EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 4, 2020, 1016-1034
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Cartan’s Approach to Second Order Ordinary
Differential Equations

Paul Bracken

Department of Mathematics, University of Texas, Edinburg, TX, 78540-2999, USA

Abstract. In his work on projective connections, Cartan discusses his theory of second order
differential equations. It is the aim here to look at how a normal projective connection can be
constructed and how it relates to the geometry of a single second order differential equation. The
calculations are presented in some detail in order to highlight the use of gauge conditions.

2020 Mathematics Subject Classifications: 34G10, 34A30, 34A26, 53B10

Key Words and Phrases: Differential equation, geometry, gauge, differential form, duality

1. Introduction

Two problems which have seen a resurgence of interest recently are two classical sub-
jects, namely, the equivalence problem under different kinds of tranformations and study
of the natural geometric structures induced on their solution spaces. In the early part of
the twentieth century, Cartan proposed a theory of the second-order ordinary differential
equation which came out of his theory of projective connections [3]. The point of interest
here is that it is possible to construct certain kinds of geometric structures on the spaces
of particular types of differential equations. In the case of the third order differential
equation, it is possible to create from the equation a conformal class of Lorentzian metrics
on the solution space, provided a function associated with the equation vanishes [6, 12, 13]
This function is a relative invariant of the equation under contact transformations. These
kinds of constructions turn out to be both useful and mathematically interesting. There
are numerous applications of this kind of work, for example, in the study of general relativ-
ity [10]. The simplest one to begin with is the second order ordinary differential equation.
A geometric structure can be constructed on the solution space of such an equation and
a function which is a relative invariant of the equation can be identified. The concept of
duality was also studied by Cartan. Considerable work has been done on the construction
of a geometric structure on the solution space of a second order differential equation. The
intention here is to discuss the theory of the second order ordinary differential equation

DOI: https://doi.org/10.29020/nybg.ejpam.v13i4.3831

Email addresses: paul.bracken@utrgv.edu (P. Bracken)

https://www.ejpam.com 1016 c© 2020 EJPAM All rights reserved.



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1017

which Cartan began and to study in particular Cartan’s notion of duality between such
types of equations [9].

The idea of what Cartan referred to as a manifold of elements with projective connec-
tion in the two-dimensional case is of great interest [2, 4, 5, 8]. By way of introduction,
some definitions of basic concepts is introduced. An element is a pair consisting of a point
of a differentiable manifold M and a one-dimensional subspace of the tangent space to M
at that point. A manifold of elements is the projective tangent bundle PTM of a two-
dimensional manifold M . Let P 2 denote real projective space with two-dimensions. Its
projective tangent bundle PTP 2 can be expressed as the homogeneous space G/H where
G = SL(2,R) and H is the subgroup of G consisting of all upper triangular elements. A
manifold of elements with projective connection is a Cartan geometry PTM modeled on
PTP 2 in which certain conditions regarding the development of curves which arise out
of the projective tangent structures of the underlying manifold and model geometry are
satisfied [1, 14, 16]. It should also be stressed that there are many physical applications
of this work especially in relativity [7, 11, 15].

2. Manifolds with Projective Connections

Local coordinates can be introduced on PTM by taking local coordinates (x, y) on M
and noting that every equivalence class of tangent vectors

u ∂x + v∂y

for which u 6= 0 has a unique representative of the form

∂x + y′ ∂y,

then (x, y, y′) are local coordinates on PTM which we work with in noting they do not
cover those equivalence classes of tangent vectors for which u = 0.

A Klein geometry is a homogeneous space G so that it is a manifold M with a transitive
action of G. Take a point x0 ∈ M and let H be the stabilizer of x0. Then M can be
identified with the coset space G/H. Then G becomes a right principle H-bundle over M
with projection g → gx0. The space (G,H) may be referred to as the Klein geometry.

Gauge changes may be made on a projective connection on such a manifold. If ω is a
connection form that is a trace-free 3× 3 matrix of local one forms on PTM , and h is an
H-valued function, then the gauged connection form is

h−1 ω h+ h−1 dh. (1)

If Ω is a curvature two-form corresponding to ω, the regauged curvature is h−1Ωh. Intro-
ducing

h =

A D F
0 B E
0 0 C

 (2)



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1018

where ABC = 1 and the inverse element is calculated to be

h−1 =

A−1 −CD DE −BF
0 B−1 −AE
0 0 0

 (3)

The effect of a change of gauge on the lower triangular terms is of most interest. Since
h−1 dh is upper triangular, it has no effect on these terms. In any event, the entire matrix
can be computed. Suppose ω is given as

ω =

 0 ω1
2 ω1

3

−ω1
2 0 ω2

3

−ω1
3 −ω2

3 0

 , ωT = −ω. (4)

Using (2) and (3), we calculate
h−1 ω h =

ACDω1
2 −A(DE −BF )ω1

3 A−1Bω1
2 + CD2ω1

2 A−1(Eω1
2 + Cω1

3) + CD(Eω1
2 − Cω2

3)
−(DE −BF )(Dω1

3 −Bω2
3) −(DE −BF )(Fω1

3 + Eω2
3)

−AB−1ω1
2 +A2Eω1

3 −B−1Dω1
2 +AE(Dω1

3 +Bω2
3) B−1(−Fω1

2 + Cω2
3)

+AE(Fω1
3 + Eω2

3)
−C−1Aω1

3 −C−1(Dω1
3 +Bω2

3) −C−1(Fω1
3 + Eω2

3)


(5)

Let Ω be the curvature form given by

Ω =

0 Ω12 Ω13

0 0 Ω23

0 0 0

 (6)

The effect of a change of gauge on (6) using (3) is found to be

h−1 Ωh =

0 A−1BΩ12 A−1(E + C)Ω13 − C2DΩ23

0 0 B−1CΩ23

0 0 0

 (7)

The equation for the development of a curve will be required so to this end, the following
mapping is proposed

(ξ, η, η′)→

1 0 0
ξ 1 0
η η′ 1

 (8)

This is a local section of SL(3,R) → PTP 2. The corresponding Maurer-Cartan form is
obtained by working out the exterior derivative of (8), At this point, the connection form
needed to calculate the development equations later is given,

ω =

ω0
0 ω0

1 ω0
2

ω1
0 ω1

1 ω1
2

ω2
0 ω2

1 ω2
2

 (9)



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1019

The development equation for a curve γ in PTM gives

aξ̇ − b(η̇ − η′ξ̇) = 〈γ̇, ω1
0〉, c(η̇ − η′ξ̇) = 〈γ̇, ω2

0〉, (10)

for some functions a(t), b(t), c(t).
Knowing these formulas, the conditions to be imposed on the development of a curve

should be considered further. There are two kinds of curve on PTM . There are vertical
curves and there are natural lifts. With respect to the coordinates (x, y, y′) introduced
above, a curve in PTM is a natural lift if its tangent vector is annihilated by the contact
form dy−y′dx. The conditions are then (i) the development into PTP 2 of a natural curve
in PTM is vertical (ii) the development into PTP 2 of a natural lift in PTM is a natural
lift. These conditions require that if γ is vertical then 〈γ, ω1

0〉 = 〈γ, ω2
0〉 = 0, while if γ is

a natural lift, then 〈γ, ω2
0〉 = 0. It follows that

ω1
0 = λ dx+ µdy, ω2

0 = ν(dy − y′ dx). (11)

for some functions λ, µ, ν on PTM .
The connection matrix ω can now be simplified by introducing a change of gauge, so

setting

A = ((λ+ µy′)ν)−1/3, B = (λ+ µy′)A, C = νA, E = µA, (12)

it can be ensured that ω1
0 = dx and ω2

0 = dy − y′ dx = ϑ. It is possible to write the form
ω2

1 as ω2
1 = k(dy′ − tdx) +mdϑ for some functions f , k and m on PTM . The function k

must be nonzero because the forms ω1
0, ω2

0 and ω2
1 have to be linearly independent. Upon

setting dy′ − f dx = ϕ, the set of forms dx, ϑ and ϕ constitute a local basis of one-forms.
It can be summarized by saying that a gauge has been chosen such that

ω =

ω2
0 ω0

1 ω0
2

dx ω1
1 ω1

2

ϑ kϕ+mϑ ω2
2

 (13)

The remaining gauge freedom involves the functions D and F . Consider a further gauge
change which is specified by taking

h =

1 D F
0 1 0
0 0 1

 , h−1 =

1 −D −F
0 1 0
0 0 1

 (14)

By using (13) and (14), we find that

h−1 ω h+ h−1 dh

=

ω0
0 −Ddx− Fϑ ω0

1 −Dω1
1 − Fω2

1 + dD ω0
2 −Dω1

2 − Fω2
2 + dF

dx ω1
1 ω1

2

ϑ ω2
1 ω2

2

 (15)



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1020

Therefore, a gauge can be chosen for any projective connectionof a manifold of elements
such that

ω1
0 = dx, ω2

0 = ϑ, ω0
0 = κϕ, (16)

for some function κ. This is referred to as the standard gauge for the projective connection.
A geodesic of this projective connection is a curve whose development satisfies the

equations η̇ − η′ξ = 0 and η̇′ = 0. Put another way, a geodesic is a curve whose tangents
are annihilated by ϑ and ϕ, and therefore is a solution of the second-order equation

d2y

dx2
= f(x, y,

dy

dx
). (17)

Geodesics are the base integral curves of the following vector field on PTM ,

Γ =
∂

∂x
+ y′

∂

∂y
+ f

∂

∂y′
. (18)

This can be referred to as the second-order differential equation field corresponding to the
projective connection. Note that Γ is determined by the conditions

〈Γ, dx〉 = 1, 〈Γ, ϑ〉 = 〈Γ, ϕ〉 = 0. (19)

Under a change of coordinates on the base manifold M , with induced change on PTM ,
the field Γ will acquire an overall factor which depends on the coordinate transformation
functions. It may be said that we are really working not with a vector field Γ, but with a
line element field.

Having fixed the gauge, the next step is to impose gauge-invariant conditions on the
curvature in order to single out a particular connection form out of the class of connections
under consideration. In fact, Cartan showed in effect that there is a unique choice of
the remaining connection forms so that the curvature Ω is upper triangular, with Ω0

1 a
multiple of dx ∧ ϑ. The unique connection obtained this way is usually referred to as the
normal projective connection on the manifold of elements associated with the second order
differential equation.

3. Model Geometry and Duality of Points and Lines in Projective
Geometry

Let M and M̄ be two-dimensional manifolds, and S a co-dimension one submanifold of
M × M̄ , which is fibered over both M and M̄ . For any p̄ ∈ M̄ the set {p ∈M |(p, p̄) ∈ S}
is a path in M , call it σp̄. For p ∈ M , {p̄ ∈ M̄ |(p, p̄) ∈ S} determines a one-parameter
family of paths σp̄ ⊂ M such that p ∈ σp̄ for all such p̄. Require that this construction
define a path space on M . For every p ∈ M and [u] ∈ PTpM , there is a unique p̄ ∈ M̄
with (p, p̄) ∈ S the direction of the tangent to σp̄ at x is [u].

Let (x, y) be coordinates on M and (x̄, ȳ) those on M̄ , and suppose the submanifold
S of M × M̄ is given by Φ(x, y, x̄, ȳ) = 0. Then σ(x̄0, ȳ0) is Φ(x, y, x̄0, ȳ0) = 0 and vector

(
∂

∂x
+ y′

∂

∂y
)(x̄0,ȳ0) (20)



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1021

is a representative of some tangent vector with u 6= 0 and is tangent to this path provided

Φx(x0, y0, x̄0, ȳ0) + y′Φy(x0, y0, x̄0, ȳ0) = 0, (21)

where Φ(x0, y0, x̄0, ȳ0) = 0. Therefore, the map S → PTM is given by (x, y, x̄, ȳ) →
(x, y, y′) such that

Φ(x, y, x̄, ȳ) = 0, Φx(x, y, x̄, ȳ) + y′Φy(x, y, x̄, ȳ) = 0. (22)

It is required that Φx and Φy do not vanish simultaneously. In a similar way, the map
S → PTM̄ is given by (x, y, x̄, ȳ)→ (x̄, ȳ, ȳ′) where

Φ(x, y, x̄, ȳ) = 0, Φx̄(x, y, x̄, ȳ) + ȳ′Φȳ(x, y, x̄, ȳ) = 0,

and Φx̄, Φȳ must not vanish at the same time.
The condition that the first pair of equations determines x̄ and ȳ in terms of x, y and

y′ is that the following matrix be nonsingular, 0 Φx Φy

Φx̄ Φxx̄ Φyx̄

Φȳ Φxȳ Φyȳ

 (23)

The same condition ensures that the second pair of equations can be solved for x and
y in terms of bared variables with bars. This assumes that Φx, Φy do not both vanish
simultaneously and the same for Φx̄ Φȳ as well.

Assume that this condition holds on S. It can be stated in the form ∆ 6= 0, where ∆
is the determinant

∆ = −ΦyΦȳ(Φxx̄ + y′Φyx̄ + ȳΦxȳ + y′ȳ′Φyȳ), ΦyΦy′ 6= 0. (24)

Take some fixed point (x̄, ȳ) ∈ M̄ such that the path it takes in M can be parametrized
by x. Then y′ and y′′ satisfy

Φxx + 2
dy

dx
Φxy + (

dy

dx
)2Φyy + (

d2y

dx2
)Φy = 0. (25)

The path then is a solution to the second order equation

d2y

dx2
= f(x, y, y′), (26)

where f(x, y, y′) is obtained by eliminating x̄ and ȳ between the equations

Φ = 0, Φx + y′Φy = 0, Φxx + 2y′Φxy + (y′)2Φyy + f Φy = 0. (27)

The right-hand side f̄(x̄, ȳ, ȳ′) of the equation giving the dual path is obtained by elimi-
nating x and y between the equations

Φ = 0, Φx̄ + ȳ′Φȳ = 0, Φx̄x̄ + 2ȳ′Φx̄ȳ + (ȳ′)2Φȳȳ + f̄Φȳ = 0. (28)



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1022

There are two Cartan normal projective connection forms associated with this structure.
There is one corresponding to d2y/dx2 = f and the other to d2ȳ/dx̄2 = f̄ . Each can
be represented by a connection form on S. The connection form associated with the bar
equation takes a lower triangular form

ω̄ =

dx̄
ϑ̄ ϕ̄− 1

3 f̄ȳϑ̄

 , ϑ̄ = dȳ − ȳ′ dx̄, ϕ̄ = dȳ′ − f̄ dx̄. (29)

Now (x, y, y′) and (x̄, ȳ, ȳ′) can be regarded as alternative coordinates on S with trans-
formation given by

Φ(x, y, x̄, ȳ) = 0, Φx(x, y, x̄, ȳ)+y′Φy(x, y, x̄, ȳ) = 0, Φx̄(x, y, x̄, ȳ)+ȳ′Φȳ(x, y, x̄, ȳ) = 0.
(30)

It can be shown that the Jacobian matrix of (x̄, ȳ, ȳ′) with respect to (x, y, y′) is nonsingular
since it is assumed that ∆ 6= 0 and moreover Φxx + 2y′Φxy + (y′)2Φyy + fΦy = 0. There
is a similar equation in f̄ of course.

Consider now everything expressed in terms of the unbarred coordinates. By taking
the exterior derivative of the equation Φ = 0 and expressing dy and dȳ in terms of ϑ and
ϑ̄, dx and dx̄, we obtain that

(Φx + y′Φy) dx+ Φyϑ+ (Φx̄ + ȳ′Φȳ) dx̄+ Φȳϑ̄ = 0, (31)

so there is the relation

ϑ̄ = −Φy

Φȳ
ϑ. (32)

In a similar way, the exterior derivative of Φx +y′Φ = 0 is worked out and the differentials
dy, dȳ are replaced yielding an expression in terms of dx, dx̄, ϑ, ϑ̄ and ϕ,

d(Φx+y′Φy) = Φxx dx+Φxy dy+Φx̄x dx̄+Φȳx dȳ+dy′Φy+y′(Φxy dx+Φyy dy+Φx̄y dx̄+Φȳy)

= (Φxx + 2y′Φxy + (y′)2Φyy + fΦy) dx+ (Φxx̄ + y′Φyx̄ + y′Φxȳ + y′ȳ′Φyȳ) dx̄ (33)

+(Φxy + y′Φyy)ϑ+ (Φxȳ + y′Φyȳ)ϑ̄+ Φyϕ = 0.

Thus,

dx̄ = (
Φ2
yΦȳ

∆
)ϕ, mod ϑ.

As ∆ is unchanged when barred and unbarred quantities are interchanged, dx can be
expressed in a similar way,

dx =
ΦyΦ̄2

ȳ

∆
ϕ̄.

Thus to briefly summarize,

dx̄ =
Φ2
yΦȳ

∆
ϕ, ϑ̄ = −Φy

Φȳ
ϑ, ϑ̄ =

∆

ΦyΦȳ
dx mod ϑ. (34)



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1023

It is desired to reduce ω̄ to a form as close as possible to the standard one for a projective
connection by means of a gauge transformation. This should involve the interchange of
the positions of the dx and kϕ + mϑ terms in a matrix of the form (13). However, for
normal projective connections we have k = 1. From the formulas of a gauge transformation
derived before, this is impossible as it would require taking A, B and C the diagonal entries
in the gauge transformation matrix to satisfy inconsistent equations. For definiteness we
suppose that

A

C
=

Φȳ

Φy
,

A

B
=

∆

Φ2
yΦȳ

,
B

C
=

ΦyΦ2
ȳ

∆
. (35)

The other coefficients of the gauge transformation are chosen so that the ϑ component of
ω̄1

2 is eliminated as well as the dx and ϑ components of ω̄2
2. There is then a unique gauge

transformation matrix h such that it has a lower triangular form,

h−1ω̄h+ h−1 dh =

ϕ+mϑ
ϑ dx κϕ

 (36)

Given a trace-free matrix valued one-form $ with $2 = −ϑ $2
1 = dx and $ a multiple

of ϕ, the remaining elements of $ are uniquely determined by the following conditions on
its curvature form Π: it must be strictly upper-triangular and Π1

2 is a multiple of dx ∧ ϑ.
The strategy is to compute the componets of Π and look at the consequences of taking
them to be zero. To calculate Π, we have to calculateω̃0

0 ω̃0
1 ω̃0

2

ω̃1
0 ω̃1

1 ω̃1
2

ω̃2
0 ω̃2

1 ω̃2
2

 ∧
ω̃0

0 ω̃0
1 ω̃0

2

ω̃1
0 ω̃1

1 ω̃1
2

ω̃2
0 ω̃2

1 ω̃2
2


Suppose µ, λ, ν are functions, then we note that

dν = Γ(ν) dx+ νyϑ+ νy′ ϕ, (37)

and moreover,

dϑ = −ϕ ∧ dx, dϕ = −df ∧ dx = −(fyϑ+ fy′ϕ) ∧ dx. (38)

The notation for the components of $ and Π follows the same system as for Ω. First we
have

Π2
1 = d$2

1 +$2 ∧$0
1 +$2

1 ∧$1
1 +$2

2 ∧$2
1 = −ϑ ∧$0

1 + dx ∧ ($1
1 ∧$2

2). (39)

This will vanish provided that,

$0
1 = λ ∧ dx, $1

1 = $2
2 + λ0 dx− λϑ. (40)

As noted λ and µ are functions which are arbitrary at first. The next component is

Π2 = d$2 +$2 ∧$0
0 +$2

1 ∧$1 +$2
2 ∧$2 = (ϕ−$1) ∧ dx− ϑ ∧ ($0

0 −$2
2). (41)



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1024

This is zero if and only if the coefficient of ϕ in $1 is one and also

$0
0 = $2

2 − µdx+ νϑ, $1 = $ + µϑ. (42)

The trace of the matrix of forms $ is to be zero. This constraint implies that

$0
0 +$1

1 +$2
2 = $2

2 − µdx+ νϑ+$2
2 + λ0 dx− λϑ+$2

2 = (λ− µ) dx+ (ν − λ)ϑ+ 3$2
2

= (λ0 − µ) dx+ (ν − λ)ϑ+ 3κϕ. (43)

Since dx, ϑ, and ϕ are independent it follows that

κ = 0, λ0 = µ, λ = ν, $2
2 = 0, $0

0 = −$1
1 = −µdx+ νϑ. (44)

The next component of Π to consider is

Π1 = d$1 +$1 ∧$2
0 +$1

1 ∧$1 +$1
2 ∧$2

= d(ϕ+µϑ)+$1∧($0
0−$1

1)+$1
2∧$2 = dϕ+dµ∧ϑ+µdϑ+2(ϕ+µϑ)∧(−µdx+νϑ)−$1

2 ∧ϑ
(45)

= dϕ+ dµ ∧ ϑ− µϕ ∧ dx+ 2(ϕ+ µϑ) ∧ (−µdx+ νϑ)−$1
2 ∧ ϑ.

The differentials dµ and df are given by

dµ = Γ(µ) dx+ µyϑ+ µy′ ϕ, df = fx dx+ fy(ϑ+ y′ dx) + fy′(ϕ+ f dx). (46)

Consequently, Π1 is

Π1 = −(fy(ϑ+ y′ dx) + fy′(ϕ+ f dx)) ∧ dx+ (Γ(µ)dx+ µyϑ+ µy′ϕ) ∧ ϑ

−µϕ ∧ dx− 2µϕ ∧ dx+ 2νϕ ∧ ϑ− 2µ2ϑ ∧ dx−$1
2 ∧ ϑ (47)

= (fy′ + 3µ) dx ∧ ϕ+ (fy + Γ(µ) + 2µ2) dx ∧ ϑ+ (µy′ + 2ν)ϕ ∧ ϑ−$1
2 ∧ ϑ.

Necessary and sufficient conditions for Π1 to be zero are the following

µ = −1

3
fy′ , $1

2 = (fy +
2

9
f2
y′ −

1

3
Γ(fy′)) dx+ (2ν − 1

3
fy′y′)ϕ+ ρϑ. (48)

The second diagonal element of Π is

Π2
2 = $2∧$0

2 +$2
1∧$1

2 = −ϑ∧$0
2 +dx∧$1

2 = ($0
2 +ρ dx)∧ϑ+(2ν− 1

3
fy′y′)dx∧ϕ. (49)

The component Π2
2 vanishes if and only if

$0
2 = −ρ dx+ σϑ, ν =

1

6
fy′y′ . (50)

Notice that the latter condition makes the ϕ component vanish in $1
2.



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1025

The next component to consider is Π0
0 which is given by

Π0
0 = d$0

0 +$0
1 ∧$1 +$0

2 ∧$2 = d$0
0 +$0

1 ∧$1 +$0
2 ∧ ϑ. (51)

Using the fact that
dfy′y′ = Γ(fy′y′) dx+ fy′y′y ϑ+ fy′y′y′ ϕ, (52)

The element Π0
0 is explicitly calculated to be

Π0
0 = d(

1

3
fy′ dx+

1

6
fy′y′ ϑ) + (

1

6
fy′y′ dx+ λ′ϑ) ∧ (ϕ− 1

3
fy′ϑ) + ρ dx ∧ ϑ

=
1

3
(fyy′ϑ+ fy′y′ ϕ)∧ dx+

1

6
(Γ(fy′y′)dx+ fy′y′yϑ+ fy′y′y′ϕ)∧ ϑ− 1

6
fy′y′ ϕ+

1

6
fy′y′ dx∧ϕ

− 1

18
fy′fy′y′ dx ∧ ϑ+ λ′ϑ ∧ ϕ+ ρ dx ∧ ϑ (53)

= (ρ− 1

3
fy′y −

1

18
fy′fy′y′ +

1

6
Γ(fy′y′)) dx ∧ ϑ+ (λ′ − 1

6
fy′y′y′)ϑ ∧ ϕ.

For Π0
0 to vanish, it is required that

ρ =
1

3
fyy′ +

1

18
fy′fy′y′ −

1

6
Γ(fy′y′)) λ′ =

1

6
fy′y′y′ . (54)

When these conditions hold, the component Π1
1 will also be zero since it is required that

the trace of Π vanish.
All of $ has been fixed, but with the exception of the coefficient σ in $0

2. This is
determined by imposing that Π1

2 be a multiple of dx ∧ ϑ hence semi-basic. Since

∂

∂y′
(
fy +

2

9
f2
y′−

1

3
(
∂fy′

∂x
+y′

∂fy′

∂y
+f

∂fy′

∂y′
)
)

= fyy′ +
4

9
fy′fy′y−

1

3
Γ(fy′y′)−

1

3
fyy′−

1

3
fy′fy′y′

=
2

3
fyy +

1

9
fy′fy′y′ −

1

3
Γ(fy′y′) = 2ρ. (55)

Using this, we find that ∂/∂y′cΠ1
2 = (ρy′ +σ)ϑ, so Π1

2 is semi-basic if and only if σ = −ρy′ .
This completes the determination of $. In fact, $ is given in terms of the first normal

projective connection by

$0
1 = −ω2

2, $0
1 = ω1

2 $0
2 = −ω0

0 $1 = ω2
1 $1

1 = −ω1
1 $1

2 = ω0
1, $2 = −ω2 $2

1 = ω1 $2
2 = −ω0

0.
(56)

This can be summarized concisely in matrix form as follows

$ = −KωTK, (57)

where ωT is the transpose of ω and matrix K is defined to be

K =

 0 0 −1
0 1 0
−1 0 0

 (58)



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1026

It may be verified that when $ and ω are related as in (57), their curvatures Π and Ω
are related in the same way. The minus signs are important for this to hold. There would
be no obvious relation between the components if that were the case. The crucial point is
that the mapping M → −KMTK = M ′ is a homomorphism of the matrix Lie algebra so
in fact

[M ′1,M
′
2] = K[MT

1 ,M
T
2 ]K = −K[M1,M2]TK = [M1,M2]′. (59)

This holds for any K such that K2 is the identity. Without the minus sign an antihomo-
morphism results instead and then

dω′ +
1

2
[ω′ ∧ ω′] = dω′ +

1

2
[ω ∧ ω]′. (60)

If Ω has the upper triangular form,0 b dx ∧ ϑ Ω0
2

0 0 aϑ ∧ ϕ
0 0 0

 (61)

Then the fact that Π = −KΩTK implies that Π is given as

Π = −KΩTK =

0 ϑ ∧ ϕ −Ω0
2

0 0 b dx ∧ ϑ
0 0 0

 . (62)

The gauged version of the second normal projective connection h−1ω̄h + h−1 dh can be
reconsidered. By calculation,

h−1Ω̄h =

0 A−1Bb̄ dx̄ ∧ ϑ̄ A−1Eb̄dx̄ ∧ ϑ̄+A−1CΩ̄0
2 + (DE −BF )Cāϑ̄ ∧ ϕ̄

0 0 B−1Cāϑ̄ ∧ ϕ̄
0 0 0

 (63)

Thus the gauge transformed version h−1ω̄h+h−1 dh satisfies the conditions that uniquely
determine $ and therefore must be $ = h−1ω̄h+ h−1 dh.

4. Duality

The condition ∆ 6= 0 inposed previously may be thought of in another way. It states
that the one-form θ = Φx dx+ Φy dy = −(Φx̄ dx̄+ Φȳ dȳ) satisfies the condition θ ∧ dθ on
S. It may be said it defines a contact structure on this three-dimensional manifold.

Let S be a three-dimensional manifold endowed with a contact structure which it may
be convenient to think of as a two-dimensional distribution D which is nonintegrable in the
following sense. For any pair of linearly independent vector fields X,Y ∈ D, [X,Y ] /∈ D.
Any one-form θ on S which is an annihilator of D satisfies the condition θ ∧ dθ 6= 0.
Suppose a basis has been given for D, and these basis vectors are denoted as X, X̄. This
means that the set {X, X̄, [X, X̄]} is a basis for vector fields on S. Let the set {φ, φ̄, θ} be
the dual basis of one-forms.



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1027

In this case, we would take X to be tangent to one of the fibers of the double fibration
of S and X̄ to the other. Then D would be the distribution spanned by X, X̄ and θ would
be a scalar multiple of Φx dx+ Φy dy. The purpose of this discussion is to reexamine the
the effect of the Cartan connection form of interchanging the roles of the fibrations while
treating them on an equal footing. This is accomplished by working in terms of the dual
basis just proposed. When X and X̄ are interchanged, the new basis of one-forms becomes
{φ̄, φ,−θ}. For the normal Cartan projective connection already described,

X = Γ, X̄ =
∂

∂y
. (64)

It is then possible to calculate the following bracket,

[X, X̄] = − ∂

∂y
− fy′

∂

∂y′
. (65)

The one-forms which reside in the lower triangular portion of the connection matrix dx,
φ − 1

3fy′θ and θ are not dual to the basis of vector fields. This is the main difference
between what has been discussed and what comes next.

The dual one-form basis is actually given as {dx, φ− fydθ,−θ} The new approach re-
quires us to take the Cartan connection form in the lower triangular part of the connection
form matrix as  φ

−θ φ̄

 (66)

In the case just discussed, this would be gauge-equivalent to the version used previously.
It is to be emphasized that now X may be any vector field tangent to the first fibration
and X̄ any vector field tangent to the second. Thus in the present version of the theory,
transformations of the form X → λX and X̄ → λ̄X̄ for any nonvanishing functions λ, λ̄,
will be allowed: such transformations induce gauge transformations of the kind discussed
previously, with coefficients given in terms of λ, λ̄ and their derivatives.

It is a consequence of these definitions that the exterior derivatives of the basis one-
forms can be written as

dφ = ψ ∧ θ, dφ̄ = −ψ̄ ∧ θ, dθ = −φ ∧ φ̄+ χ ∧ θ̄, (67)

where ψ, ψ̄ and χ are certain one-forms which are linear combinations of φ and φ̄.
Changing the notation proves to be useful, so the connection form has the structure,

ω =

 α β γ
φ −α− α′ β′

−θ φ̄ α′

 (68)

It is assumed that X and X̄ and hence φ, φ̄ and θ have been fixed. Therefore, the only
remaining gauge freedom is the one coming from a gauge transformation of the form

h =

1 0 F
0 1 0
0 0 1

 , h−1 =

1 0 −F
0 1 0
0 0 1

 (69)



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1028

For such a choice of h, it is determined that

h−1ωh+ h−1 dh

=

α− Fθ β − Fφ̄ F (α− Fθ)
φ −α− α′ Fφ− β′
θ φ̄ Fθ + α′

 (70)

By pursuing an argument similar to the one used before, that α, α′, β, β′ and γ are uniquely
determined in terms of ψ, ψ̄ and χ and their derivatives by the requirement that the
curvature Ω of ω takes the form,

Ω =

0 B 0
0 0 B′

0 0 0

 (71)

with B a multiple of φ ∧ θ. The constraints that require the torsion to vanish amount to
the following system:

dφ−(2α+α′)∧φ−β′∧θ = 0, dφ̄+(α+2α′)∧γ̄+β∧θ = 0, dθ−(α−α′)∧θ+φ∧φ̄ = 0.
(72)

It follows from the last of these together with the gauge-fixing assumption that α−α′ = χ.
The first two equations in (72) determine the φ and φ̄ components of α. Therefore α′ is
in terms of ψ, ψ̄ and χ with the φ, φ̄ components of β and β′ in terms of α and α′. The
conditions that the diagonal elements of Ω must vanish results in the pair

dα+ β ∧ φ+ γ ∧ θ = 0, dα′ − β′ ∧ φ̄+ γ ∧ θ = 0. (73)

This are equivalent under linear combinations to the following,

d(α+ α′) + β ∧ φ− β′ ∧ φ̄ = 0, d(α− α′) + β ∧ φ+ β′ ∧ φ̄ = 2γ ∧ θ. (74)

The φ ∧ φ̄ component of these determines the θ component of α + α′ and therefore of
α and α′ since they have the same θ component. The remaining components determine
the θ components of β and β′. The φ ∧ φ̄ component of the second equation is satisfied
identically, and the other two components yield the φ and φ̄ components of γ. Given that
ω has the form (68) the curvature form Ω is given by dα dβ dγ

dφ −d(α+ α′) dβ′

−dθ dφ̄ dα′



+

 β ∧ φ− γ ∧ θ α ∧ β − β ∧ (α+ α′) + γ ∧ φ̄ α ∧ γ + β ∧ β′ + γ ∧ α′
φ ∧ α− (α+ α′) ∧ φ− β′ ∧ θ φ ∧ β + β′ ∧ φ̄ φ ∧ γ − (α+ α′) ∧ β′ + β′ ∧ α′
−θ ∧ α+ φ̄ ∧ φ− α′ ∧ θ −θ ∧ β − φ̄ ∧ (α+ α′) + α′ ∧ φ −θ ∧ β − φ̄ ∧ (α+ α′) + α′ ∧ φ


(75)



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1029

It is required that the curvature have the following structure

Ω =

0 B 0
0 0 B′

0 0 0

 (76)

The requirement that the lower left entries vanish gives rise to the following three equations

dφ+φ∧(2α∧α′)−β′∧θ = 0, dφ̄+(α+2α′)∧φ̄+β∧θ = 0, dθ−(α−α′)∧θ+φ∧φ̄ = 0. (77)

These may be referred to as the torsion equations.
The diagonal elements of the curvature Ω yield the constraints

dφ+ β ∧φ− γ ∧ θ = 0, dα′− θ∧ γ+ φ̄∧ β′ = 0, −d(α+α′) +φ∧ β+ β′ ∧φ = 0, (78)

Adding together the first two in (78) the negative of the third one results and everything
is consistent. As well,

d(α− α′) + β ∧ φ− φ̄ ∧ β′ = 2γ ∧ θ. (79)

It follows from their basic definitions that the exterior derivatives of the basic one-forms
can be expressed as

dφ = ψ ∧ θ, dφ̄ = −ψ̄ ∧ θ, dθ = −φ ∧ φ̄+ χ ∧ θ. (80)

The one-forms ψ, ψ̄ and χ are linear combinations of φ and φ̄.
Substituting these derivatives into the torsion equations, the following are obtained

(2α+α′)∧φ+(β′−ψ)∧θ = 0, (α+2α′)∧ φ̄+(β− φ̄)∧θ = 0, (χ−(α−α′))∧θ = 0.
(81)

From the first two of these, 2α + α′ can be taken proportional to φ, β′ − ψ proportional
to θ, α+ 2α′ proportional to φ̄ and β − ψ̄ proportional to θ, hence

2α+ α′ = c1φ, α+ 2α′ = c2φ̄. (82)

Taking the constants c1 = c2 = 1, these imply that

α =
2

3
φ− 1

3
φ̄, α′ = −1

3
φ+

2

3
φ̄. (83)

The third constraint in (81) can be satisfied by taking χ− (α− α′) = 0 independent of θ
as a gauge condition so

χ = α− α′ = φ− φ̄. (84)

Using (80), it is found that this implies that

dχ = d(α− α′) = dφ− dφ̄ = (ψ − ψ̄) ∧ θ. (85)

It makes sense to take β′ = ψ and β = ψ̄ in which case, the other two constraints are
satisfied.



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1030

Returning to Ω in (74) the components B and B′ are given as

B = dβ + (2α+ α′) ∧ β + γ ∧ φ̄, B′ = dβ′ − (α− 2α′) ∧ β′ + φ ∧ γ. (86)

Upon differentiating the equation dφ̄+ (α+ 2α′) ∧ φ̄+ β ∧ θ = 0 it follows that

d(β ∧ θ) + d((α+ 2α′) ∧ φ̄) = 0,

dβ ∧ θ − β ∧ dθ + d(α+ 2α′) ∧ φ̄− (α− 2α′) ∧ dφ̄ = 0,

dβ ∧ θ − β ∧ dθ + (−β ∧ φ+ γ ∧ θ + 2β′ ∧ φ̄− 2γ ∧ θ) ∧ φ̄− (α+ 2α′) ∧ dφ̄ = 0,

dβ ∧ θ − β ∧ dθ − β ∧ φ ∧ φ̄− γ ∧ θ ∧ φ̄− (α+ 2α′) ∧ dφ̄ = 0, (87)

(dβ + (2α+ α′) ∧ β + γ ∧ φ̄) ∧ θ = 0.

The expression which remains inside the brackets is exactly B, so this result can be written
more concisely as

B ∧ θ = 0. (88)

Differentiate exteriorly dφ = β′ ∧ θ − φ ∧ (2α+ α) to obtain

dβ′ ∧ θ − β′ ∧ θ − dφ ∧ (2α+ α′) + φ ∧ (−2β ∧ φ+ 2γ ∧ θ + θ ∧ γ − φ̄ ∧ β′) = 0.

dβ′ ∧ θ − β′ ∧ (dθ + θ ∧ (2α+ α′) + φ ∧ φ̄) + φ ∧ γ ∧ θ = 0.

Finally after simplifying, we have,

(dβ′ + β′ ∧ (α+ 2α′) + φ ∧ γ) ∧ θ = 0. (89)

The form inside the bracket is exactly B′, so (89) is just the equation

B′ ∧ θ = 0. (90)

Suppose B is a scalar multiple of φ ∧ θ so

B = C φ ∧ θ = dβ + (2α+ α′) ∧ β + γ ∧ φ̄. (91)

This holds if and only if

dβ ∧ φ+ (2α+ α′) ∧ β ∧ φ = γ ∧ φ ∧ φ̄. (92)

However, dβ ∧ φ = β ∧ dφ+ d(γ ∧ θ) and so consequently,

dβ∧φ+(2α+α′)∧β∧φ = β∧dφ+d(γ∧θ)+(2α+α′)∧β∧φ = β∧β′∧θ+d(γ∧θ). (93)

Since B′ is a multiple of φ̄ ∧ θ, or explicitly,

B′ = C̃ φ̄ ∧ θ = dβ′ − (α+ 2α′) ∧ β′ − γ ∧ φ. (94)



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1031

This occurs if and only if

(dβ′ − (α+ 2α′) ∧ β′) ∧ φ̄ = γ ∧ φ ∧ φ̄. (95)

Since dβ′ ∧ φ̄ = β′ ∧ dφ̄+ d(γ ∧ θ) we have

β′ ∧ dφ̄+ d(γ ∧ θ)− (α+ 2α′) ∧ β′ ∧ φ̄ = γ ∧ φ ∧ φ̄. (96)

Therefore, it follows that

dβ′ ∧φ− (α+ 2α′)∧ β ∧ φ̄ = −β′ ∧ (α+ 2α′)∧ φ̄− β′ ∧ β ∧ θ+ d(γ ∧ θ)− (α+ 2α′)∧ β′ ∧ φ̄

= −β′ ∧ β ∧ θ + d(γ ∧ θ). (97)

The same condition arises both ways

β ∧ β′ ∧ θ + d(γ ∧ θ) = γ ∧ φ ∧ φ̄. (98)

Therefore, it holds that B ∧ φ = 0 if and only if B′ ∧ φ̄ = 0, so the condition that B be a
multiple of φ ∧ θ and B′ a multiple of φ̄ ∧ θ are the same.

Finally, let us discuss the effects of the duality transformation X → X̄ and X̄ → X.
This is supposed to suggest the idea that this action look like complex conjugation so θ
can be thought of as purely imaginary and χ as real. The dual connection form is ω̄ and
takes the form

ω̄ =

ᾱ β̄ γ̄
φ̄ −ᾱ− ᾱ′ β̄′

θ φ ᾱ′

 (99)

It is assumed to be gauged so that ᾱ − ᾱ′ is independent of θ as before. Using (99) the
curvature form is found to be dᾱ dβ̄ dγ̄

dφ̄ −d(ᾱ+ ᾱ′) dβ̄′

dθ dφ dᾱ′



+

 β̄ ∧ φ̄+ γ̄ ∧ θ ᾱ ∧ β̄ − β̄ − β̄ ∧ (ᾱ+ ᾱ′) + γ̄ ∧ φ ᾱ ∧ γ̄ + β̄ ∧ β̄′ + γ̄′ ∧ ᾱ′
φ̄ ∧ ᾱ− (ᾱ+ ᾱ′) ∧ φ+ β̄′ ∧ θ φ̄ ∧ β̄ + β̄′ ∧ φ φ̄ ∧ γ̄ − (ᾱ+ ᾱ′) ∧ β + β̄′ ∧ ᾱ′

θ ∧ ᾱ+ φ ∧ φ̄+ ᾱ′ ∧ θ θ ∧ β̄ − φ ∧ (ᾱ+ ᾱ′) + ᾱ′ ∧ φ θ ∧ γ̄ + φ ∧ β̄′


(100)

The connection is then uniquely determined by the conditions for the torsion to vanish

dφ̄+φ̄∧ᾱ∧φ̄+β̄′∧θ = 0, dφ+θ∧β̄−φ∧(ᾱ+2ᾱ′) = 0, dθ+θ∧ᾱ+φ∧φ̄+ᾱ′∧θ = 0.
(101)

Vanishing of the diagonal elements of Ω̄ gives

dᾱ+ β̄ ∧ φ̄+ γ̄ ∧ θ = 0, dα′ − β′ ∧ φ− γ ∧ θ = 0. (102)



P. Bracken / Eur. J. Pure Appl. Math, 13 (4) (2020), 1016-1034 1032

At last, we have

γ̄ ∧ φ ∧ φ̄ = −(dβ̄ + (2ᾱ+ ᾱ′) ∧ φ̄ = −(dβ̄′ − (ᾱ+ 2ᾱ′) ∧ β′) ∧ φ. (103)

This is just B̄ ∧ φ̄ = 0 or equivalently, B̄′ ∧ φ = 0.
Subtracting the two expressions dθ−(α−α′)∧θ+φ∧φ̄ = 0 and dθ−(ᾱ−ᾱ′)∧θ+φ∧φ̄ = 0,

it is found that
(α− α′) ∧ θ = (ᾱ− ᾱ′) ∧ θ. (104)

This implies that α− α′ = ᾱ− ᾱ′ which is equivalent to α+ ᾱ′ = ᾱ+ α′. Grouping from
both sets, there are four torsion equations which remain, and these are

dφ− (2α+ α′) ∧ φ− β′ ∧ θ = 0 dφ̄+ (α+ 2α′) ∧ φ̄+ β ∧ θ = 0

dφ+ (ᾱ+ 2ᾱ′) ∧ φ− β̄ ∧ θ = 0 dφ̄− (2ᾱ+ ᾱ′) ∧ φ̄+ β̄′ ∧ θ = 0.
(105)

Subtracting the equations in (105) pairwise, the following linear combinations are pro-
duced,

(2(α+ᾱ′)+α′+ᾱ)∧φ+(β′−β̄)∧θ = 0, (α+ᾱ′+2(α′+ᾱ))∧φ̄+(β+β̄′)∧θ = 0. (106)

Forming the wedge product with θ, the results in (106) imply that

(2(α+ ᾱ′) + α′ + ᾱ) ∧ φ ∧ θ = 0, (α+ ᾱ′ + 2(α′ + ᾱ)) ∧ φ̄ ∧ θ = 0. (107)

Based on the fact that α+ α′ is a multiple of θ we can write this as κ θ, κ a scalar. Also
reasoning in a similar way, α + ᾱ′ = κθ. Applying these new results in (106), it is found
that

(β′ − β) ∧ θ = −3κθ ∧ φ. (108)

On the other hand , it follows that

d(α′ + ᾱ) = dκ ∧ θ + κ(φ ∧ φ̄− χ ∧ θ) = (β′ − β) ∧ φ̄− (γ + γ̄) ∧ θ. (109)

Taking the exterior product of this with θ we obtain κφ∧ φ̄∧ θ = 0 which leads to κ = 0.
Therefore, it is concluded that

α′ = −ᾱ, ᾱ′ = −α. (110)

Thus β′ − β̄ is in fact a scalar multiple of θ, and similarly as well, so is β − β̄′. Since
α′ + ᾱ = 0, it is concluded that

(β′ − β̄) ∧ φ̄ = (γ + γ̄) ∧ θ = (β − β̄′) ∧ φ. (111)

It follows that β′ = β̄ and β̄′ = β with γ ∧ θ = −γ̄ ∧ θ. The conditions on B and B′ yield
the final conclusion,

γ̄ ∧ φ ∧ φ̄ = −(dβ + α′ + 2α) ∧ β) ∧ φ = −γ ∧ φ ∧ φ̄. (112)



REFERENCES 1033

This implies that γ̄ = −γ. Finally, it follows that the form ω can be summarized in matrix
form,

ω =

 α β γ
φ −(α− ᾱ) β̄
−θ φ̄ −ᾱ

 (113)

where γ̄ = −γ and ω̄ is calculated from ω through

ω̄ = −KωT K. (114)

It may be concluded that Ω = −K ΩT K, and it follows that B′ = B̄ and B̄′ = B hold.

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