribution of Type ₄ by Singular Curves Article published earlier |
| spellingShingle | Prolongation of (8, 15)-Distribution of Type ₄ by Singular Curves Ishikawa, Goo Machida, Yoshinori |
| title | Prolongation of (8, 15)-Distribution of Type ₄ by Singular Curves |
| title_full | Prolongation of (8, 15)-Distribution of Type ₄ by Singular Curves |
| title_fullStr | Prolongation of (8, 15)-Distribution of Type ₄ by Singular Curves |
| title_full_unstemmed | Prolongation of (8, 15)-Distribution of Type ₄ by Singular Curves |
| title_short | Prolongation of (8, 15)-Distribution of Type ₄ by Singular Curves |
| title_sort | prolongation of (8, 15)-distribution of type ₄ by singular curves |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214092 |
| work_keys_str_mv | AT ishikawagoo prolongationof815distributionoftype4bysingularcurves AT machidayoshinori prolongationof815distributionoftype4bysingularcurves |