Flat (2,3,5)-Distributions and Chazy's Equations
n the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or (2,3,5)-distributions determined by a single function of the form F(q), the vanishing condition for the...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2016 |
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Інститут математики НАН України
2016
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| Цитувати: | Flat (2,3,5)-Distributions and Chazy's Equations / M. Randall // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859678191749169152 |
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| author | Randall, M. |
| author_facet | Randall, M. |
| citation_txt | Flat (2,3,5)-Distributions and Chazy's Equations / M. Randall // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | n the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or (2,3,5)-distributions determined by a single function of the form F(q), the vanishing condition for the curvature invariant is given by a 6th order nonlinear ODE. Furthermore, An and Nurowski showed that this ODE is the Legendre transform of the 7th order nonlinear ODE described in Dunajski and Sokolov. We show that the 6th order ODE can be reduced to a 3rd order nonlinear ODE that is a generalised Chazy equation. The 7th order ODE can similarly be reduced to another generalised Chazy equation, which has its Chazy parameter given by the reciprocal of the former. As a consequence of solving the related generalised Chazy equations, we obtain additional examples of flat (2,3,5)-distributions not of the form F(q)=qm. We also give 4-dimensional split signature metrics where their twistor distributions via the An-Nurowski construction have split G₂ as their group of symmetries.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 029, 28 pages
Flat (2, 3, 5)-Distributions and Chazy’s Equations
Matthew RANDALL
Department of Mathematics and Statistics, Faculty of Science, Masaryk University,
Kotlářská 2, 611 37 Brno, Czech Republic
E-mail: randallm@math.muni.cz
Received September 23, 2015, in final form March 14, 2016; Published online March 18, 2016
http://dx.doi.org/10.3842/SIGMA.2016.029
Abstract. In the geometry of generic 2-plane fields on 5-manifolds, the local equivalence
problem was solved by Cartan who also constructed the fundamental curvature invariant.
For generic 2-plane fields or (2, 3, 5)-distributions determined by a single function of the
form F (q), the vanishing condition for the curvature invariant is given by a 6th order non-
linear ODE. Furthermore, An and Nurowski showed that this ODE is the Legendre transform
of the 7th order nonlinear ODE described in Dunajski and Sokolov. We show that the 6th
order ODE can be reduced to a 3rd order nonlinear ODE that is a generalised Chazy equation.
The 7th order ODE can similarly be reduced to another generalised Chazy equation, which
has its Chazy parameter given by the reciprocal of the former. As a consequence of solving
the related generalised Chazy equations, we obtain additional examples of flat (2, 3, 5)-
distributions not of the form F (q) = qm. We also give 4-dimensional split signature metrics
where their twistor distributions via the An–Nurowski construction have split G2 as their
group of symmetries.
Key words: generic rank two distribution in dimension five; conformal geometry; Chazy’s
equations
2010 Mathematics Subject Classification: 58A30; 53A30; 34A05; 34A34
1 Introduction
The following 6th order nonlinear ODE
10F (6)F ′′
3 − 80F ′′
2
F (3)F (5) − 51F ′′
2
F (4)2 + 336F ′′F (3)2F (4) − 224F (3)4 = 0, (1.1)
arises in [6, Corollary 2.1] in the study of generic 2-plane fields on 5-manifolds. The generi-
city condition here means F ′′(q) 6= 0 in (1.1). This ODE arises as the integrability condition
for generic 2-plane fields on 5 manifolds determined by a function of a single variable of the
form F (q). A generic 2-plane field D on a 5-manifold M is a maximally non-integrable rank 2
distribution. For further details, see [6, 18, 24, 25]. This determines a filtration of the tangent
bundle given by
D ⊂ [D,D] ⊂ [[D,D],D] = TM.
The distribution [D,D] has rank 3 while the full tangent space TM has rank 5, hence such a geo-
metry is also known as a (2, 3, 5)-distribution. Let Mxyzpq denote the 5-dimensional manifold
with local coordinates given by (x, y, z, p, q). The generic 2-plane field or rank 2 distribution
determined by a function F (q) of a single variable with F ′′(q) 6= 0 is given by
D = span{∂q, ∂x + p∂y + q∂p + F (q)∂z}.
The fundamental Cartan curvature invariant of this distribution is computed in [6] and is found
to be the term in the left hand side of (1.1). It is known that equation (1.1) vanishes when
mailto:randallm@math.muni.cz
http://dx.doi.org/10.3842/SIGMA.2016.029
2 M. Randall
F (q) = qm and m ∈
{
−1, 13 ,
2
3 , 2
}
. In these cases, the vanishing of the fundamental curvature
invariant associated to the distribution D on Mxyzpq implies that the group of local symmetries
is the maximal possible given by the split real form of G2. This is the result of [6, Corollary 2.1].
The authors of [6] call such generic 2-plane fields with vanishing Cartan curvature invariant
symmetric and it is known that such symmetric or flat distributions are locally equivalent to
the flat model F (q) = q2. Nonetheless we are interested in the general solution to (1.1) and
it turns out that the ODE can be solved completely and is related to the generalised Chazy
equation. To see this, let E(q) = F ′′(q) so that the ODE becomes 4th order:
10E(4)E3 − 80E2E′E′′′ − 51E2E′′
2
+ 336EE′
2
E′′ − 224E′
4
= 0.
Working locally on an open set of Mxyzpq, we may assume that E(q) is positive on that open
set. Making the substitution E(q) = eG(q) (if E(q) < 0, take E(q) = −eG(q) instead) gives
e4G(q)
(
10G′′′′ − 40G′′′G′ − 21(G′′)2 + 54G′′(G′)2 − 9(G′)4
)
= 0
and taking G′(q) = j(q) gives a 3rd order ODE
10j′′′ − 40j′′j − 21(j′)2 + 54j′j2 − 9j4 = 0.
Rescaling the ODE by taking j(q) = I(q)
2 , we can put it into the normal form for the generalised
Chazy equation (see [14])
I ′′′ − 2I ′′I + 3(I ′)2 − 4
36−
(
2
3
)2 (6I ′ − I2)2 = 0 (1.2)
with the Chazy parameter k2 =
(
2
3
)2
= 4
9 . The generalised Chazy equation can be solved com-
pletely and the solutions give us new families of flat (2, 3, 5)-distributions that are not of the
form F (q) = qm. In this article we first review the solutions to Chazy’s equations in Sections 2
and 3. In Section 4 we discuss the relationship between (1.1) and a 7th order ODE studied by
Dunajski and Sokolov in [16] and also exhibit a Legendre transform that relates equation (1.2)
to another generalised Chazy equation with the Chazy parameter given by k2 =
(
3
2
)2
= 9
4 . We
compute the solutions to (1.2) in Section 5 and present examples of flat (2, 3, 5)-distributions in
Section 6 using Nurowski’s metric. These examples are all explicit. In [5], the authors associated
to split signature conformal structures on a 4-manifold a circle bundle with the natural structure
of a (2, 3, 5)-distribution. This construction encapsulates the configuration space of 2 surfaces
rolling along one another without slipping and twisting. The authors in [5] then found new exam-
ples of flat (2, 3, 5)-distributions that arise from rolling bodies, prompting further search in [8].
The solutions to (1.2) give examples of 4-dimensional split signature metrics that have their
An–Nurowski twistor distributions having split G2 as its group of symmetries and we exhibit
them in Section 7. Let us recall some facts about Chazy’s equation and its generalised version.
2 Chazy’s equation
The study of Chazy’s equation is a very rich subject and has received alot of attention because
of its connection to other diverse fields such as integrable systems and modular forms. See for
instance [2, 10, 11, 14]. We will review here some facts about Chazy’s equation we need for the
paper. Chazy [12, 13] studied the nonlinear 3rd order ODE
y′′′(x)− 2y(x)y′′(x) + 3(y′(x))2 = 0 (2.1)
Flat (2, 3, 5)-Distributions and Chazy’s Equations 3
in the context of investigating its Painlevé property. Solutions to equation (2.1) turn out to
depend on hypergeometric functions. For further details, see [2] or [14]. Treat x as a dependent
variable of s so that
x(s) =
z2(s)
z1(s)
,
where z1(s), z2(s) are linearly independent solutions to the second order hypergeometric diffe-
rential equation
s(1− s)z′′ + (c− (a+ b+ 1)s)z′ − abz = 0. (2.2)
Here a, b, c are constants to be determined. The general solution to this ODE (2.2) is given by
hypergeometric functions
z(s) = µ 2F1(a, b; c; s) + ν 2F1(a− c+ 1, b− c+ 1; 2− c; s)s1−c.
Here µ, ν are constants. A computation gives
dx =
z1ż2 − z2ż1
(z1)2
ds,
where dot denotes derivative with respect to s. We deduce that
d
dx
=
(z1)
2
z1ż2 − z2ż1
d
ds
.
Applying the derivative to Chazy’s solution for y given by
y = 6
d
dx
log z1 =
6z1ż1
z1ż2 − z2ż1
, (2.3)
we find that (2.1) is satisfied precisely when (a, b, c) is one of(
1
12
,
1
12
,
1
2
)
,
(
1
12
,
1
12
,
2
3
)
,
(
1
6
,
1
6
,
2
3
)
,
provided both a and b are non-zero. The equations (2.2) for the first two values of (a, b, c) are
related by a linear transformation of the form s 7→ 1− s, while the solutions for the second and
third values are related by a quadratic transformation (see [23, equation (2)]) given by
2F1
(
1
6
,
1
6
;
2
3
; s
)
= 2F1
(
1
12
,
1
12
;
2
3
; 4s(1− s)
)
.
The general solution to (2.1) thus depend on hypergeometric functions. If either one of a or b is
zero (say b = 0), solutions to (2.1) can be easily and explicitly described. The solutions to (2.2)
with b = 0 are given by
z(s) = µ+ ν 2F1(1− c, a+ 1− c; 2− c; s)s1−c = µ− νπ(c− 1)
sin(πc)
P
(1−c, a−c)
c−1 (1− 2s)s1−c,
where P
(a1,b1)
n is the Jacobi polynomial. Taking z1(s) = ν 2F1(1− c, a+ 1− c; 2− c; s)s1−c and
z2(s) = µ, a computation shows that
x(s) =
µ
ν 2F1(1− c, a+ 1− c; 2− c; s)s1−c
4 M. Randall
and
y(x(s)) = −6
ν
µ
2F1(1− c, a+ 1− c; 2− c; s)s1−c.
Switching back to the original independent variable x, this gives
y(x) = −6
x
as one solution to (2.1). This solution is invariant under translations of the form x 7→ x + C.
In [12, 13], Chazy also observed that
y = − 6
x+ C
− B
(x+ C)2
is a solution to (2.1). It is well-known that Chazy’s equation and its generalised version can be
rewritten as a first order system. This provides different parametrisations of y, in addition to
the solution (2.3) originally given by Chazy. This will be discussed in Section 5. The method
discussed here can also be applied to the generalised Chazy equation.
3 Generalised Chazy equations
The generalised Chazy equation is given by
y′′′(x)− 2y′′(x)y(x) + 3(y′(x))2 − 4
36− k2
(6y′(x)− y(x)2)2 = 0 (3.1)
for k 6= ±6. We have the following:
Proposition 3.1. Let x(s) = z2(s)
z1(s)
where z1(s), z2(s) are linearly independent solutions to the
hypergeometric differential equation (2.2) where (a, b, c) is one of(
k − 6
12k
,
k + 6
12k
,
1
2
)
,
(
k − 6
6k
,
k + 6
6k
,
2
3
)
,
(
k − 6
12k
,
k + 6
12k
,
2
3
)
.
Then
y(x(s)) = 6
d
dx
log z1 =
6z1ż1
z1ż2 − z2ż1
satisfies equation (3.1).
Proof. Analogous to solving Chazy’s equation (2.1), we find that the generalised equation (3.1)
holds provided
6ab(z1)
8
(
6((a− b)k − 6(a+ b))((a− b)k + 6(a+ b))s2
+
(
(24ab− 12(a+ b)c+ 5(a+ b) + (2c− 1))k2 + 432(a+ b)c− 180(a+ b)− 72c+ 36
)
s
+ (k − 6)(k + 6)(2c− 1)(3c− 2)
)
= 0.
For a, b 6= 0, solving the system of equations
6((a− b)k − 6(a+ b))((a− b)k + 6(a+ b)) = 0,
(24ab− 12(a+ b)c+ 5(a+ b) + (2c− 1))k2 + 432(a+ b)c− 180(a+ b)− 72c+ 36 = 0,
(k − 6)(k + 6)(2c− 1)(3c− 2) = 0,
gives the list of (a, b, c) as above. We exclude the case where (a, b, c) =
(
0, 0, 12
)
. Note that
interchanging a and b gives the same solution so that the full list is symmetric in a and b. �
Flat (2, 3, 5)-Distributions and Chazy’s Equations 5
When either a or b is zero, we again get y(x) = − 6
x as a solution. In [12, 13], Chazy noted that
y =
k − 6
2(x+ C)
− k + 6
2(x+B)
(3.2)
is also a solution to (3.1). As a corollary to Proposition 3.1, we have
Corollary 3.2. Let q(s) = z2(s)
z1(s)
where z1(s), z2(s) are linearly independent solutions to the
hypergeometric differential equation (2.2) with (a, b, c) one of(
−2
3
,
5
6
,
1
2
)
,
(
−4
3
,
5
3
,
2
3
)
,
(
−2
3
,
5
6
,
2
3
)
.
Then
I(q(s)) = 6
d
dq
log z1 =
6z1ż1
z1ż2 − z2ż1
satisfies equation (1.2).
A Painlevé type analysis of equation (3.1) as done in [14] shows that the leading orders for
analytic solutions to (3.1) occur at −6, −3 + k
2 or −3− k
2 . This corresponds to solutions of (3.1)
given by
y(x) = −6
x
, y(x) =
−3 + k
2
x
, y(x) =
−3− k
2
x
.
These solutions are invariant under translations of the form x 7→ x+ C. In the case of k = ±2
3
obtained in (1.2), we have
I(q) = −6
q
, I(q) = −10
3q
, I(q) = − 8
3q
.
Along with the zero solution I(q) = 0, these solutions correspond respectively (modulo constants
of integration) to the well-known explicit solutions to (1.1):
F (q) = q−1, F (q) = q
1
3 , F (q) = q
2
3 , F (q) = q2.
For these functions of a single variable q the associated (2, 3, 5)-distributions have vanishing
Cartan invariant and therefore have G2 as their local symmetry.
4 Relationship to ODE studied by Dunajski and Sokolov
For the function y = y(t), the 7th order nonlinear ODE studied in [16] is given by
10
(
y(3)
)3
y(7) − 70
(
y(3)
)2
y(4)y(6) − 49
(
y(3)
)2(
y(5)
)2
+ 280
(
y(3)
)(
y(4)
)2
y(5) − 175
(
y(4)
)4
= 0. (4.1)
This is the unique 7th order ODE admitting the submaximal contact symmetry group of dimen-
sion ten (see [16, 21]) and its relationship to equation (1.1) was originally explored in [6]. It is
instructive to consider the 6th order ODE (for the Legendre transformation later on):
10
(
H(2)
)3
H(6) − 70
(
H(2)
)2
H(3)H(5) − 49
(
H(2)
)2(
H(4)
)2
+ 280
(
H(2)
)(
H(3)
)2
H(4) − 175
(
H(3)
)4
= 0 (4.2)
6 M. Randall
with H(t) = y′(t). Let us show that this ODE can be reduced to a generalised Chazy equation.
Again working locally in an open set where y′′′(t) is non-zero, and assuming y′′′(t) to be positive,
we can make the substitution ep(t) = y(3) to get
ep(t)
(
10p(4) − 30p′p′′′ − 19(p′′)2 + 32(p′)2p′′ − 4(p′)4
)
= 0. (4.3)
We note that this 4th order ODE historically appears in [9, Section XII, formula (12)], where it
first arises as the obstruction to integrability for (2, 3, 5)-distributions of the form DF (q). This
will be made clear below once we show that (4.2) is the Legendre transform of (1.1) [6] and we
will discuss this further in Section 6. Thus, for v(t) = p′(t), we obtain the third order ODE
10v′′′ − 30vv′′ − 19(v′)2 + 32v′v2 − 4v4 = 0. (4.4)
Rescaling v(t) by u(t) = 3
2v(t), we put (4.4) into the normal form
u′′′ − 2u′′u+ 3(u′)2 − 4
36−
(
3
2
)2 (6u′ − u2)2 = 0. (4.5)
We therefore see that the ODE that Dunajski and Sokolov study in [16] reduces to a generalised
Chazy equation (4.5) with parameter k′ = ±3
2 , related to the generalised Chazy equation (1.2)
just by taking the reciprocals (k′)2 = 1
k2
of the corresponding parameters.
Let t(s) = w2(s)
w1(s)
where w1(s), w2(s) are linearly independent solutions to the hypergeometric
differential equation (2.2) with (a, b, c) one of(
−1
4
,
5
12
,
1
2
)
,
(
−1
4
,
5
12
,
2
3
)
,
(
−1
2
,
5
6
,
2
3
)
.
The solution to (4.5) is then given by u = 6 d
dt logw1. A similar leading order analysis as before
shows that the leading orders occur at
−6, −9
8
, −15
8
.
This corresponds to solutions of (4.5) given by
u(t) = −6
t
, u(t) = − 9
4t
, u(t) = −15
4t
.
Along with the zero solution u(t) = 0, these correspond respectively (modulo constants of
integration) to solutions of (4.2) given by
H(t) = t−2, H(t) = t
1
2 , H(t) = t−
1
2 , H(t) = t2.
In [6, Proposition 2.2], it is shown that a Legendre transformation takes (1.1) to (4.1). Hence
we may hypothesise that amongst all 3rd order generalised Chazy equations, only those with
the parameters k′ = ±3
2 , k = ±2
3 have in addition solutions that can be obtained from the dual
equation via a Legendre transform.
Proposition 4.1 ([6, Proposition 2.2]). Consider the Legendre transformation
F (q) +H(t) = qt.
Then F (q) satisfies the ODE (1.1) iff H(t) satisfies the ODE (4.2).
Flat (2, 3, 5)-Distributions and Chazy’s Equations 7
Proof. Applying the exterior derivative to the relation gives
(F ′ − t)dq + (H ′ − q)dt = 0,
so that we take F ′ = t, H ′ = q and applying
d
dq
=
1
H ′′
d
dt
we obtain F ′′ = 1
H′′ , F
(3) = − H(3)
(H′′)3 , etc. A computation shows that the 6th order ODE (1.1)
holds for F iff (4.2) holds for H. �
In light of the solutions obtained by solving the generalised Chazy equations, we can pass to
F (q) =
∫∫
e
∫ I(q)
2
dqdqdq, (4.6)
where q = z2(s)
z1(s)
and I(q) = 6 d
dq log z1 are given in Corollary 3.2. This gives
F (q) =
∫∫
(z1)
3dqdq.
Similarly, for the dual equation (4.2) under the Legendre transform we pass to
H(t) =
∫∫
e
∫ 2u(t)
3
dtdtdt,
where t = w2(s)
w1(s)
and u(t) = 6 d
dt logw1 are solutions to (4.5). This gives
H(t) =
∫∫
(w1)
4dtdt.
We have
Lemma 4.2. There exists a Legendre transformation between Chazy’s solutions of (1.2)
and (4.5) given by taking
w1(s) = z
− 3
4
1 , w2(s) = (z1)
− 3
4
∫
(z1)(ż2z1 − ż1z2)ds.
This defines a mapping
q =
z2(s)
z1(s)
7→ t =
w2(s)
w1(s)
=
∫
z1(ż2z1 − ż1z2)ds.
If I(q) = 6 d
dq log z1 solves (1.2) where z1(s) and z2(s) are given in Corollary 3.2, then u(t) =
6 d
dt logw1 solves the dual ODE (4.5). Consequently, if F (q) =
∫∫
(z1)
3dqdq solves (1.1), then
H(t) =
∫∫
(w1)
4dtdt =
∫∫
(z1)
−2(ż2z1 − ż1z2)ds z1(ż2z1 − ż1z2)ds
solves the 6th order ODE (4.2). For the converse, the Legendre transform is given by
z1(s) = w
− 4
3
1 , z2(s) = (w1)
− 4
3
∫
(w1)
2(ẇ2w1 − ẇ1w2)ds.
8 M. Randall
This sends
t =
w2(s)
w1(s)
7→ q =
z2(s)
z1(s)
=
∫
(w1)
2(ẇ2w1 − ẇ1w2)ds.
In particular, if u(t) = 6 d
dt logw1 solves the dual ODE (4.5), then I(q) = 6 d
dq log z1 solves (1.2).
Hence, if H(t) =
∫∫
(w1)
4dtdt solves the 6th order ODE (4.2), then
F (q) =
∫∫
(z1)
3dqdq =
∫∫
(w1)
−2(ẇ2w1 − ẇ1w2)ds (w1)
2(ẇ2w1 − ẇ1w2)ds
solves (1.1).
Proof. We observe that as a consequence of Chazy’s solutions, the Legendre transform in
Proposition 4.1 gives
w2
w1
= t = F ′ =
∫
(z1)
3dq =
∫
z1(ż2z1 − ż1z2)ds,
z2
z1
= q = H ′ =
∫
(w1)
4dt =
∫
(w1)
2(ẇ2w1 − ẇ1w2)ds,
and therefore
ẇ2w1 − ẇ1w2
(w1)2
= z1(ż2z1 − ż1z2),
ż2z1 − ż1z2
(z1)2
= (w1)
2(ẇ2w1 − ẇ1w2).
Together this yields (z1)
3 = (w1)
−4, from which we deduce
w1 = z
− 3
4
1 and w2 = w1
∫
z1(ż2z1 − ż1z2)ds = z
− 3
4
1
∫
z1(ż2z1 − ż1z2)ds.
For the converse, we find
z1 = w
− 4
3
1 and z2 = z1
∫
(w1)
2(ẇ2w1 − ẇ1w2)ds = w
− 4
3
1
∫
(w1)
2(ẇ2w1 − ẇ1w2)ds.
The rest follows from a routine computation. �
In [16, formula (8)], a family of solutions to (4.1) is found to be given by the algebraic curve
(y + f(t))2 = (t− a)(t− b)3,
with a 6= b, and f(t) a quadratic. This gives
y = ±
√
(t− a)(t− b)3 − f(t).
We obtain a solution to (4.1) with
y(3) = ±3
8
(t− b)6(a− b)3
(y + f)5
= ± 3(a− b)3
8(t− a)2(y + f)
.
We find that for this solution, it yields
u(t) =
3
4
5b+ 3a− 8t
(t− a)(t− b)
= − 15
4(t− a)
− 9
4(t− b)
as a solution to the generalized Chazy’s equation with parameter k2 = 9
4 . This corresponds to
the solution given by Chazy in (3.2). It will be interesting to determine the solutions of (4.5)
from the general solution given by [16, formula (13)].
Flat (2, 3, 5)-Distributions and Chazy’s Equations 9
5 First order system and different parametrisations
of Chazy’s equations
In this section, we first show that the generalised Chazy equation is equivalent to solving a third
order differential equation involving the Schwarzian derivative and a potential term V (s). It is
well-known that solutions to the generalised Chazy equation (3.1) can be rewritten as a first order
system. For further details, see [2]. The first order system provides different parametrisations of
the solutions, in addition to the one given by (2.3). We compute the solutions to the generalised
Chazy equation (1.2) with k = ±2
3 for the different parametrisations below and present them
in Tables 1, 2 and 3. We also show how these solutions are related to one another by algebraic
transformation of hypergeometric functions.
Let Ω1, Ω2, Ω3 be functions of q. Let dot denote differentiation with respect to q. Then
consider
Ω̇1 = Ω2Ω3 − Ω1(Ω2 + Ω3) + τ2,
Ω̇2 = Ω3Ω1 − Ω2(Ω3 + Ω1) + τ2,
Ω̇3 = Ω1Ω2 − Ω3(Ω1 + Ω2) + τ2, (5.1)
where
τ2 = α2(Ω1 − Ω2)(Ω3 − Ω1) + β2(Ω2 − Ω3)(Ω1 − Ω2) + γ2(Ω3 − Ω1)(Ω2 − Ω3)
and α, β, γ are constants. Introducing the parameter
s(q) =
Ω1 − Ω3
Ω2 − Ω3
,
we find that
Ω1 = −1
2
d
dq
log
ṡ
s(s− 1)
, Ω2 = −1
2
d
dq
log
ṡ
s− 1
, Ω3 = −1
2
d
dq
log
ṡ
s
.
The system of equations (5.1) are satisfied iff
{s, q}+
ṡ2
2
V (s) = 0, (5.2)
where
{s, q} =
d
dq
(
s̈
ṡ
)
− 1
2
(
s̈
ṡ
)2
is the Schwarzian derivative of s(q) and
V (s) =
1− β2
s2
+
1− γ2
(s− 1)2
+
β2 + γ2 − α2 − 1
s(s− 1)
.
Switching independent and dependent variables in (5.2), we have
{s, q} = −(ṡ)2{q, s},
so that the dual of (5.2) is
{q, s} − 1
2
V (s) = 0.
10 M. Randall
The general solution is given by
q(s) =
u2(s)
u1(s)
where u1, u2 are linearly independent solutions of the second order ODE
u′′ +
1
4
V (s)u = 0. (5.3)
The general solution of (5.3) suggests taking
u(s) = (s− 1)
1−γ
2 s
1−β
2 z(s)
(cf. [3]) to transform (5.3) into the hypergeometric differential equation (2.2) with
a =
1
2
(1− α− β − γ), b =
1
2
(1 + α− β − γ), c = 1− β.
In [2], it was determined that taking
y = −2(Ω1 + Ω2 + Ω3) =
d
dq
log
ṡ3
s2(s− 1)2
(5.4)
gives solutions to the generalised Chazy equation (3.1) with variable q whenever α = β = γ = 2
k
or α = 2
k , β = γ = 1
3 and its cyclic permutations. For α = β = γ = 2
k this gives (a, b, c) =(
k−6
2k ,
k−2
2k ,
k−2
k
)
. For α = 2
k , β = γ = 1
3 with cyclic permutations, this gives respectively
(a, b, c) =
(
k − 6
6k
,
k + 6
6k
,
2
3
)
,
(
k − 6
6k
,
k − 2
2k
,
k − 2
k
)
,
(
k − 6
6k
,
k − 2
2k
,
2
3
)
,
with only the first coinciding with the list in Proposition 3.1. This suggests that the solutions
to (5.2) are more general than the solution of the form y = 6 d
dq log z1 given by Chazy [13]. We
can express Chazy’s solution in terms of s(q) as follows. A computation of the Wronskian of
linearly independent solutions z1, z2 to (2.2) gives
W (z1, z2) = z1ż2 − z2ż1 = w0(s− 1)c−a−b−1s−c
for some non-zero constant w0. The latter equality holds by solving the first order differential
equation
W ′ = −c− (a+ b+ 1)s
s(1− s)
W.
See for instance [4]. From q(s) = z2(s)
z1(s)
, we find that d
dq = (z1)2
z1ż2−z2ż1
d
ds . Applying this derivative
to s(q), we obtain
s′(q) =
d
dq
s(q) =
(z1)
2
z1ż2 − z2ż1
=
(z1)
2
W (z1, z2)
.
Hence
s′′(q) =
d
dq
s′(q) = s′(q)
d
ds
(
(z1)
2
W (z1, z2)
)
= s′(q)
(
2z1ż1
W (z1, z2)
− (z1)
2W ′(z1, z2)
W (z1, z2)2
)
= s′(q)
(
2s′(q)
ż1
z1
− s′(q)W
′(z1, z2)
W (z1, z2)
)
= s′(q)
(
2s′(q)
ż1
z1
+ s′(q)
c− (a+ b+ 1)s(q)
s(q)(1− s(q))
)
= s′(q)
(
2s′(q)
ż1
z1
+ s′(q)
c(1− s(q))− (a+ b+ 1− c)s(q)
s(q)(1− s(q))
)
,
Flat (2, 3, 5)-Distributions and Chazy’s Equations 11
and we get
s̈
ṡ
= 2ṡ
ż1
z1
+ c
ṡ
s
− (a+ b+ 1− c) ṡ
1− s
.
Therefore we have
y = 6
d
dq
log z1 = 6
(z1)
2
W (z1, z2)
ż1
z1
= 6ṡ
ż1
z1
= 3
s̈
ṡ
− 3c
ṡ
s
+ 3(a+ b+ 1− c) ṡ
1− s
= 3
d
dq
log ṡ− 3c
d
dq
log s− 3(a+ b+ 1− c) d
dq
log(1− s)
= 3
d
dq
log
ṡ
sc(1− s)a+b+1−c =
1
2
d
dq
log
ṡ6
s6c(1− s)6(a+b+1−c) .
A comparison of Chazy’s formula for y = 6 d
dq log z1 with the formula for y in (5.4) suggests
taking c = 2
3 , a+ b = 1
3 . This is satisfied by (a, b, c) =
(
k−6
6k ,
k+6
6k ,
2
3
)
in Proposition 3.1.
For k = 2
3 as in (1.2), we get the following solutions for u given in Table 1. For this
Table 1.
(α, β, γ) (a, b, c) General solution to u′′ + 1
4V (s)u = 0
(3, 3, 3) (−4,−1,−2) c1
2s−1
s(s−1) + c2
s2(s−2)
s−1(
3, 13 ,
1
3
) (
−4
3 ,
5
3 ,
2
3
)
c1(3s− 2)s
2
3 (s− 1)
1
3 + c2(3s− 1)s
1
3 (s− 1)
2
3(
1
3 , 3,
1
3
) (
−4
3 ,−1,−2
)
c1
2s−3
s (s− 1)
1
3 + c2
s−3
s (s− 1)
2
3(
1
3 ,
1
3 , 3
) (
−4
3 ,−1, 23
)
c1
2s+1
s−1 s
1
3 + c2
s+2
s−1s
2
3
Chazy parameter, the hypergometric series truncate and the solutions to (5.3) can be given by
elementary functions. We have
2F1(−4,−1;−2; s) = 1− 2s, 2F1
(
−4
3
,
5
3
;
2
3
; s
)
=
3s2 − 4s+ 1
(1− s)
2
3
,
2F1
(
−4
3
,−1;−2; s
)
= 1− 2
3
s, 2F1
(
−4
3
,−1;
2
3
; s
)
= 1 + 2s.
Moreover, the solutions in each row are related to one another by algebraic transformations
of hypergeometric functions. The solutions in the second, third and fourth rows of Table 1
can be obtained from the first by a cubic transformation of hypergeometric functions (see [23,
formula (23)]). Explicitly, to show how the solution in the third row is related to the first, let ω
be a cube root of unity (solution to ω2 + ω + 1 = 0) and consider the map
s 7→ t =
3(2ω + 1)s(s− 1)
(s+ ω)3
and the relation
z̃(t) = (1 + ωs)−4z(s).
Then z̃(t) is a solution to the hypergeometric differential equation (2.2) with (a, b, c) =
(
−4
3 ,−1,
−2
)
iff z(s) solves (2.2) with (a, b, c) = (−4,−1,−2). The solutions in the last three rows
are related to one another by fractional linear transformations. The symmetry s 7→ 1 − s
interchanges β and γ in V (s) and hence transforms the solution in the 3rd row to the solution
12 M. Randall
given in the 4th row while s 7→ t = s
s−1 and the relation z̃(t) = (1 − s)−
4
3 z(s) transforms the
solution in the 4th row to those in the 2nd.
A different parametrisation of Chazy’s equations (cf. [10]) is also given by
y = −Ω1 − 2Ω2 − 3Ω3 =
1
2
d
dq
log
ṡ6
s4(s− 1)3
. (5.5)
A comparison with Chazy’s formula (2.3) yields (a, b, c) =
(
k−6
12k ,
k+6
12k ,
2
3
)
from Proposition 3.1.
The solution of the form (5.5) solves the generalised Chazy equation (3.1) whenever (α, β, γ)
in (5.2) is given by(
1
k
,
1
3
,
1
2
)
,
(
1
k
,
2
k
,
1
2
)
or
(
1
k
,
1
3
,
3
k
)
.
The solution to (5.2) with (α, β, γ) =
(
1
k ,
1
3 ,
1
2
)
is given by the Schwarz function J (see [2, 3]).
Considering the symmetry s = 1−K brings (5.5) to
y = −Ω1 − 3Ω2 − 2Ω3 =
1
2
d
dq
log
−K̇6
(1−K)4K3
(5.6)
and comparing with Chazy’s formula gives (a, b, c) =
(
k−6
12k ,
k+6
12k ,
1
2
)
from Proposition 3.1. The
solution of the form (5.6) solves (3.1) whenever (α, β, γ) in (5.2) is given by(
1
k
,
1
2
,
1
3
)
,
(
1
k
,
1
2
,
2
k
)
or
(
1
k
,
3
k
,
1
3
)
.
The symmetry s = 1 −K permutes β and γ in the formula for V (s) in (5.2). For k = 2
3 , the
solutions to (5.3) with the parametrisation by (5.5) are presented in Table 2.
Table 2.
(α, β, γ) (a, b, c) General solution to u′′ + 1
4V (s)u = 0(
3
2 ,
1
3 ,
1
2
) (
5
6 ,−
2
3 ,
2
3
)
c1(s− 1)
1
4 s
1
2P
1
3
1 (
√
1− s) + c2(s− 1)
1
4 s
1
2Q
1
3
1 (
√
1− s)(
3
2 , 3,
1
2
) (
−1
2 ,−2,−2
)
c1
(s−1)
3
4
s + c2
(s−1)
1
4
s (s2 + 4s− 8)(
3
2 ,
1
3 ,
9
2
) (
−7
6 ,−
8
3 ,
2
3
)
c1s
2
3 (s− 1)
11
4 2F1
(
13
6 ,
11
3 ; 4
3 ; s
)
+ c2s
1
3 (s− 1)
11
4 2F1(
11
6 ,
10
3 ; 2
3 ; s)
We also note that we have
2F1
(
5
6
,−2
3
;
2
3
; s
)
=
Γ
(
2
3
)3√
3
π
P
(− 1
3
,− 1
2)
2
3
(1− 2s),
2F1
(
−1
2
,−2;−2; s
)
=
1
8
(
8− 4s− s2
)
,
2F1
(
−7
6
,−8
3
;
2
3
; s
)
=
10
7
Γ
(
2
3
)3√
3
π
P
(− 1
3
,− 9
2)
8
3
(1− 2s).
The algebraic transformations relating the solutions between the rows are given as follows. The
map that takes the solution from the second row to the solution in the first row of Table 2
is a composition of fractional linear transformations and cubic transformation due to Goursat
(see [23, formulas (20)–(22)] and [17, formula (123)]). To see this, we first consider the map
s 7→ t =
s(9− s)2
(s+ 3)3
Flat (2, 3, 5)-Distributions and Chazy’s Equations 13
and the relation
z̃(t) =
(
1 +
s
3
)−2
z(s).
Then z̃(t) satisfies (2.2) with (a, b, c) =
(
−2
3 ,−
1
3 ,
1
2
)
iff z(s) satisfies (2.2) with (a, b, c) =(
−2,−1
2 ,
1
2
)
. Now
2F1
(
−2
3
,−1
3
;
1
2
; t
)
= (1− t)
2
3 2F1
(
−2
3
,
5
6
;
1
2
;
t
t− 1
)
while the map s 7→ 1 − s takes the differential equation (2.2) with (a, b, c) =
(
−2
3 ,
5
6 ,
1
2
)
to
(a, b, c) =
(
−2
3 ,
5
6 ,
2
3
)
. Moreover, we have
2F1
(
−2,−1
2
;
1
2
; 1− s
)
=
1
3
(
8− 4s− s2
)
=
8
3
2F1
(
−2,−1
2
;−2; s
)
,
so that all together the composition gives the map
s 7→ t̃ = −(s− 4)3
27s2
and the relation
z̃(t̃) = s−
4
3 z(s).
Thus z̃(t̃) satisfies (2.2) with (a, b, c) =
(
−2
3 ,
5
6 ,
2
3
)
iff z(s) satisfies (2.2) with (a, b, c) =
(
−2,
−1
2 ,−2
)
.
To obtain the solution given in the third row from those in the first row requires a trans-
formation of degree 4 (see [23, formulas (25)–(27)] and [17, equation (131)]). Consider the
map
s 7→ t = − s(s+ 8)3
64(1− s)3
and the relation
z̃(t) = (1− s)
5
2 z(s).
Then z̃(t) satisfies (2.2) with (a, b, c) =
(
5
6 ,−
2
3 ,
2
3
)
iff z(s) satisfies (2.2) with (a, b, c) =
(
11
6 ,
10
3 ,
2
3
)
.
Finally, we use the Euler transformation,
2F1
(
−7
6
,−8
3
;
2
3
; s
)
= (1− s)
9
2 2F1
(
11
6
,
10
3
;
2
3
; s
)
to obtain the solution in the 3rd row.
There is furthermore a degree 6 transformation relating the solution in the first row of Table 1
to the solution in the first row of Table 2 (cf. [23, equation (28)], [17, equation (134)] and [1,
Table 1]). Consider the map
s 7→ t =
27s2(s− 1)2
4(s2 − s+ 1)3
and the relation
z̃(t) = (1− s+ s2)−2z(s).
14 M. Randall
Then z̃(t) satisfies (2.2) with (a, b, c) =
(
−1
3 ,−
2
3 ,−
1
2
)
iff z(s) satisfies (2.2) with (a, b, c) =
(−4,−1,−2). Furthermore s 7→ 1 − s takes the solution to (2.2) with (a, b, c) =
(
−1
3 ,−
2
3 ,−
1
2
)
to the solution with (a, b, c) =
(
−1
3 ,−
2
3 ,
1
2
)
and an Euler transformation followed by s 7→ 1 − s
again takes this to the solution with (a, b, c) =
(
5
6 ,−
2
3 ,
2
3
)
as given in row 1 of Table 2.
Finally, consider the parametrisation given by
y = −4Ω1 − Ω2 − Ω3 =
d
dq
log
ṡ3
(s− 1)
5
2 s
5
2
. (5.7)
We find that (α, β, γ) is one of(
4
k
,
1
k
,
1
k
)
,
(
2
3
,
1
k
,
1
k
)
,
(
2
3
,
1
6
,
1
6
)
.
For (α, β, γ) =
(
4
k ,
1
k ,
1
k
)
, we find that (a, b, c) =
(
k−6
2k ,
k+2
2k ,
k−1
k
)
. For k = 2
3 , we obtain
(a, b, c) = (−4, 2,−1
2). Let us relate the solution to the differential equation (2.2) with (a, b, c) =
(−4, 2,−1
2) to the solution given in the second row of Table 2. For (α, β, γ) =
(
1
2 ,
1
k ,
2
k
)
, we find
that (a, b, c) =
(
k−6
4k ,
3k−6
4k , k−1k
)
. We have
2F1
(
1
2
− 3
k
,
1
2
+
1
k
; 1− 1
k
; s
)
= 2F1
(
1
4
− 3
2k
,
1
4
+
1
2k
; 1− 1
k
; 4s(1− s)
)
= (1− 4s(1− s))−
1
4
+ 3
2k 2F 1
(
1
4
− 3
2k
,
3
4
− 3
2k
; 1− 1
k
;
4s(1− s)
4s(1− s)− 1
)
and thus for k = 2
3 , we have
2F1
(
−4, 2;−1
2
; s
)
= (1− 4s(1− s))2 2F1
(
−2,−3
2
;−1
2
;
4s(1− s)
4s(1− s)− 1
)
.
Again s 7→ 1 − s brings the solution to (2.2) with (a, b, c) =
(
−2,−3
2 ,−
1
2
)
to the solution with
(a, b, c) =
(
−2,−3
2 ,−2
)
and an Euler transform gives the solution with (a, b, c) =
(
−2,−1
2 ,−2
)
.
For (α, β, γ) =
(
2
3 ,
1
k ,
1
k
)
, we find that (a, b, c) =
(
k−6
6k ,
5k−6
6k , k−1k
)
. For k = 2
3 , this gives
(a, b, c) =
(
−4
3 ,−
2
3 ,−
1
2
)
. We have a degree 2 transformation of the solution to (2.2) with this
value to the solution in the first row of Table 2 given by the map
s 7→ t = − 1
4s(s− 1)
and the relation
z̃(t) =
1
4
(s(1− s))−
2
3 z(s).
We have z̃(t) satisfying (2.2) with (a, b, c) =
(
−2
3 ,
5
6 ,
2
3
)
iff z(s) satisfies (2.2) with (a, b, c) =(
−4
3 ,−
2
3 ,−
1
2
)
. Let us summarise the solutions to (5.3) with parametrisation given by (5.7) in
Table 3.
We also note that we have
2F1
(
−4, 2;−1
2
; s
)
= −128
5
P
(− 3
2
,− 3
2)
4 (1− 2s) = −128s4 + 256s3 − 144s2 + 16s+ 1,
2F1
(
−4
3
,−2
3
;−1
2
; s
)
= −16
27
π22
2
3
Γ
(
2
3
)3P (− 3
2
,− 3
2)
4
3
(1− 2s),
2F1
(
0,
2
3
;
5
6
; s
)
= 1.
Flat (2, 3, 5)-Distributions and Chazy’s Equations 15
Table 3.
(α, β, γ) (a, b, c) General solution to u′′ + 1
4V (s)u = 0(
6, 32 ,
3
2
) (
−4, 2,−1
2
)
c1(2s− 1)(s(s− 1))
5
4 + c2
128s4−256s3+144s2−16s−1
(s(s−1))
1
4(
2
3 ,
3
2 ,
3
2
) (
−4
3 ,−
2
3 ,−
1
2
)
c1(s(s− 1))
5
4 2F1
(
5
3 ,
7
3 ; 5
2 ; s
)
+ c2
(s−1)
5
4
s
1
4
2F1
(
1
6 ,
5
6 ;−1
2 ; s
)
(
2
3 ,
1
6 ,
1
6
) (
0, 23 ,
5
6
)
c1(s(s− 1))
7
12 2F1
(
1
3 , 1; 7
6 ; s
)
+ c2s
5
12 (1− s)
5
12
Comparing the parametrisation (5.7) with solutions of the form (2.3), we obtain (a, b, c) =(
0, 23 ,
5
6
)
. For this solution given in the third row, we obtain solutions of the form y(q) = −6
q .
Up to fractional linear transformations in the variable s, we have 7 classes of solutions to (1.2)
determined by the solutions to (5.2). They are given by the solutions with (α, β, γ) either (3, 3, 3)
or
(
3, 13 ,
1
3
)
for I(q) given by (5.4), (α, β, γ) one of
(
3
2 ,
1
3 ,
1
2
)
,
(
3
2 ,
1
3 ,
9
2
)
or
(
3
2 , 3,
1
2
)
for I(q) given
by (5.5) and (α, β, γ) either
(
6, 32 ,
3
2
)
or
(
2
3 ,
3
2 ,
3
2
)
for I(q) given by (5.7).
When α = β = γ = 0, we have τ = 0. The first order system (5.1) is the classical Darboux–
Halphen system and y = −2(Ω1 + Ω2 + Ω3) satisfies Chazy’s equation (2.1). This system arises
as the anti-self-dual Ricci-flat equations for Bianchi-IX metrics (see [7, 22]).
6 Examples of flat (2, 3, 5)-distributions
In [19], Nurowski associated to (2, 3, 5)-distributions a conformal class of metrics of signatu-
re (2, 3). The fundamental curvature invariant of (2, 3, 5)-distributions appears as the Weyl ten-
sor of Nurowski’s metric. For (2, 3, 5)-distributions DF (q) determined by a single function F (q),
the metric is described in [6, 19]. The distribution DF (q) on Mxyzpq is encoded by the annihilator
of the three 1-forms
ω1 = dy − pdx, ω2 = dp− qdx, ω3 = dz − F (q)dx, (6.1)
and supplemented by the 1-forms
ω4 = dq, ω5 = dx.
The coframe on Mxyzpq is given by
θ1 = ω1 −
1
F ′′
(
F ′ω2 − ω3
)
, θ2 =
1
F ′′
(
F ′ω2 − ω3
)
,
θ3 =
(
1− F ′F (3)
4(F ′′)2
)
ω2 +
F (3)
4(F ′′)2
ω3,
θ4 =
(
7(F (3))2 − 4F ′′F (4)
40(F ′′)3
)(
F ′ω2 − ω3
)
+ ω4 − ω5, θ5 = −ω4, (6.2)
(cf. [6]) and Nurowski’s metric
gDF (q)
= 2θ1θ5 − 2θ2θ4 +
4
3
θ3θ3 (6.3)
has vanishing Weyl tensor (and hence conformally flat) iff DF (q) has the split real form of G2
as its group of local symmetries. There is a more elegant way to present Nurowski’s metric for
16 M. Randall
the distribution DF (q). Equivalently, we can encode the distribution by
ω̃1 = ω1 = dy − pdx,
ω̃2 =
1
F ′′
(F ′ω2 − ω3) =
1
F ′′
(
F ′(dp− qdx)− (dz − F (q)dx)
)
,
ω̃3 = ω2 = dp− qdx, (6.4)
with ω̃4 = ω4 and ω̃5 = ω5. The coframe is then given by
θ1 = ω̃1 − ω̃2, θ2 = ω̃2, θ3 = ω̃3 −
F (3)
4F ′′
ω̃2,
θ4 =
(
7(F (3))2 − 4F ′′F (4)
40(F ′′)2
)
ω̃2 + ω̃4 − ω̃5, θ5 = −ω̃4.
Let us take
F (q) =
∫∫
e
1
2
∫
I(q)dqdqdq,
as in (4.6). This gives F ′′ = e
1
2
∫
I(q)dq. Nurowski’s metric now has a very simple form given by
gDF (q)
= 2θ1θ5 − 2θ2θ4 +
4
3
θ3θ3
= 2ω̃2ω̃5 − 2ω̃1ω̃4 +
4
3
(ω̃3)
2 − I
3
ω̃2ω̃3 +
1
10
(
I ′ − I2
6
)
(ω̃2)
2. (6.5)
The Ricci tensor of the above metric is
Rabθ
aθb =
9
120
(
6I ′ − I2
)
(ω̃4)
2.
We can consider conformal rescalings of the metric such that ĝDF (q)
= Ω2gDF (q)
is Ricci flat. It
turns out that if we take Ω = ν(q)−1 > 0, then the Ricci tensor of the rescaled metric ĝDF (q)
is
given by
Rabθ
aθb =
3
40ν
(
40ν ′′ +
(
6I ′ − I2
)
ν
)
(ω̃4)
2, (6.6)
so the appropriate conformal scale ν(q) can be found by solving the differential equation in (6.6)
(cf. [24, Proposition 35]). In the first part of this section we consider the conformally flat met-
rics (6.5) obtained by solving (1.2) using the solution (2.3). Next, we then consider the solutions
obtained from different parametrisations of the generalised Chazy equation given by (5.4), (5.5)
and (5.7). We also consider conformally flat metrics obtained from solving the Legendre trans-
form of (1.2). This involves computing the coframe for the metric under the Legendre transform.
Finally, we consider the metrics obtain from Chazy’s solutions given by (3.2). The metrics asso-
ciated to (2, 3, 5)-distributions DF (q) of the form F (q) = qm where m ∈
{
−1, 13 ,
2
3 , 2
}
are given
in [18].
6.1 Chazy’s solution
In order to express Nurowski’s metric associated to flat (2, 3, 5)-distributions obtained from
solving (1.2), we have to switch independent variable s and dependent variable q. In other
words we pass to coordinates (x, y, z, p, s) with q(s) = z2(s)
z1(s)
where z1(s), z2(s) are given in
Corollary 3.2.
Flat (2, 3, 5)-Distributions and Chazy’s Equations 17
Let us first consider solutions of the form
I(q(s)) = 6
d
dq
log z1(s)
as in Corollary 3.2. Observe that∫
I(q)dq = 6 log z1
and so
F (q) =
∫∫
e
1
2
∫
I(q)dqdqdq =
∫∫
(z1)
3dqdq.
For this parametrisation we have
dq =
z1ż2 − z2ż1
(z1)2
ds,
so that
F (q(s)) =
∫ (∫
z1(z1ż2 − z2ż1)ds
)
z1ż2 − z2ż1
(z1)2
ds.
Let us denote
K(s) =
∫
z1(z1ż2 − z2ż1)ds.
Theorem 6.1. Let z1(s), z2(s) be two linearly independent solutions to (2.2) with (a, b, c) given
by one of the list in Corollary 3.2. Let DC denote the (2, 3, 5)-distribution on Mxyzps associated
to the annihilator of
ω̃1 = dy − pdx,
ω̃2 =
K(s)
(z1)3
(
dp− z2
z1
dx
)
− 1
(z1)3
(
dz −
(∫
K(s)
z1ż2 − z2ż1
(z1)2
ds
)
dx
)
,
ω̃3 = dp− z2
z1
dx.
Supplement by the 1-forms
ω̃4 =
z1ż2 − z2ż1
(z1)2
ds, ω̃5 = dx.
Then Nurowski’s metric (6.5)
gDC = 2ω̃2ω̃5 − 2ω̃1ω̃4 +
4
3
(ω̃3)
2 − 2z1ż1
z1ż2 − z2ż1
ω̃2ω̃3
+
3
5
(
(z1)
3z̈1
(z1ż2 − z2ż1)2
− (z1)
3ż2(z1z̈2 − z2z̈1)
(z1ż2 − z2ż1)3
)
(ω̃2)
2
has vanishing Weyl tensor (and hence conformally flat) and DC has the split real form of G2 as
its group of local symmetries.
18 M. Randall
Let us provide an explicit example given by Corollary 3.2. It turns out for the values of
(a, b, c) obtained in Corollary 3.2, the solutions can be given by elementary functions. For
(a, b, c) =
(
−2
3 ,
5
6 ,
1
2
)
, the solutions to the hypergeometric differential equation (2.2) are given by
z(s) = µ(s− 1)
1
6P
1
3
1 (
√
s) + ν(s− 1)
1
6Q
1
3
1 (
√
s),
where Pm
` and Qm
` are the associated Legendre functions. This suggest passing further to the
variable r =
√
s, in which case (2.2) with (a, b, c) =
(
−2
3 ,
5
6 ,
1
2
)
becomes
1
4
(1− r2)z′′(r)− 1
3
rz′(r) +
5
9
z(r) = 0. (6.7)
The general solution is now given by the elementary functions
z(r) = c1(r − 1)
1
3 (3r + 1) + c2(r + 1)
1
3 (3r − 1).
There is thus a 4-dimensional space of solutions given by
z1(r) = c1(r − 1)
1
3 (3r + 1) + c2(r + 1)
1
3 (3r − 1),
z2(r) = c3(r − 1)
1
3 (3r + 1) + c4(r + 1)
1
3 (3r − 1),
where c1c4− c2c3 6= 0. We also note that in the case (a, b, c) =
(
−2
3 ,
5
6 ,
2
3
)
, the change of variable
r =
√
1− s brings (2.2) to (6.7), so that the two hypergeometric ODEs with (a, b, c) =
(
−2
3 ,
5
6 ,
1
2
)
and
(
−2
3 ,
5
6 ,
2
3
)
can be brought to the same equation (6.7) by a coordinate transformation.
Moreover, since
2F1
(
−2
3
,
5
6
;
2
3
; 4s(1− s)
)
= 2F1
(
−4
3
,
5
3
;
2
3
; s
)
,
if we take r =
√
1− 4s(1− s) = 2s − 1, we can pass from (2.2) with (a, b, c) =
(
−4
3 ,
5
3 ; 2
3
)
to
equation (6.7). Also compare with the second row of Table 1.
We now pass to coordinates on Mxyzpr and take for a simple example
z1(r) = c1(r − 1)
1
3 (3r + 1), z2(r) = c2(r + 1)
1
3 (3r − 1)
where c1, c2 are non-zero constants. We also make use of
ds = 2rdr.
Here we have K(r) = −16(c1)
2c2(r − 1)
2
3 (r + 1)
1
3 . We have
Proposition 6.2. Let c1, c2 be non-zero constants. Let D0 denote the (2, 3, 5)-distribution on
Mxyzpr associated to the annihilator of
ω̃1 = dy − pdx,
ω̃2 =
−16c2(r + 1)
1
3
c1(r − 1)
1
3 (3r + 1)3
dp− 1
(c1)3(r − 1)(3r + 1)3
dz +
16(c2)
2(r + 1)
2
3
(c1)2(r − 1)
2
3 (3r + 1)3
dx,
ω̃3 = dp− c2(r + 1)
1
3 (3r − 1)
c1(r − 1)
1
3 (3r + 1)
dx,
and supplemented by the 1-forms
ω̃4 = − 16c2
3c1(3r + 1)2(r − 1)
4
3 (r + 1)
2
3
dr, ω̃5 = dx.
Flat (2, 3, 5)-Distributions and Chazy’s Equations 19
Then Nurowski’s metric (6.5)
gD0 = 2ω̃2ω̃5 − 2ω̃1ω̃4 +
4
3
(ω̃3)
2 +
c1(3r − 2)(3r + 1)(r − 1)
1
3 (r + 1)
2
3
2c2
ω̃2ω̃3
+
3(c1)
2(r − 1)
5
3 (3r + 1)4(r + 1)
1
3
64(c2)2
(ω̃2)
2
has vanishing Weyl tensor (and hence conformally flat) and D0 has the split real form of G2 as
its group of local symmetries. The Ricci tensor for this metric is
Rabθ
aθb =
6
r2 − 1
drdr.
Rescaling this metric by
Ω =
1
ν
=
4
3
(3r + 1)(r − 1)
1
3
a1(r − 1)
1
3 − a2(r + 1)
1
3
,
where a1 and a2 are constants, the conformally rescaled metric ĝD0 = Ω2gD0 given by
ĝD0 =
16(3r + 1)2(r − 1)
2
3
9(a1(r − 1)
1
3 − a2(r + 1)
1
3 )2
(
2ω̃2ω̃5 − 2ω̃1ω̃4 +
4
3
(ω̃3)
2
+
c1(3r − 2)(3r + 1)(r − 1)
1
3 (r + 1)
2
3
2c2
ω̃2ω̃3 +
3(c1)
2(r − 1)
5
3 (3r + 1)4(r + 1)
1
3
64(c2)2
(ω̃2)
2
)
is both Ricci-flat and conformally flat.
6.2 Other parametrisations of the generalised Chazy equation
Instead of choosing I(q) = 6 d
dq log z1, we can consider the parametrisation
I(q) =
d
dq
log
ṡ3
s2(s− 1)2
given in (5.4). In this case
F (q) =
∫∫
e
1
2
∫
I(q)dqdqdq =
∫∫
ṡ
3
2
s(s− 1)
dqdq
and the corresponding metric associated to the (2, 3, 5)-distribution DF (q) gives the following
Theorem 6.3. Let s(q) be a solution to (5.2) with (α, β, γ) given by one of
(3, 3, 3),
(
3,
1
3
,
1
3
)
,
(
1
3
, 3,
1
3
)
,
(
1
3
,
1
3
, 3
)
.
Let Ds denote the (2, 3, 5)-distribution on Mxyzpq associated to the annihilator of
ω1 = dy − pdx, ω2 = dp− qdx, ω3 = dz −
(∫∫
ṡ
3
2
s(s− 1)
dqdq
)
dx.
Supplement by the 1-forms
ω4 = dq, ω5 = dx,
20 M. Randall
and take the coframe on Mxyzpq to be given by (θ1, θ2, θ3, θ4, θ5) as in (6.2) where
F (q) =
∫∫
ṡ
3
2
s(s− 1)
dqdq.
Then Nurowski’s metric
gDs = 2θ1θ5 − 2θ2θ4 +
4
3
θ3θ3
has vanishing Weyl tensor (and hence conformally flat) and Ds has the split real form of G2 as
its group of local symmetries.
To obtain explicit examples, it is useful to switch the independent variable q and the depen-
dent variable s. We pass to the variables (x, y, z, p, s) with q = u2(s)
u1(s)
where u1 and u2 are linearly
independent solutions of (5.3) given in Table 1. Note that up to fractional linear transforma-
tions in the variable s, we only need to consider the solutions to (5.2) with the values of (α, β, γ)
given by either (3, 3, 3) or
(
3, 13 ,
1
3
)
for the parametrisation given by (5.4). Note the symmetry
permuting β and γ.
A computation shows that W (u1, u2) = u1u̇2 − u̇1u2 is constant, which we can normalise to
set W (u1, u2) = 1. We have
dq =
u1u̇2 − u2u̇1
(u1)2
ds =
1
(u1)2
ds
and
ṡ =
1
q′(s)
= (u1)
2,
so that
F (q(s)) =
∫ (∫
(u1)
3
s(s− 1)
1
(u1)2
ds
)
1
(u1)2
ds =
∫ (∫
u1
s(s− 1)
ds
)
1
(u1)2
ds.
Let us denote
K(s) =
∫
u1
s(s− 1)
ds.
Theorem 6.4. Let u1(s), u2(s) be two linearly independent solutions to (5.3) subject to the
constraint W (u1, u2) = 1 with (α, β, γ) given by Table 1. Let Ds denote the (2, 3, 5)-distribution
on Mxyzps associated to the annihilator of
ω1 = dy − pdx, ω2 = dp− u2(s)
u1(s)
dx, ω3 = dz −
(∫
K(s)
(u1)2
ds
)
dx.
We pass to the annihilator 1-forms given by (6.4) to obtain
ω̃1 = ω1 = dy − pdx, ω̃2 =
s(s− 1)
(u1)3
(Kω2 − ω3), ω̃3 = ω2 = dp− u2(s)
u1(s)
dx,
with
ω̃4 = ω4 =
1
(u1)2
ds, ω̃5 = ω5 = dx.
Flat (2, 3, 5)-Distributions and Chazy’s Equations 21
Take the coframe on Mxyzps to be given by
θ1 = ω̃1 − ω̃2, θ2 = ω̃2, θ3 = ω̃3 −
(u1)
2
4s(s− 1)
(
3
u̇1
u1
s(s− 1)− (2s− 1)
)
ω̃2,
θ4 =
(u1)
4
40
(
4s2 − 4s− 1
s2(s− 1)2
+ 3V (s)− 10(2s− 1)
s(s− 1)
u̇1
u1
+ 15
(
u̇1
u1
)2
)
ω̃2 + ω̃4 − ω̃5,
θ5 = −ω̃4.
Then Nurowski’s metric
gDs = 2θ1θ5 − 2θ2θ4 +
4
3
θ3θ3
= 2ω̃2ω̃5 − 2ω̃1ω̃4 +
4
3
(ω̃3)
2 +
2u1(3s(1− s)u′1 + (2s− 1)u1)
3s(s− 1)
ω̃2ω̃3
− (u1)
4(9V (s)s2(s− 1)2 − 8(s2 − s+ 1))
60(s− 1)2s2
(ω̃2)
2
has vanishing Weyl tensor (and hence conformally flat) and Ds has the split real form of G2 as
its group of local symmetries.
The analogous results hold for I(q) given by the formulas in (5.5), (5.6) and (5.7). If we take
I(q) =
d
dq
log
ṡ3
s2(s− 1)
3
2
,
as in (5.5), we have the corresponding (2, 3, 5)-distribution associated to
F (q) =
∫∫
ṡ
3
2
s(s− 1)
3
4
dqdq.
Theorem 6.5. Let s(q) be a solution to (5.2) with (α, β, γ) given by
(
3
2 ,
1
3 ,
1
2
)
,
(
3
2 , 3,
1
2
)
or(
3
2 ,
1
3 ,
9
2
)
. Let Ds1 denote the (2, 3, 5)-distribution on Mxyzpq associated to the annihilator of
ω1 = dy − pdx, ω2 = dp− qdx, ω3 = dz −
(∫∫
ṡ
3
2
s(s− 1)
3
4
dqdq
)
dx.
Supplement by the 1-forms
ω4 = dq, ω5 = dx,
and take the coframe on Mxyzpq to be given by (θ1, θ2, θ3, θ4, θ5) as in (6.2). Then Nurowski’s
metric (6.3) has vanishing Weyl tensor (and hence conformally flat) and Ds1 has the split real
form of G2 as its group of local symmetries.
Similarly, for I(q) given by (5.7), that is to say
I(q) =
d
dq
log
ṡ3
s
5
2 (s− 1)
5
2
,
we have the corresponding (2, 3, 5)-distribution associated to
F (q) =
∫∫
ṡ
3
2
s
5
4 (s− 1)
5
4
dqdq.
22 M. Randall
Theorem 6.6. Let s(q) be a solution to (5.2) with (α, β, γ) given by
(
6, 32 ,
3
2
)
,
(
2
3 ,
3
2 ,
3
2
)
or(
2
3 ,
1
6 ,
1
6
)
. Let Ds2 denote the (2, 3, 5)-distribution on Mxyzpq associated to the annihilator of
ω1 = dy − pdx, ω2 = dp− qdx, ω3 = dz −
(∫∫
ṡ
3
2
s
5
4 (s− 1)
5
4
dqdq
)
dx.
Supplement by the 1-forms
ω4 = dq, ω5 = dx,
and take the coframe on Mxyzpq to be given by (θ1, θ2, θ3, θ4, θ5) as in (6.2). Then Nurowski’s
metric (6.3) has vanishing Weyl tensor (and hence conformally flat) and Ds2 has the split real
form of G2 as its group of local symmetries.
6.3 Legendre transformed coframe
The Legendre transform of Proposition 4.1 takes the 1-forms (6.1) to
ω1 = dy − pdx, ω2 = dp−H ′dx, ω3 = dz − (tH ′ −H)dx,
ω4 = H ′′dt, ω5 = dx,
where H = H(t) with H ′′ 6= 0 on Mxyzpt and the coframe (6.2) to
θ1 = ω1 −H ′′(tω2 − ω3), θ2 = H ′′(tω2 − ω3), θ3 =
(
1 + t
H ′′′
4H ′′
)
ω2 −
H ′′′
4H ′′
ω3,
θ4 =
4H ′′H ′′′′ − 5(H ′′′)2
40(H ′′)3
(tω2 − ω3) + ω4 − ω5, θ5 = −ω4.
Note that our H(t) is related to Θ(x5) of [6] via Θ55 = −H, t = x5. The Nurowski metric
g = 2θ1θ5 − 2θ2θ4 +
4
3
θ3θ3
has the only non-vanishing component of the Weyl tensor given by the left hand side term of
the dual ODE (4.2). This accounts for the appearance of equation (4.3) in [9]. The solutions
of the dual generalised Chazy ODE (4.5) with parameter ±3
2 give us further examples of flat
(2, 3, 5)-distributions. We pass to (x, y, z, p, s) as before, with t(s) = w2(s)
w1(s)
where w1(s), w2(s)
are linearly independent solutions to (2.2) with (a, b, c) one of(
−1
4
,
5
12
,
1
2
)
,
(
−1
4
,
5
12
,
2
3
)
,
(
−1
2
,
5
6
,
2
3
)
.
Here we have taken k = 3
2 . Note that the equations (2.2) for (a, b, c) =
(
−1
4 ,
5
12 ,
1
2
)
and(
−1
4 ,
5
12 ,
2
3
)
are equivalent by a linear transformation and thus the solutions to each equation can
be expressed as linear combinations of the other, while the solutions for (a, b, c) =
(
−1
4 ,
5
12 ,
2
3
)
and
(
−1
2 ,
5
6 ,
2
3
)
are equivalent by a quadratic transformation as before. However, the author does
not know if the solutions in these cases can be expressed by elementary functions. We consider
once again solutions to (4.5) of the form u(t) = 6 d
dt logw1. This gives
H(t) =
∫∫
(w1)
4dtdt, H ′(t) =
∫
(w1)
4dt, H ′′(t) = (w1)
4.
For this parametrisation we have
dt =
w1ẇ2 − w2ẇ1
(w1)2
ds
Flat (2, 3, 5)-Distributions and Chazy’s Equations 23
and so
H =
∫∫
(w1)
2(w1ẇ2 − w2ẇ1)ds
w1ẇ2 − w2ẇ1
(w1)2
ds, H ′ =
∫
(w1)
2(w1ẇ2 − w2ẇ1)ds.
We therefore obtain
ω1 = dy − pdx, ω2 = dp−
∫
(w1)
2(w1ẇ2 − w2ẇ1)dsdx,
ω3 = dz −
(
w2
w1
∫
(w1)
2(w1ẇ2 − w2ẇ1)ds
−
∫∫
(w1)
2(w1ẇ2 − w2ẇ1)ds
w1ẇ2 − w2ẇ1
(w1)2
ds
)
dx,
ω4 = (w1)
2(w1ẇ2 − w2ẇ1)ds, ω5 = dx
and the adapted coframe for Nurowski’s metric is
θ1 = ω1 − (w1)
4
(
w2
w1
ω2 − ω3
)
, θ2 = (w1)
4
(
w2
w1
ω2 − ω3
)
,
θ3 =
(
1 +
w2ẇ1
w1ẇ2 − w2ẇ1
)
ω2 −
w1ẇ1
w1ẇ2 − w2ẇ1
ω3,
θ4 =
2(ẅ1ẇ2 − ẅ2ẇ1)
5(w1ẇ2 − w2ẇ1)3
(
w2
w1
ω2 − ω3
)
+ ω4 − ω5, θ5 = −ω4. (6.8)
Equivalently, we can take
ω̃1 = ω1, ω̃2 = (w1)
4
(
w2
w1
ω2 − ω3
)
, ω̃3 = ω2
with ω̃4 = ω4 and ω̃5 = ω5. The coframe is then given by
θ1 = ω̃1 − ω̃2, θ2 = ω̃2, θ3 = ω̃3 +
ẇ1
(w1)3(w1ẇ2 − w2ẇ1)
ω̃2,
θ4 =
2(ẅ1ẇ2 − ẅ2ẇ1)
5(w1)4(w1ẇ2 − w2ẇ1)3
ω̃2 + ω̃4 − ω̃5, θ5 = −ω̃4. (6.9)
Proposition 6.7. The Nurowski metric
g = 2θ1θ5 − 2θ2θ4 +
4
3
θ3θ3
given by the above coframe (6.8) for w1(s), w2(s) linearly independent solutions to the hyperge-
ometric differential equation (2.2) with (a, b, c) one of(
−1
4
,
5
12
,
1
2
)
,
(
−1
4
,
5
12
,
2
3
)
,
(
−1
2
,
5
6
,
2
3
)
are all conformally flat. For each (a, b, c) there is a 4-dimensional family of solutions. Up to
fractional linear transformation in the variable s there are 2 distinct classes given by the last
two entries.
In addition, the Legendre transformation of Lemma 4.2 given by
w1 = (z1)
− 3
4 , w2 = (z1)
− 3
4
∫
z1(z1ż2 − z2ż1)ds
24 M. Randall
takes the coframe (6.9) to the coframe in Theorem 6.1 and conversely so. The Legendre transform
also applies to the coframes given in Theorems 6.3, 6.5 and 6.6. We also have the analogous
results of Section 6.2. Up to fractional linear transformations in s we have 7 classes of solutions
to the generalised Chazy equation (4.5) determined by s(t) satisfying (5.2). These are given by
the parametrisations in Section 5 and the corresponding values for (α, β, γ) can be computed
for the parameter k = 3
2 . For the parametrisation
H(t) =
∫∫
ṡ2
s
4
3 (s− 1)
4
3
dtdt,
the values for (α, β, γ) are given by either
(
4
3 ,
4
3 ,
4
3
)
or
(
4
3 ,
1
3 ,
1
3
)
. For
H(t) =
∫∫
ṡ2
s
4
3 (s− 1)
dtdt,
(α, β, γ) takes the values of
(
2
3 ,
1
2 ,
1
3
)
,
(
2
3 ,
1
2 ,
4
3
)
or
(
2
3 , 2,
1
3
)
. For
H(t) =
∫∫
ṡ2
s
5
3 (s− 1)
5
3
dtdt,
we obtain
(
8
3 ,
2
3 ,
2
3
)
or
(
2
3 ,
2
3 ,
2
3
)
.
The Legendre transform therefore provides seven further classes of flat Nurowski metrics up
to fractional linear transformations in s.
6.4 Additional examples
The solution (3.2) for k = ±2
3 gives I(q) = − 8
3(q+C) −
10
3(q+B) . Hence, the metric (6.5) on Mxyzpq
given by
gDF (q)
= 2ω̃2ω̃5 − 2ω̃1ω̃4 +
4
3
(ω̃3)
2
+
(
8
9(q + C)
+
10
9(q +B)
)
ω̃2ω̃3 +
4(B − C)2
27(q +B)2(q + C)2
(ω̃2)
2
is conformally flat. In the dual coframe the flat metric (6.5) on Mxyzpt is given by
g = 2ω̃2ω̃5 − 2ω̃1ω̃4 +
4
3
(ω̃3)
2
+
4
9
u(t)e
∫
− 2
3
u(t)dtω̃2ω̃3 −
2
135
(9u̇(t)− 4u(t)2)e
∫
− 4
3
u(t)dt(ω̃2)
2, (6.10)
where u(t) satisfies the generalised Chazy equation (4.5) with parameter k = ±3
2 . The solu-
tion (3.2) for k = ±3
2 gives u(t) = − 15
4(t−a) −
9
4(t−b) and substituting this into (6.10) gives the
conformally flat metric
g = 2ω̃2ω̃5 − 2ω̃1ω̃4 +
4
3
(ω̃3)
2 − 256
3
(8t− 5b− 3a)(t− a)
3
2 (t− b)
1
2 ω̃2ω̃3
+
65536
3
(t− b)2(t− a)3(4t− 3a− b)(ω̃2)
2.
To summarise the results of this section, we first presented different examples of Nurowski
metrics that are conformally flat up to fractional linear transformation in the variable s. Two
examples are given in Theorem 6.3, three examples are given in Theorem 6.5 and two more
in Theorem 6.6. Seven additional examples are obtained from the Legendre transform as in
Proposition 4.1. We also have 2 additional examples from the solutions of the form (3.2). Finally,
there are examples associated to distributions of the form F (q) = qm, where m ∈
{
−1, 13 ,
2
3 , 2
}
and passing to the dual coframe, distributions of the form H(t) = tm, where m ∈
{
−2,−1
2 ,
1
2 , 2
}
.
Flat (2, 3, 5)-Distributions and Chazy’s Equations 25
7 An–Nurowski circle twistor bundle
In [5], An and Nurowski showed how to associate to a split signature conformal structure [g]
on a 4-manifold M4 a natural (2, 3, 5)-distribution. 4-dimensional split signature conformal
structures admit real self-dual totally null 2-planes. The bundle of such 2-planes is a circle
bundle over M4 with fibres S1 [5]. This is called the circle twistor bundle T(M4) and it has
a rank 2 distribution given by lifting horizontally the null 2-planes on M4. This distribution is
non-integrable, i.e., defines a (2, 3, 5)-distribution whenever the self-dual part of the Weyl tensor
of g on M4 is non-vanishing. Moreover in [6], the authors presented split signature conformal
structures on M4 that give rise to (2, 3, 5)-distributions of the form DF (q) on T(M4). Such split
signature metrics are called Plebański’s second heavenly metrics in [6].
Following [6, Section 3], we can find these metrics that have a flat circle twistor bundle.
Such circle twistor bundles have split G2 as their group of symmetries. Let (w, x, y, z) be
local coordinates on M4. Let Θ = Θ(w, x, y, z) be an arbitrary function of 4 variables (second
heavenly function of Plebański). Let (ei) be an orthonormal frame on M4 and (θj) the dual
coframe satisfying θj(ei) = δj i. The split signature Plebański metric is given by
g = gijθ
i ⊗ θj = 2θ1θ2 + 2θ3θ4,
where θiθj = 1
2θ
i ⊗ θj + 1
2θ
j ⊗ θi and
θ1 = dx−Θyydw + Θxydz, θ2 = dw, θ3 = dy −Θxxdz + Θxydw, θ4 = dz.
Hence g12 = g34 = 1 and all other components are zero. Such split signature metrics admit a real
parallel spinor [15]. A computation shows that the connection 1-forms we need are given by
Γ1
1 = −Θyyxθ
2 + Θyxxθ
4, Γ1
3 = −Θyyyθ
2 + Θyyxθ
4,
Γ3
1 = Θyxxθ
2 −Θxxxθ
4, Γ3
3 = Θyyxθ
2 −Θyxxθ
4.
Using [6] and Nurowski’s notes [20], we find that the (2, 3, 5)-distribution on T(M4) is annihi-
lated by the following three 1-forms:
ω3 = dξ + Γ3
1 +
(
Γ3
3 − Γ1
1
)
ξ − Γ1
3ξ
2
= dξ +
(
Θyxx + 2Θyyxξ + Θyyyξ
2
)
θ2 −
(
Θxxx + 2Θyxxξ +Hyyxξ
2
)
θ4
= dξ +
(
Θyxx + 2Θyyxξ + Θyyyξ
2
)
dw −
(
Θxxx + 2Θyxxξ + Θyyxξ
2
)
dz
and
ω4 = ξθ4 + θ2 = ξdz + dw,
ω5 = θ3 − ξθ1 = dy −Θxxdz + Θxydw − ξ(dx−Θyydw + Θxydz)
= dy − ξdx− (Θxx + ξΘxy)dz + (Θxy + ξΘyy)dw.
The distribution is therefore annihilated by the 1-forms
ω̃3 = dξ −
(
Θxxx + 3Θyxxξ + 3Θyyxξ
2 + Θyyyξ
3
)
dz, ω̃4 = ξdz + dw,
ω̃5 = dy − ξdx−
(
Θxx + 2ξΘxy + ξ2Θyy
)
dz.
Following [6], we now pass to the new coordinates (x̃, ỹ, z̃, p̃, t̃) on T(M4)
x 7→ t̃, w 7→ ỹ z 7→ x̃, −ξ 7→ p̃, y 7→ z̃ − p̃t̃.
26 M. Randall
We obtain the distribution annihilated by the following 1-forms:
ω̃3 = −dp̃− Ãdx̃, ω̃4 = −p̃dx̃+ dỹ,
ω̃5 = dz̃ − p̃dt̃− t̃dp̃+ p̃dt̃− B̃dx̃ = dz̃ − t̃dp̃− B̃dx̃ = dz̃ + (t̃Ã− B̃)dx̃,
where à and B̃ are coordinate transforms of the functions
A(w, x, y, z, ξ) = Θxxx + 3Θyxxξ + 3Θyyxξ
2 + Θyyyξ
3,
B(w, x, y, z, ξ) = Θxx + 2ξΘxy + ξ2Θyy
respectively. This suggests taking
à = −H ′(t), B̃ = −H(t)
to obtain the Legendre transformed 1-forms in Section 6.3. Passing back to coordinates (w, x, y,
z, ξ) on T(M4), this gives
−H ′(x) = Θxxx + 3Θyxxξ + 3Θyyxξ
2 + Θyyyξ
3,
−H(x) = Θxx + 2ξΘxy + ξ2Θyy,
so that
Θ(x) = −
∫∫
H(x)dxdx
will satisfy the condition. We have Θxx = −H(x). We have the following theorem.
Theorem 7.1. The An–Nurowski twistor distribution D on the circle twistor bundle T(M4)→
M4 of (M4, g) with the Plebański metric
g = dwdx+ dzdy +H(x)dz2
and the function H(x) has split G2 as its group of local symmetries provided that H(x) is one
of the following up to fractional linear transformations in s:
1. The function H(x) is given by
H(x) =
∫∫
ṡ2
s
4
3 (s− 1)
4
3
dxdx,
where s(x) is a solution to the 3rd order ODE (5.2)
{s, x}+
ṡ2
2
V (s) = 0
with (α, β, γ) given by either
(
4
3 ,
4
3 ,
4
3
)
or
(
4
3 ,
1
3 ,
1
3
)
.
2. The function H(x) is given by
H(x) =
∫∫
ṡ2
s
4
3 (s− 1)
dxdx,
where s(x) is a solution to (5.2) with (α, β, γ) one of
(
2
3 ,
1
2 ,
1
3
)
,
(
2
3 ,
1
2 ,
4
3
)
or
(
2
3 , 2,
1
3
)
.
Flat (2, 3, 5)-Distributions and Chazy’s Equations 27
3. The function H(x) is given by
H(x) =
∫∫
ṡ2
s
5
3 (s− 1)
5
3
dxdx,
where s(x) is a solution to (5.2) with (α, β, γ) either
(
8
3 ,
2
3 ,
2
3
)
or
(
2
3 ,
2
3 ,
2
3
)
.
4. The function H(x) is given by H(x) = xm where m ∈
{
−2,−1
2 ,
1
2 , 2
}
.
5. The function H(x) is given by
H(x) = − 1
192
√
x+ C(4x+ 3B + C)√
x+B(B − C)3
.
This corresponds to the solution obtained from (3.2).
6. The function H(x) is the Legendre transform of the function F (q) with q = H ′(x) and
H(x) = qx− F (q) = xH ′(x)− F (H ′(x)).
In this case F (q) can be given by one of the following:
(a) F (q) =
∫∫
ṡ
3
2
s(s− 1)
dqdq
where s(q) is again a solution to the 3rd order ODE (5.2) with (α, β, γ) one of (3, 3, 3) or(
3, 13 ,
1
3
)
.
(b) F (q) =
∫∫
ṡ
3
2
s(s− 1)
3
4
dqdq,
where s(q) is a solution to (5.2) with (α, β, γ) one of
(
3
2 ,
1
3 ,
1
2
)
,
(
3
2 , 3,
1
2
)
or
(
3
2 ,
1
3 ,
9
2
)
.
(c) F (q) =
∫∫
ṡ
3
2
s
5
4 (s− 1)
5
4
dqdq,
where s(q) is a solution to (5.2) with (α, β, γ) one of
(
6, 32 ,
3
2
)
or
(
2
3 ,
3
2 ,
3
2
)
.
(d) F (q) = qm,
where m ∈
{
−1, 13 ,
2
3 , 2
}
.
(e) F (q) = −1
6
(q +B)
1
3 (q + C)
2
3
(B − C)2
,
again corresponding to the solution (3.2).
Acknowledgements
This work is inspired by the paper of [6]. The author would like to thank Daniel An, Pawe l
Nurowski, Travis Willse and the anonymous referees for comments. Part of this work is sup-
ported by the Grant agency of the Czech Republic P201/12/G028.
28 M. Randall
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1 Introduction
2 Chazy's equation
3 Generalised Chazy equations
4 Relationship to ODE studied by Dunajski and Sokolov
5 First order system and different parametrisations of Chazy's equations
6 Examples of flat (2,3,5)-distributions
6.1 Chazy's solution
6.2 Other parametrisations of the generalised Chazy equation
6.3 Legendre transformed coframe
6.4 Additional examples
7 An–Nurowski circle twistor bundle
References
|
| id | nasplib_isofts_kiev_ua-123456789-147724 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-30T17:03:05Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Randall, M. 2019-02-15T18:43:43Z 2019-02-15T18:43:43Z 2016 Flat (2,3,5)-Distributions and Chazy's Equations / M. Randall // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 25 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58A30; 53A30; 34A05; 34A34 DOI:10.3842/SIGMA.2016.029 https://nasplib.isofts.kiev.ua/handle/123456789/147724 n the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or (2,3,5)-distributions determined by a single function of the form F(q), the vanishing condition for the curvature invariant is given by a 6th order nonlinear ODE. Furthermore, An and Nurowski showed that this ODE is the Legendre transform of the 7th order nonlinear ODE described in Dunajski and Sokolov. We show that the 6th order ODE can be reduced to a 3rd order nonlinear ODE that is a generalised Chazy equation. The 7th order ODE can similarly be reduced to another generalised Chazy equation, which has its Chazy parameter given by the reciprocal of the former. As a consequence of solving the related generalised Chazy equations, we obtain additional examples of flat (2,3,5)-distributions not of the form F(q)=qm. We also give 4-dimensional split signature metrics where their twistor distributions via the An-Nurowski construction have split G₂ as their group of symmetries. This work is inspired by the paper of [6]. The author would like to thank Daniel An, Pawe l Nurowski, Travis Willse and the anonymous referees for comments. Part of this work is supported by the Grant agency of the Czech Republic P201/12/G028. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Flat (2,3,5)-Distributions and Chazy's Equations Article published earlier |
| spellingShingle | Flat (2,3,5)-Distributions and Chazy's Equations Randall, M. |
| title | Flat (2,3,5)-Distributions and Chazy's Equations |
| title_full | Flat (2,3,5)-Distributions and Chazy's Equations |
| title_fullStr | Flat (2,3,5)-Distributions and Chazy's Equations |
| title_full_unstemmed | Flat (2,3,5)-Distributions and Chazy's Equations |
| title_short | Flat (2,3,5)-Distributions and Chazy's Equations |
| title_sort | flat (2,3,5)-distributions and chazy's equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147724 |
| work_keys_str_mv | AT randallm flat235distributionsandchazysequations |