# A global invariant for path structures and second order differential
equations

E. Falbel and J. M. Veloso

###### Abstract

We study a global invariant for path structures. The invariant is obtained as
a secondary invariant from a Cartan connection on a canonical bundle
associated to a path structure. It is computed in examples which are defined
in terms of reductions of the path structure. In particular we give a formula
for this global invariant for second order differential equations defined on a
torus $T^{2}$.

##  1 Introduction

Path structures on a 3-manifold are defined by a choice of contact structure
and a decomposition of the contact plane bundle as a direct sum of two line
bundles. This structure was througly studied in the 19th century (see in
particular [T]) as it appears in the description of second order differential
equations and their equivalence under certain transformations (see Section 2
and references [A, IL, BGH]).

In Section 2 we collect definitions and examples. In particular we explain the
relation with ordinary second order equations. In the following section we
define the most important reductions of path structures. The first one is
obtained by fixing a global contact form and it is called strict path
structure. There exists a Cartan bundle $Y_{1}$ and a connection adapted to
that structure (see 2.5) which was used in ([FMMV]) to obtain a classification
of compact 3-manifolds with non-compact automorphism group preserving the
strict path structure. We recall the construction in Proposition 2.3. The
second one, we call enriched path structure following [MM] which were used by
Mion-Mouton to classify certain classes of partial-hyperbolic diffeomorphisms
of three manifolds. It consists of path structures where we fix a line
transverse to the contact distribution. We define an adapted Cartan bundle
$Y_{2}$ and a canonical connection adapted to this structure (see 2.6 and
Proposition 2.5). There exists a natural embedding $Y_{1}\to Y_{2}$ (Section
2.6.2, Proposition 2.7).

In Section 3 we recall the construction of the Cartan bundle $Y$ and the
canonical adapted connection to a path structure on a 3-manifold (see
Proposition 3.3). This construction is due to Cartan in [Car]. Although one
can find modern treatments of this topic in several references (in particular
[IL, BGH]), we include this section for the sake of completeness and because
the conventions we use might differ from others. We obtain a natural embedding
$Y_{2}\to Y$ (see 3.4, Proposition 3.4) and compute the curvature of the
bundle $Y$ in terms of the curvature of $Y_{2}$ (see 3.4.1). The formulas are
used in the computation of the global invariant in the next section. We also
recall the computations by Cartan of the invariants of a second order
differential equation.

In the following section we define the global invariant when $Y_{2}$ admits a
global section (see Definition 4.2). This construction is inspired by an
analogous construction of a Chern-Simons invariant in the case of CR manifolds
given in [BE] (see also [CL] for a relative version which does not depend on
the existence of a global section). In [FV] we defined a similar invariant for
flag structures. Those are manifolds equipped with a decomposition of a
complex contact structure defined on the complexified tangent bundle of a
3-manifold. In this paper we restrict the definition to path structures. We
obtain the expression of the invariant in terms of a reduction $Y_{2}$ or
$Y_{1}$ of the Cartan bundle $Y$ of the path structure (see Proposition 4.5).
We also give a formula of the invariant in the case of a second order
differential equation on the torus (Proposition 4.10). It involves an
integration of fifth order derivatives of the function defining the ordinary
equation in the form $y^{\prime\prime}=F(x,y,y^{\prime})$. We use coordinates
in the projective cotangent bundle over a surface as explained in section 4.1.
We characterize certain families of differential equations on the torus which
have vanishing global invariant in Corollary 4.11. We then compute the
invariant for a family of path structures on tight contact structures on the
torus (see Proposition 5.3) and characterize those structures with vanishing
global invariant, they turn to be flat. Finally we compute the global
invariant for homogeneous path structures on $\hbox{\bf SU}(2)$ (see
Proposition 6.1) and identify the flat structure on the sphere where the
global invariant is maximal.

The authors thank Martin Mion-Mouton for useful discussions.

##  2 Path structures in dimension 3

Path geometries are very related to the theory of second order differential
equations. See a modern treatment in section 8.6 of [IL] and in [BGH] where
the relation to second order differential equations is also explained. Le $M$
be a real three dimensional manifold and $TM$ be its tangent bundle.

###### Definition 2.1

A path structure on $M$ is a choice of two sub-bundles $T^{1}$ and $T^{2}$ in
$TM$ such that $T^{1}\cap T^{2}=\\{0\\}$ and such that $T^{1}\oplus T^{2}$ is
a contact distribution.

The condition that $T^{1}\oplus T^{2}$ be a contact distribution means that,
locally, there exists a one form $\theta\in{T^{*}M}$ such that
$\ker\theta=T^{1}\oplus T^{2}$ and $d\theta\wedge\theta$ is never zero.

One can choose a contact form $\theta$ up to a scalar function. One can
interpret this as follows: one has a ${\mathbb{R}}^{*}$-bundle over the
manifold given by the choice of $\theta$ at each point (one might keep only
positive multiples for simplicity). Over this line bundle one defines the
tautological form $\omega_{x}=\pi^{*}(\theta_{\pi(x)})$. This bundle is
trivial if and only if there exists a global contact form $\theta$. If the
contact distribution is oriented, then there exists a global contact form.
Indeed, using a global metric on the distribution one can define locally a
transversal vector to the distribution taking a Lie bracket of orthonormal
vectors in the distribution. This defines a global 1-form.

Fix $\theta$ and local forms $Z^{1}$ and $Z^{2}$ defining the lines as above
such that $d\theta=Z^{1}\wedge Z^{2}$. There exists global forms $Z^{1}$ and
$Z^{2}$ if and only if there exists global vector fields along the lines.
Clearly, if the contact distribution is oriented, it suffices that there
exists a global vector field along one of the foliations by lines.

Local equivalence (also called point equivalence) between path structures
happens when there exists a local diffeomorphism which gives a correspondence
between the lines defining each structure.

###  2.1 The flat model space

Flat path geometry is the geometry of real flags in ${\mathbb{R}}^{3}$. That
is the geometry of the space of all couples $(p,l)$ where
$p\in{\mathbb{R}}P^{2}$ and $l$ is a real projective line containing $p$. The
space of flags is identified to the quotient

$\hbox{\bf SL}(3,{\mathbb{R}})/B$

where $B$ is the Borel group of all real upper triangular matrices.

The Lie algebra of $\hbox{\bf SL}(3,{\mathbb{R}})$ decomposes into the
following direct sum of vector subspaces:

${\mathfrak{sl}}(3,{\mathbb{R}})={\mathfrak{g}}^{-2}\oplus{\mathfrak{g}}^{-1}\oplus{\mathfrak{g}}^{0}\oplus{\mathfrak{g}}^{1}\oplus{\mathfrak{g}}^{2},$

where

${\mathfrak{g}}^{-2}=\left\\{\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
z&0&0\end{array}\right)\right\\},\ \ \
{\mathfrak{g}}^{-1}=\left\\{\left(\begin{array}[]{ccc}0&0&0\\\ x&0&0\\\
0&y&0\par\end{array}\right)\right\\},$
${\mathfrak{g}}^{0}=\left\\{\left(\begin{array}[]{ccc}u+v&0&0\\\ 0&-2v&0\\\
0&0&-u+v\par\end{array}\right)\right\\},$
${\mathfrak{g}}^{1}=\left\\{\left(\begin{array}[]{ccc}0&a&0\\\ 0&0&b\\\
0&0&0\par\end{array}\right)\right\\},\ \ \
{\mathfrak{g}}^{2}=\left\\{\left(\begin{array}[]{ccc}0&0&c\\\ 0&0&0\\\
0&0&0\par\end{array}\right)\right\\}.$

That is the graded decomposition of ${\mathfrak{sl}}(3,{\mathbb{R}})$ where
${\mathfrak{b}}={\mathfrak{g}}^{0}\oplus{\mathfrak{g}}^{1}\oplus{\mathfrak{g}}^{2}$
corresponds to upper triangular matrices with null trace. The tangent space of
$\hbox{\bf SL}(3,{\mathbb{R}})/B$ at $[B]$ is identified to

${\mathfrak{sl}}(3,{\mathbb{R}})/{\mathfrak{b}}={\mathfrak{g}}^{-2}\oplus{\mathfrak{g}}^{-1}.$

###  2.2 Examples

Example I Consider the Heisenberg group

$\mathbf{Heis}(3)=\\{\ (z,t)\ |\ z\in{\mathbb{C}},\,t\in{\mathbb{R}}\ \\}$

with multiplication defined by
$(z_{1},t_{1})\star(z_{2},t_{2})=(z_{1}+z_{2},t_{1}+t_{2}+2\hbox{\rm
Im}\,{z_{1}\overline{z_{2}}})$. The contact form

$\theta=dt-xdy-ydx$

is invariant under left multiplications (also called Heisenberg translations).
If $\Lambda\subset\mathbf{Heis}(3)$ is a lattice then the quotient
$\Lambda\setminus\mathbf{Heis}(3)$ is a circle bundle over the torus with a
globaly defined contact form.

A lattice $\Lambda$ determines a lattice $\Gamma\subset{\mathbb{C}}$
corresponding to projection in the exact sequence

$0\rightarrow{\mathbb{R}}\rightarrow\mathbf{Heis}(3)\rightarrow{\mathbb{C}}\rightarrow
0.$

There are many global vector fields in the distribution defined by $\theta$
invariant under $\Lambda$, it suffices to lift an invariant vector field on
${\mathbb{C}}$ under $\Gamma$. All circle bundles obtained in this way are not
trivial and the fibers are transverse to the distribution.

Example II. Here we consider the torus $T^{3}$ with coordinates $(x,y,t)$
($\mod 1$) and the global contact form

$\theta_{n}=\cos(2\pi nt)dx-\sin(2\pi nt)dy.$

There are two canonical global vector fields on the distribution given by
$\frac{\partial}{\partial t}$ and $\sin(2\pi nt)\frac{\partial}{\partial
x}+\cos(2\pi nt)\frac{\partial}{\partial y}$. In this example, the fiber given
by the coordinate $t$ has tangent space contained in the distribution.

Example III. An homogeneous example is the Lie group $\hbox{\bf SU}(2)$ with
left invariant vector fields $X$ and $Y$ with $Z=[X,Y]$ and cyclic commutation
relations. The vector fields $X$ and $Y$ define a path structure on $\hbox{\bf
SU}(2)$.

Example IV. Another homogeneous example is the Lie group $\hbox{\bf
SL}(2,{\mathbb{R}})$ with left invariant vector fields $X$ and $Y$ with
$Z=[X,Y]$ with $[Z,X]=X$ and $[Z,Y]=-Y$ given by generators

$X=\left(\begin{array}[]{cc}0&1\\\ 0&0\par\end{array}\right),\
Y=\left(\begin{array}[]{cc}0&0\\\ 1&0\par\end{array}\right),\
Z=\left(\begin{array}[]{cc}1&0\\\ 0&-1\par\end{array}\right).$

The path structure defined by $X$ and $Y$ induces a path structure on the
quotient $\Gamma\setminus\hbox{\bf SL}(2,{\mathbb{R}})$ by a discrete torsion
free subgroup $\Gamma\subset\hbox{\bf SL}(2,{\mathbb{R}})$. This structure is
invariant under the flow defined by right multiplication by $e^{tZ}$.

Example V. Let $\Sigma$ be a surface equipped with a Riemannian metric. The
geodesic flow on the unit tangent bundle $T^{1}\Sigma$ defines a distribution
which, together with the distribution defined by the vertical fibers of the
projection of the unit tangent bundle on $\Sigma$, defines a path structure
which is not invariant under the geodesic flow. For
$\Sigma=H^{2}_{\mathbb{R}}$, the hyperbolic space, we obtain
$T^{1}\Sigma=\hbox{\bf PSL}(2,{\mathbb{R}})$ with distributions defined by the
left invariant distributions $X-Y$ and $Z$ (using the same generators of the
Lie algebra as in the previous example).

Example VI Let $M$ be a three manifold equipped with a path structure
$D=T^{1}\oplus T^{2}\subset TM$. Suppose $D$ is orientable and choose a
section $u$ of $T^{1}$. Each section $v$ of $T^{2}$ such that $(u,v)$ is
positive gives rise to a CR structure. Indeed we define $Ju=v$ and $Jv=-u$.
The choice of $v$ corresponds to a section of an ${\mathbb{R}}^{*}_{+}$-bundle
over $M$. Reciprocally given a CR structure on $M$, defined by $J:D\rightarrow
D$, one can associate path structures corresponding to a choice $T^{1}\subset
D$ and defining then $T^{2}=J(T^{1})$.

###  2.3 Path structures and second order differential equations

This is studied since a long time (see [T], [IL] and [BGH]). It turns out that
path structures can be obtained putting together second order differential
equations in one variable. Indeed, a second order differential equation in one
variable is described locally as

$\frac{d^{2}y}{dx^{2}}=F(x,y,\frac{dy}{dx}).$

This defines a path structure on a neighborhood of a point in
${\mathbb{R}}^{3}$ with coordinates $(x,y,p)$:

$L_{1}=\ker\\{dp-Fdx\\}\cap\ker\\{dy-pdx\\},\ \ \ L_{2}=\ker dx\cap\ker dy.$

The contact structure is defined by the form

$\theta=dy-pdx.$

Defining the forms $Z^{1}=dx$ and $Z^{2}=dp-Fdx$, one has that
$d\theta=Z^{1}\wedge Z^{2}$.

One can show easily that every path structure is, in fact, locally equivalent
to a second order equation. That is, there exists local coordinates such that
$L_{1}$ and $L_{2}$ are defined via a second order ODE as above.

###  2.4 Reductions of path structures

We will describe two reductions of path geometry corresponding to subgroups
$G_{1}\subset G_{2}\subset SL(3,{\mathbb{R}})$ where

$G_{1}=\left\\{\left(\begin{array}[]{ccc}a&0&0\\\ \star&\frac{1}{a^{2}}&0\\\
\star&\star&a\par\end{array}\right)\right\\}\ \ \ $

and

$G_{2}=\left\\{\left(\begin{array}[]{ccc}a&0&0\\\ \star&\frac{1}{ab}&0\\\
\star&\star&b\par\end{array}\right)\right\\}.$

The models are $G_{1}/{\mathbb{R}}^{*}$ and $G_{2}/{{\mathbb{R}}^{*}}^{2}$ and
correspond to the Heisenberg group where in the first model we fix a contact
form and, in the second, a transverse line to the contact distribution.

Other reductions of the $G_{2}$-structure might occur, namely by choosing
other embeddings of ${\mathbb{R}}^{*}$ into $G_{2}$. They appear naturally
when certain components of the curvature of the Cartan connections on $Y_{2}$
or $Y$ are non-vanishing.

We will construct coframe bundles $Y_{1},Y_{2}$ and a principal bundle $Y$
over $M$ with structure groups ${\mathbb{R}}^{*},{{\mathbb{R}}^{*}}^{2}$ and
the Borel group $B$ together with Cartan connections and canonical embeddings

$Y_{1}\rightarrow Y_{2}\rightarrow Y.$

They correspond to a strict path structure, an enriched path structure (see
next sections for definitions) and finally, a path structures on the manifold
$M$.

###  2.5 Path structures with a fixed contact form: strict path structures.

In this section we fix a contact form and recall the reduction of the
structure group of a path geometry obtained in [FV] where we called the path
structure with a fixed contact form a pseudo flag structure. This structure is
called strict path structure in [FMMV].

$G_{1}$ denotes from now on the subgroup of $\hbox{\bf SL}(3,{\mathbb{R}})$
defined by

$G_{1}=\left\\{\left(\begin{array}[]{ccc}a&0&0\\\ x&\frac{1}{a^{2}}&0\\\
z&y&a\par\end{array}\right)\ |\
{a\in{\mathbb{R}}^{*},(x,y,z)\in{\mathbb{R}}^{3}}\right\\}$

and $P_{1}\subset G_{1}$ the subgroup defined by

$P_{1}=\left\\{\left(\begin{array}[]{ccc}a&0&0\\\ 0&\frac{1}{a^{2}}&0\\\
0&0&a\par\end{array}\right)\right\\}.\ \ \ $

Writing the Maurer-Cartan form of $G_{1}$ as the matrix

$\left(\begin{array}[]{ccc}w&0&0\\\ \theta^{1}&-2w&0\\\
\theta&\theta^{2}&w\par\end{array}\right)$

one obtains the Maurer-Cartan equations:

$d\theta+\theta^{2}\wedge\theta^{1}=0$ $d\theta^{1}-3w\wedge\theta^{1}=0$
$d\theta^{2}+3w\wedge\theta^{2}=0$ $dw=0.$

$G_{1}$ is the automorphism group of the canonical left-invariant strict path
structure of $\mathbf{Heis}(3)$, and that its action induces an identification
of $\mathbf{Heis}(3)$ with the homogeneous space $X=G_{1}/P_{1}$.

Let $M$ be a three-manifold equipped with a strict path structure
$(E^{1},E^{2},\theta)$ defined by two one dimensional bundles $E^{1}$ and
$E^{2}$ and contact form $\theta$. We let $R$ be the associated Reeb vector
field (satisfying $\iota_{R}d\theta=0$ and $\theta(R)=1$). Now let $X_{1}\in
E^{1}$, $X_{2}\in E^{2}$ be such that $d\theta(X_{1},X_{2})=1$. The dual
coframe of $(X_{1},X_{2},R)$ is $(\theta^{1},\theta^{2},\theta)$, for two
1-forms $\theta 1$ and $\theta_{2}$ verifying
$d\theta=\theta^{1}\wedge\theta^{2}$.

At any point $x\in M$, one can look at the coframes of the form

$\omega^{1}={a^{3}}\theta^{1}(x),\ \omega^{2}=\frac{1}{a^{3}}\theta^{2}(x),\
\omega=\theta(x)$

for $a\in{\mathbb{R}}^{*}$.

###### Definition 2.2

We denote by $p_{1}:Y_{1}\rightarrow M$ the ${\mathbb{R}}^{*}$-coframe bundle
over $M$ given by the set of coframes $(\omega,\omega^{1},\omega^{2})$ of the
above form.

We will denote the tautological forms defined by
$\omega^{1},\omega^{2},\omega$ using the same letters. That is, we write
$\omega^{i}$ at the coframe $(\omega^{1},\omega^{2},\omega)$ to be
$p_{1}^{*}(\omega^{i})$.

###### Proposition 2.3

There exists a unique Cartan connection on $Y_{1}$

$\pi_{1}=\left(\begin{array}[]{ccc}w&0&0\\\ \omega^{1}&-2w&0\\\
\omega&\omega^{2}&w\par\end{array}\right)$

such that its curvature form is of the form

$\Pi_{1}=d\pi_{1}+\pi_{1}\wedge\pi_{1}=\left(\begin{array}[]{ccc}dw&0&0\\\
\omega\wedge\tau^{1}&-2dw&0\\\
0&-\omega\wedge\tau^{2}&dw\par\end{array}\right)$

with $\tau^{1}\wedge\omega^{2}=\tau^{2}\wedge\omega^{1}=0$.

Observe that the condition
$\tau^{1}\wedge\omega^{2}=\tau^{2}\wedge\omega^{1}=0$ implies that we may
write $\tau^{1}=\tau^{1}_{2}\omega^{2}$ and $\tau^{2}=\tau^{2}_{1}\omega^{1}$.
The structure equations are

$d\omega+\omega^{2}\wedge\omega^{1}=0,$
$d\omega^{1}-3w\wedge\omega^{1}=\omega\wedge\tau^{1},$
$d\omega^{2}+3w\wedge\omega^{2}=-\omega\wedge\tau^{2}.$

The proof of the proposition is given in [FMMV] and [FV].

Bianchi identities are obtained differentiating the structure equations. They
are described in the following equations:

$dw=C\omega\wedge\omega^{1}+D\omega\wedge\omega^{2}+S\omega^{1}\wedge\omega^{2},$
(1)
$d\tau^{1}_{2}-6\tau^{1}_{2}w+3D\omega^{1}=\tau^{1}_{20}\omega+\tau^{1}_{22}\omega^{2}$
(2)
$d\tau^{2}_{1}+6\tau^{2}_{1}w+3C\omega^{2}=\tau^{2}_{10}\omega+\tau^{2}_{11}\omega^{1}$
(3)

###  2.6 Path structures with a fixed transverse line: enriched path
structures.

In this section we introduce a coframe bundle and a Cartan connection
associated to a path structure with a fixed transverse line to the the contact
distribution.

The model space is the homogeneous space which is the quotient of the group of
lower triangular matrices in $SL(3,{\mathbb{R}})$ by the subgroup of diagonal
matrices. The Maurer-Cartan form is the Lie algebra valued form which can be
represented by

$\pi=\left(\begin{array}[]{ccc}\varphi+w&0&0\\\ \omega^{1}&-2w&0\\\
\omega&\omega^{2}&-\varphi+w\end{array}\right)$

The Maurer-Cartan equations $d\pi+\pi\wedge\pi=0$ are given by

$d\omega=2\varphi\wedge\omega+\omega^{1}\wedge\omega^{2}$
$d\omega^{1}=\varphi\wedge\omega^{1}+3w\wedge\omega^{1}$
$d\omega^{2}=\varphi\wedge\omega^{2}-3w\wedge\omega^{2}.$

Let $M$ be a three manifold equipped with a path structure $D=T^{1}\oplus
T^{2}\subset TM$. We fix a transverse line $L$ so that $TM=T^{1}\oplus
T^{2}\oplus L$.

We suppose $X_{1}\in T^{1}$, $X_{2}\in T^{2}$ and $X\in L$ is a frame. The
dual coframe is $\theta^{1}$, $\theta^{2}$ and $\theta$. Observe that $\theta$
is simply a form with $\ker\theta=D$. One can define a coframe bundle defined
by all coframes:

$\omega^{1}=a^{1}\theta^{1},\ \omega^{2}=a^{2}\theta^{2},\
\omega=\lambda\theta.$

where we will suppose, for simplicity, that $a^{1},a^{2},\lambda>0$.

A reduction of this coframe bundle is obtained by imposing that each coframe
verifies

$d\omega_{|D}=(\omega^{1}\wedge\omega^{2})_{|D}$

for an extension of the 1-form such that $\ker\omega=D$. This relation does
not depend on the particular extension of a form $\omega$ defined at a point
because $d\omega_{|D}(X,Y)=-\omega([X,Y])$ for any vector fields $X$ and $Y$
which are sections of the distribution $D$.

###### Definition 2.4

We denote by $p_{2}:Y_{2}\rightarrow M$ the ${{\mathbb{R}}^{*}}^{2}$-coframe
bundle over $M$ given by the set of 1-forms $(\omega,\omega^{1},\omega^{2})$
defined above. The structure group ${{\mathbb{R}}^{*}}^{2}$ acts as follows

$(\omega^{\prime},\omega^{\prime 1},\omega^{\prime
2})=(\omega,\omega^{1},\omega^{2})\left(\begin{array}[]{cccc}\lambda&0&0\\\
0&a^{1}&0\\\ 0&0&a^{2}\\\ \end{array}\right)$

where $\lambda,a^{1},a^{2}\in{\mathbb{R}}_{+}^{*}$ with $a^{1}a^{2}=\lambda$.

In order to define a Cartan connection on $Y_{2}$ we start taking the
tautological forms corresponding to the forms $\omega,\omega^{1},\omega^{2}$,
which we will denote by the same letters by abuse of notation.

Using a coframe section $(\theta,\theta^{1},\theta^{2})$ on $M$ one can
express the tautological forms as

$\omega=\lambda p_{2}^{*}(\theta),\ \omega^{1}=a^{1}p_{2}^{*}(\theta^{1}),\
\omega^{2}=a^{2}p_{2}^{*}(\theta^{2}),$

with $a^{1}a^{2}=\lambda$.

We need to define two forms $\varphi$ and $w$ corresponding to the vertical
directions

Observe first that we have

$d\omega=\frac{d\lambda}{\lambda}\wedge\omega+\omega^{1}\wedge\omega^{2}\ \
\mbox{mod}(\omega)$

and therefore one may write

$d\omega=2\varphi\wedge\omega+\omega^{1}\wedge\omega^{2}$ (4)

where $\varphi$ restricted to the vertical fiber is
$\frac{d\lambda}{2\lambda}$. The form $\varphi$ is not yet fixed and any other
form $\varphi^{\prime}$ satisfying the equation satisfies

$\varphi-\varphi^{\prime}=s\omega$

where $s$ is a funtion on $Y_{2}$.

Differentiating the forms $\omega^{1}$ and $\omega^{2}$ we obtain new forms
which correspond to the coordinates $a^{1},a^{2}$ :

$d\omega^{1}=\frac{da^{1}}{a^{1}}\wedge\omega^{1}+a^{1}d\theta^{1}$ and
$d\omega^{2}=\frac{da^{2}}{a^{2}}\wedge\omega^{2}+a^{2}d\theta^{2}$.

Observing that

$\frac{d\lambda}{\lambda}=\frac{da^{1}}{a^{1}}+\frac{da^{2}}{a^{2}}$

we can write

$d\omega^{1}=\frac{d\lambda}{2\lambda}\wedge\omega^{1}+\frac{1}{2}\left(\frac{da^{1}}{a^{1}}-\frac{da^{2}}{a^{2}}\right)\wedge\omega^{1}+a^{1}d\theta^{1}$
$d\omega^{2}=\frac{d\lambda}{2\lambda}\wedge\omega^{2}-\frac{1}{2}\left(\frac{da^{1}}{a^{1}}-\frac{da^{2}}{a^{2}}\right)\wedge\omega^{2}+a^{2}d\theta^{2}$

Now we can make the first right hand term of each equation to be
$\varphi\wedge\omega^{1}$ and $\varphi\wedge\omega^{2}$ respectively by adding
terms in $\omega,\omega^{1},\omega^{2}$ to $\frac{d\lambda}{2\lambda}$. The
terms in $\omega^{1}\wedge\omega^{2}$ not appearing in these first terms can
be absorbed in the second term in each equation. It remains a last term in
each equation that we denote by $\omega\wedge\tau^{1}$ and
$-\omega\wedge\tau^{2}$ respectively. We proved the following:

###### Lemma 2.1

There exists forms $w,\tau^{1},\tau^{2}$ defined on $Y_{2}$ such that

$d\omega^{1}=\varphi\wedge\omega^{1}+3w\wedge\omega^{1}+\omega\wedge\tau^{1}\
\mbox{and}\ \
d\omega^{2}=\varphi\wedge\omega^{2}-3w\wedge\omega^{2}-\omega\wedge\tau^{2}.$
(5)

The forms $\tau^{1}$ and $\tau^{2}$ are horizontal, that is, they vanish on
vectors tangent to the fibers of $Y_{2}\rightarrow M$. Moreover, writing
$\omega^{1}=a^{1}\theta^{1}$, $\omega^{2}=a^{2}\theta^{2}$,
$\omega=\lambda\theta$ for a choice of sections on $M$, one has
$\varphi=\frac{d\lambda}{2\lambda}$ and
$6w=\frac{da^{1}}{a^{1}}-\frac{da^{2}}{a^{2}}$ modulo the tautological forms
of the fiber bundle $Y_{2}$.

Let $\varphi^{\prime},w^{\prime},{{\tau^{\prime}}^{1}}$ and ${\tau^{\prime
2}}$ be other forms satisfying equations above. Taking the difference we
obtain

$0=(\varphi-\varphi^{\prime})\wedge\omega^{1}+3(w-w^{\prime})\wedge\omega^{1}+\omega\wedge(\tau^{1}-\tau^{\prime
1})$

and

$0=(\varphi-\varphi^{\prime})\wedge\omega^{2}-3(w-w^{\prime})\wedge\omega^{2}-\omega\wedge(\tau^{2}-\tau^{\prime
2})$

Therefore, as $\varphi-\varphi^{\prime}=s\omega$, we can write

$0=-3\omega^{1}\wedge(w-w^{\prime})+\omega\wedge(s\omega^{1}+\tau^{1}-\tau^{\prime
1})$

and

$0=3\omega^{2}\wedge(w-w^{\prime})-\omega\wedge(-s\omega^{2}+\tau^{2}-\tau^{\prime
2}).$

By Cartan’s lemma we obtain

$w-w^{\prime}=a\omega,$ $\tau^{1}-\tau^{\prime
1}=-3a\omega^{1}-{s}\omega^{1}+b^{1}\omega,$ $\tau^{2}-\tau^{\prime
2}=-3a\omega^{2}+{s}\omega^{2}+b^{2}\omega.$

Now, we can impose that
$\tau^{1}=\tau^{1}_{1}\omega^{1}+\tau^{1}_{2}\omega^{2}$ and
$\tau^{2}=\tau^{2}_{1}\omega^{1}+\tau^{2}_{2}\omega^{2}$ by choosing
convenient $b^{1}$ and $b^{2}$ (or by simply considering, from the beginning,
$\tau^{1}$ and $\tau^{2}$ with no terms in $\omega$). Moreover, one can choose
unique $a$ and $s$ so that $\tau^{1}_{1}=0$ and $\tau^{2}_{2}=0$. We conclude
that

###### Lemma 2.2

There exists unique forms $\varphi,w,\tau^{1},\tau^{2}$ defined on $Y_{2}$
such that

$d\omega=2\varphi\wedge\omega+\omega^{1}\wedge\omega^{2}$
$d\omega^{1}=\varphi\wedge\omega^{1}+3w\wedge\omega^{1}+\omega\wedge\tau^{1}$
$d\omega^{2}=\varphi\wedge\omega^{2}-3w\wedge\omega^{2}-\omega\wedge\tau^{2}$

with $\tau^{1}\wedge\omega^{2}=\tau^{2}\wedge\omega^{1}=0$.

Bianchi identities are obtained differentiating the above equations:

###### Lemma 2.3

There exists a 1-form $\psi$ such that

$d\varphi=\omega\wedge\psi$ (6)

The form $\psi$ may be chosen satisfying $\psi=A\omega^{1}+B\omega^{2}$ and
$d\psi=-2\varphi\wedge\psi+\omega\wedge\alpha$ where $A,B$ are functions on
$Y_{2}$ and $\alpha$ is a 1-form on $Y_{2}$.

Proof. Differentiating equation
$d\omega=2\varphi\wedge\omega+\omega^{1}\wedge\omega^{2}$ one obtains, using
equations 5, that $d\varphi\wedge\omega=0$, that is,

$d\varphi=\omega\wedge\psi$ (7)

for a 1-form $\psi$ defined on $Y_{2}$.

Differentiating $d\varphi=\omega\wedge\psi$ one has

$0=d\omega\wedge\psi-\omega\wedge
d\psi=(2\varphi\wedge\omega+\omega^{1}\wedge\omega^{2})\wedge\psi-\omega\wedge
d\psi=\omega^{1}\wedge\omega^{2}\wedge\psi-\omega\wedge(d\psi+2\varphi\wedge\psi).$

Using Cartan’s lemma, $\psi=A\omega^{1}+B\omega^{2}$ modulo $\omega$, and we
certainly can choose $\psi$ satisfying $d\varphi=\omega\wedge\psi$ with
$\psi=A\omega^{1}+B\omega^{2}$. We conclude that

$d\psi+2\varphi\wedge\psi=\omega\wedge\alpha.$

$\Box$

Equation $dd\omega^{1}=0$ gives after simplifications

$\displaystyle
0=d(\varphi+3w)\wedge\omega^{1}+\omega\wedge\omega^{2}(d\tau^{1}_{2}+2\tau^{1}_{2}(\varphi-3w)).$
(8)

Analogously, $dd\omega^{2}=0$ simplifies to

$\displaystyle
0=d(\varphi-3w)\wedge\omega^{2}-\omega\wedge\omega^{1}(d\tau^{2}_{1}+2\tau^{2}_{1}(\varphi+3w)).$
(9)

Using the previous lemma we may write

$dw=C\omega\wedge\omega^{1}+D\omega\wedge\omega^{2}+S\omega^{1}\wedge\omega^{2},$

where $C,D$ and S are functions on $Y_{2}$.

We can represent the equations above as a matrix equation whose entries are
differential forms. The forms are disposed in the Lie algebra
${\mathfrak{b}}\subset{\mathfrak{sl}}(3,{\mathbb{R}})$ (the Lie algebra of
lower triangular matrices) and we obtain the following Proposition.

###### Proposition 2.5

Let $Y_{2}$ be the adapted principal bundle constructed above associated to an
enriched path structure on a manifold $M$. Then there exists a unique Cartan’s
connection with values in ${\mathfrak{b}}$:

$\pi_{2}=\left(\begin{array}[]{ccc}\varphi+w&0&0\\\ \omega^{1}&-2w&0\\\
\omega&\omega^{2}&-\varphi+w\end{array}\right)$

with curvature:

$\Pi_{2}=d\pi_{2}+\pi_{2}\wedge\pi_{2}=\left(\begin{array}[]{ccc}\omega\wedge\psi+W&0&0\\\
\tau^{1}_{2}\omega\wedge\omega^{2}&-2W&0\\\
0&-\tau^{2}_{1}\omega\wedge\omega^{1}&-\omega\wedge\psi+W\end{array}\right)$
(10)

where
$W=C\omega\wedge\omega^{1}+D\omega\wedge\omega^{2}+S\omega^{1}\wedge\omega^{2}$
and $\psi=A\omega^{1}+B\omega^{2}$.

####  2.6.1 More Bianchi identities

  * •

Substituting the expressions above in equations 8 and 9 we obtain

$d\tau^{1}_{2}+2\tau^{1}_{2}(\varphi-3w)+(B+3D)\omega^{1}=\tau^{1}_{20}\omega+\tau^{1}_{22}\omega^{2}.$
(11)

  * •

Analogously we obtain

$d\tau^{2}_{1}+2\tau^{2}_{1}(\varphi+3w)-(A-3C)\omega^{2}=\tau^{2}_{10}\omega+\tau^{2}_{11}\omega^{1}.$
(12)

From the last two equations we obtain the following

###### Proposition 2.6

If the adapted connection of $Y_{2}$ has nul torsion and

$dw=S\omega^{1}\wedge\omega^{2},$

then $d\varphi=0$.

  * •

Analogously, $dd\varphi=0$ simplifies to

$0=\omega\wedge\omega^{1}(dA+3A(\varphi+w))+\omega\wedge\omega^{2}(dB+3B(\varphi-w))$

and we obtain

$\displaystyle dA+3A(\varphi+w)=A_{0}\omega+A_{1}\omega^{1}+A_{2}\omega^{2},$
(13) $\displaystyle
dB+3B(\varphi-w)=B_{0}\omega+B_{1}\omega^{1}+B_{2}\omega^{2},$ (14)

with $A_{2}=B_{1}$.

  * •

Also, $ddw=0$ simplifies to

$0=\omega\wedge\omega^{1}(dC+3C(\varphi+w))+\omega\wedge\omega^{2}(dD+3D(\varphi-w))+\omega^{1}\wedge\omega^{2}(dS+2S\varphi)$

and we obtain

$\displaystyle dC+3C(\varphi+w)=C_{0}\omega+C_{1}\omega^{1}+C_{2}\omega^{2},$
(15) $\displaystyle
dD+3D(\varphi-w)=D_{0}\omega+D_{1}\omega^{1}+D_{2}\omega^{2},$ (16)
$\displaystyle dS+2S\varphi=S_{0}\omega+S_{1}\omega^{1}+S_{2}\omega^{2},$ (17)

with $C_{2}-D_{1}+S_{0}=0$.

###### Lemma 2.4

If $\tau^{1}=\tau^{2}=C=D=0$

$d\varphi=0.$

Proof. From the last formulae we obtain that $\psi$ is a multiple of $\omega$
and the result follows.  $\Box$

####  2.6.2 The embedding $\iota_{1}:Y_{1}\to Y_{2}$

Given a path structure with a fixed contact form $\omega$ we obtained first a
coframe bundle $Y_{1}$ and one can also obtain a canonical transverse
direction by considering the Reeb vector field associated to $\omega$. One
obtains then a coframe bundle $Y_{2}$ of last section.

Given a coframe $(\omega,\omega^{1},\omega^{2})\in Y_{1}$ one can view the
same coframe as a coframe of $Y_{2}$. This gives an embedding

$\iota_{1}:Y_{1}\rightarrow Y_{2}.$

By abuse of language we may write the connection forms of $Y_{1}$ and $Y_{2}$
using the same letters and then obtain:

###### Proposition 2.7

There exists a unique embedding $\iota_{1}:Y_{1}\rightarrow Y_{2}$ satisfying
$\iota_{1}^{*}(\omega)=\omega$, $\iota_{1}^{*}(\omega^{1})=\omega^{1}$ and
$\iota_{1}^{*}(\omega^{2})=\omega^{2}$. Moreover, for this embedding,
$\iota_{1}^{*}(\varphi)=0$ and $\iota_{1}^{*}(w)=w$.

Proof. If unicity is not satisfied one can obtain the same forms pulling back
a different coframe. But from the transformations of the coframe,

$\displaystyle\tilde{\omega}$ $\displaystyle=\frac{a}{b}\,\omega$
$\displaystyle\tilde{\omega}^{1}$ $\displaystyle=a^{2}b\,\omega^{1}$
$\displaystyle\tilde{\omega}^{2}$
$\displaystyle=\frac{1}{ab^{2}}\,\omega^{2}.$

We obtain then that $a=b=1$ and the embedding is uniquely determined by the
conditions.

Comparing the structure equations of both structures we further get the
equations $\iota_{1}^{*}(\varphi)=0$ and $\iota_{1}^{*}(w)=w$.  $\Box$

##  3 The Cartan connection of a path structure

We review in this section the construction of a Cartan connection. The
construction is due to E. Cartan in [Car] and one can read a modern
description of it in [IL]. We include this section in order to fix our
conventions and describe the embedding of $Y_{2}$ into the corresponding fiber
bundle associated to a path geometry (see 3.4 and 3.4.1) which will be used to
define the global invariant.

The Maurer-Cartan form on $SL(3,{\mathbb{R}})$ is given by a form with values
in the Lie algebra ${\mathfrak{sl}}(3,{\mathbb{R}})$ :

$\pi=\left(\begin{array}[]{ccc}\varphi+w&\varphi^{2}&\psi\\\
\omega^{1}&-2w&\varphi^{1}\\\ \omega&\omega^{2}&-\varphi+w\end{array}\right)$

satisfying the equation $d\pi+\pi\wedge\pi=0$. That is

$d\omega=\omega^{1}\wedge\omega^{2}+2\varphi\wedge\omega$
$d\omega^{1}=\varphi\wedge\omega^{1}+3w\wedge\omega^{1}+\omega\wedge\varphi^{1}$
$d\omega^{2}=\varphi\wedge\omega^{2}-3w\wedge\omega^{2}-\omega\wedge\varphi^{2}$
$dw=-\frac{1}{2}\varphi^{2}\wedge\omega^{1}+\frac{1}{2}\varphi^{1}\wedge\omega^{2}$
$d\varphi=\omega\wedge\psi-\frac{1}{2}\varphi^{2}\wedge\omega^{1}-\frac{1}{2}\varphi^{1}\wedge\omega^{2}$
$d\varphi^{1}=\psi\wedge\omega^{1}-\varphi\wedge\varphi^{1}+3w\wedge\varphi^{1}$
$d\varphi^{2}=-\psi\wedge\omega^{2}-\varphi\wedge\varphi^{2}-3w\wedge\varphi^{2}$
$d\psi=\varphi^{1}\wedge\varphi^{2}+2\psi\wedge\varphi.$

###  3.1 The ${\mathbb{R}}^{*}$-bundle of contact forms and an adapted coframe
bundle

We recall the construction of the ${\mathbb{R}}^{*}$-bundle of contact forms.
Define ${E}$ to be the ${\mathbb{R}}^{*}$-bundle of all forms $\theta$ on $TM$
such that $\ker\theta=T^{1}\oplus T^{2}$. Remark that this bundle is trivial
if and only if there exists a globally defined non-vanishing form $\theta$.
Define the set of forms $\theta^{1}$ and $\theta^{2}$ on $M$ satisfying

$\theta^{1}(T^{1})\neq 0\ \ {\mbox{and}}\ \theta^{2}(T^{2})\neq 0.$
$\ker\theta^{1}_{|\ker\theta}=T^{2}\ \ {\mbox{and}}\ \
\ker\theta^{2}_{|\ker\theta}=T^{1}.$

Fixing one choice, all others are given by $\theta^{\prime
i}=a^{i}\theta^{i}+v^{i}\theta$.

On $E$ we define the tautological form $\omega$. That is
$\omega_{\theta}=\pi^{*}(\theta)$ where $\pi:{E}\rightarrow M$ is the natural
projection. We also consider the tautological forms defined by the forms
$\theta^{1}$ and $\theta^{2}$ over the line bundle $E$. That is, for each
$\theta\in E$ we let $\omega^{i}_{\theta}=\pi^{*}(\theta^{i})$. At each point
$\theta\in E$ we have the family of forms defined on $E$:

$\omega^{\prime}=\omega$ $\omega^{\prime 1}=a^{1}\omega^{1}+v^{1}\omega$
$\omega^{\prime 2}=a^{2}\omega^{2}+v^{2}\omega$

We may, moreover, suppose that

$d\theta=\theta^{1}\wedge\theta^{2}\ \ \mbox{modulo}\ \theta$

and therefore

$d\omega=\omega^{1}\wedge\omega^{2}\ \ \mbox{modulo}\ \omega.$

This imposes that $a^{1}a^{2}=1$.

Those forms vanish on vertical vectors, that is, vectors in the kernel of the
map $TE\rightarrow TM$. In order to define non-horizontal 1-forms we let
$\theta$ be a section of $E$ over $M$ and introduce the coordinate
$\lambda\in{\mathbb{R}}^{*}$ in $E$. By abuse of notation, let $\theta$ denote
the tautological form on the section $\theta$. We write then the tautological
form $\omega$ over $E$ is

$\omega_{\lambda\theta}=\lambda\theta.$

Differentiating this formula we obtain

$d\omega=2\varphi\wedge\omega+\omega^{1}\wedge\omega^{2}$ (18)

where $\varphi=\frac{d\lambda}{2\lambda}$ modulo
$\omega,\omega^{1},\omega^{2}$. Here $\frac{d\lambda}{2\lambda}$ is a form
intrinsically defined on $E$ up to horizontal forms. We obtain in this way a
coframe bundle satisfying equation 18 over $E$:

$\omega^{\prime}=\omega$ $\omega^{\prime 1}=a^{1}\omega^{1}+v^{1}\omega$
$\omega^{\prime 2}=a^{2}\omega^{2}+v^{2}\omega$
$\varphi^{\prime}=\varphi-\frac{1}{2}a^{1}v^{2}\omega^{1}+\frac{1}{2}a^{2}v^{1}\omega^{2}+s\omega$

$v^{1},v^{2},s\in{\mathbb{R}}$ and $a^{1},a^{2}\in{\mathbb{R}}^{*}$ such that
$a^{1}a^{2}=1$.

###### Definition 3.1

We denote by $Y$ the coframe bundle $Y\rightarrow E$ given by the set of
1-forms $\omega,\omega^{1},\omega^{2},\varphi$ as above. Two coframes are
related by

$(\omega^{\prime},\omega^{\prime 1},\omega^{\prime
2},\varphi^{\prime})=(\omega,\omega^{1},\omega^{2},\varphi)\left(\begin{array}[]{cccc}1&v^{1}&v^{2}&s\\\
0&a^{1}&0&-\frac{1}{2}a^{1}v^{2}\\\ 0&0&a^{2}&\frac{1}{2}a^{2}v^{1}\\\
0&0&0&1\par\end{array}\right)$

where and $s,v^{1},v^{2}\in{\mathbb{R}}$ and $a^{1},a^{2}\in{\mathbb{R}}^{*}$
satisfy $a^{1}a^{2}=1$.

The bundle $Y$ can also be fibered over the manifold $M$. In order to describe
the bundle $Y$ as a principal fiber bundle over $M$ observe that choosing a
local section $\theta$ of $E$ and forms $\theta^{1}$ and $\theta^{2}$ on $M$
such that $d\theta=\theta^{1}\wedge\theta^{2}$ one can write a trivialization
of the fiber

$\omega=\lambda\theta$ $\omega^{1}=a^{1}\theta^{1}+v^{1}\lambda\theta$
$\omega^{2}=a^{2}\theta^{2}+v^{2}\lambda\theta$
$\varphi=\frac{d\lambda}{2\lambda}-\frac{1}{2}a^{1}v^{2}\theta^{1}+\frac{1}{2}a^{2}v^{1}\theta^{2}+s\theta,$

where $v^{1},v^{2},s\in{\mathbb{R}}$ and $a^{1},a^{2}\in{\mathbb{R}}^{*}$ such
that $a^{1}a^{2}=\lambda$. Here the coframe
$\omega,\omega^{1},\omega^{2},\varphi$ is seen as composed of tautological
forms.

The group $H$ acting on the right of this bundle is

$H=\left\\{\left(\begin{array}[]{cccc}\lambda&v^{1}\lambda&v^{2}\lambda&s\\\
0&a^{1}&0&-\frac{1}{2}a^{1}v^{2}\\\ 0&0&a^{2}&\frac{1}{2}a^{2}v^{1}\\\
0&0&0&1\par\end{array}\right)\mbox{ where $s,v^{1},v^{2}\in{\mathbb{R}}$ and
$a^{1},a^{2}\in{\mathbb{R}}^{*}$ satisfy $a^{1}a^{2}=\lambda$ }\right\\}.$

Consider the homomorphism from the Borel group $B\subset\hbox{\bf
SL}(3,{\mathbb{R}})$ of upper triangular matrices with determinant one into
$H$

$j:B\rightarrow H$

given by

$\left(\begin{array}[]{ccc}a&c&e\\\ 0&\frac{1}{ab}&f\\\
0&0&b\par\end{array}\right)\longrightarrow\left(\begin{array}[]{cccc}\frac{a}{b}&-a^{2}f&\frac{c}{b}&-eb+\frac{1}{2}acf\\\
0&a^{2}b&0&-\frac{1}{2}abc\\\ 0&0&\frac{1}{ab^{2}}&-\frac{f}{2b}\\\
0&0&0&1\par\end{array}\right)$

One verifies that the homomorphism is surjective so that $H$ is isomorphic to
the Borel group of upper triangular matrices in $\hbox{\bf
SL}(3,{\mathbb{R}})$.

###### Proposition 3.2

The bundle $Y\rightarrow M$ is a principal bundle with structure group $H$.

###  3.2 Construction of connection forms on the bundle $Y$

The goal of this section is to review the construction of canonical forms
defined on the coframe bundle $Y\rightarrow E$ as in [FV]. They give rise to a
Cartan connection on $Y$ with values in ${\mathfrak{sl}}(3,{\mathbb{R}})$.

A local section of the coframe bundle over $M$ may be given by three forms

$\theta,\ \ \theta^{1},\ \ \theta^{2}$

satisfying $d\theta=\theta^{1}\wedge\theta^{2}$, with
$\ker\theta^{1}_{|\ker\theta}=T^{2}$ and $\ker\theta^{2}_{|\ker\theta}=T^{1}.$
They give coordinates on the cotangent bundle over $E$. Indeed, at
$\lambda\theta\in E$, the coframes of $Y$ are parametrized as follows:

$\omega=\lambda\theta$ $\omega^{i}=a^{i}\theta^{i}+v^{i}\lambda\theta$

with $a^{1}a^{2}=\lambda$ and

$d\omega=2\varphi\wedge\omega+\omega^{1}\wedge\omega^{2},$ (19)

where $\varphi=\frac{d\lambda}{2\lambda}\ \mod{\omega^{1},\omega^{2},\omega}$.

Differentiating the forms $\omega^{1}$ and $\omega^{2}$ we obtain new forms
which correspond to the coordinates $a^{1},v^{1},a^{2},v^{2}$ (recall that
$a^{1}$ and $a^{2}$ are not independent):

###### Lemma 3.1

There exists linearly independent forms $w,\varphi^{1},\varphi^{2}$ defined on
$T^{*}Y$ such that

$d\omega^{1}=\varphi\wedge\omega^{1}+3w\wedge\omega^{1}+\omega\wedge\varphi^{1}\
\mbox{and}\ \
d\omega^{2}=\varphi\wedge\omega^{2}-3w\wedge\omega^{2}-\omega\wedge\varphi^{2}$
(20)

with $w=\frac{1}{6}(\frac{da^{1}}{a^{1}}-\frac{da^{2}}{a^{2}})$ mod
$(\omega,\omega^{1},\omega^{2})$ and $\varphi^{1}=-dv^{1}$,
$\varphi^{2}=dv^{2}$ mod $(\omega,\omega^{1},\omega^{2})$.

The coordinate $s$ in the bundle $Y$ is associated to a new form:

###### Lemma 3.2

There exists a 1-form $\psi$ such that

$d\varphi=\omega\wedge\psi-\frac{1}{2}(\varphi^{2}\wedge\omega^{1}+\varphi^{1}\wedge\omega^{2})$
(21)

The forms $w,\varphi^{1},\varphi^{2}$ and $\psi$ are not yet determined.
Define

$W=dw+\frac{1}{2}\omega^{2}\wedge\varphi^{1}-\frac{1}{2}\omega^{1}\wedge\varphi^{2}$
$\Phi^{1}=d\varphi^{1}+3\varphi^{1}\wedge
w+\omega^{1}\wedge\psi+\varphi\wedge\varphi^{1}$
$\Phi^{2}=d\varphi^{2}-3\varphi^{2}\wedge
w-\omega^{2}\wedge\psi+\varphi\wedge\varphi^{2}$

###### Lemma 3.3

There exists unique forms $w,\varphi^{1},\varphi^{2}$ and $\psi$ such that
$W=0$, $\Phi^{1}=Q^{1}\omega\wedge\omega^{2}$ and
$\Phi^{2}=Q^{2}\omega\wedge\omega^{1}$ where $Q^{1}$ and $Q^{2}$ are functions
on $Y$.

We can represent the structure equations 19, 20, 21 as a matrix equation whose
entries are differential forms. The forms are disposed in the Lie algebra
${\mathfrak{sl}}(3,{\mathbb{R}})$ and define a Cartan connection on $Y$.

###### Proposition 3.3

There exists a unique Cartan connection
$\pi:TY\to{\mathfrak{sl}}(3,{\mathbb{R}})$ defined on $Y$ of the form

$\pi=\left(\begin{array}[]{ccc}\varphi+w&\varphi^{2}&\psi\\\
\omega^{1}&-2w&\varphi^{1}\\\ \omega&\omega^{2}&-\varphi+w\end{array}\right).$

such that its curvature satisfies

$\Pi=d\pi+\pi\wedge\pi=\left(\begin{array}[]{ccc}0&\Phi^{2}&\Psi\\\
0&0&\Phi^{1}\\\ 0&0&0\end{array}\right)$

with $\Phi^{1}=Q^{1}\omega\wedge\omega^{2},\ \
\Phi^{2}=Q^{2}\omega\wedge\omega^{1}\ \mbox{and}\ \
\Psi=\left(U_{1}\omega^{1}+U_{2}\omega^{2}\right)\wedge\omega.$

###  3.3 Curvature forms and Bianchi identities

Curvature forms appear as differentials of connection forms and are used
implicitly in order to fix the connection forms.

We recall:

$W=dw-\frac{1}{2}\omega^{2}\wedge\varphi^{1}+\frac{1}{2}\omega^{1}\wedge\varphi^{2}=0,$
(22) $\Phi^{1}=d\varphi^{1}+3\varphi^{1}\wedge
w+\omega^{1}\wedge\psi+\varphi\wedge\varphi^{1}=Q^{1}\omega\wedge\omega^{2},$
(23) $\Phi^{2}=d\varphi^{2}-3\varphi^{2}\wedge
w-\omega^{2}\wedge\psi+\varphi\wedge\varphi^{2}=Q^{2}\omega\wedge\omega^{1},$
(24)
$\Psi:=d\psi-\varphi^{1}\wedge\varphi^{2}-2\varphi\wedge\psi=(U_{1}\omega^{1}+U_{2}\omega^{2})\wedge\omega.$
(25)

where $Q^{1},Q^{2},U^{1}$ and $U^{2}$ are functions on $Y$.

#### 3.3.1

Equation $d(d\varphi^{1})=0$ obtained differentiating $\Phi^{1}$ above implies

$dQ^{1}-6Q^{1}w+4Q^{1}\varphi=S^{1}\omega+U_{2}\omega^{1}+T^{1}\omega^{2},$
(26)

‘where we introduced functions $S^{1}$ and $T^{1}$.

#### 3.3.2

Analogously, equation $d(d\varphi^{2})=0$ obtained differentiating $\Phi^{2}$
above implies

$dQ^{2}+6Q^{2}w+4Q^{2}\varphi=S^{2}\omega-U_{1}\omega^{2}+T^{2}\omega^{1},$
(27)

where we introduced new functions $S^{2}$ and $T^{2}$.

#### 3.3.3

Equation $d(d\psi)=0$ obtained from 25 implies

$dU_{1}+5U_{1}\varphi+3U_{1}w+Q^{2}\varphi^{1}=A\omega+B\omega^{1}+C\omega^{2}$
(28)

and

$dU_{2}+5U_{2}\varphi-3U_{2}w-Q^{1}\varphi^{2}=D\omega+C\omega^{1}+E\omega^{2}.$
(29)

###  3.4 Embedding $\iota_{2}:Y_{2}\rightarrow Y$

The goal now is to obtain an immersion $\iota_{2}:Y_{2}\rightarrow Y$. One can
construct the bundle $Y_{2}$ using the bundle $E$ of contact forms as a first
step. Than $Y_{2}$ is a coframe bundle over $E$ obtained by the tautological
forms $\omega,\omega^{1},\omega^{2}$ corresponding to forms
$\theta,\theta^{1},\theta^{2}$ satisfying
$d\theta=\theta^{1}\wedge\theta^{2}+2\varphi\wedge\omega$ with an appropriate
$\varphi$.

By abuse of language again as for $\iota_{1}:Y_{1}\rightarrow Y_{2}$, we may
write the connection forms of $Y_{1}$ and $Y$ using the same letters and then
obtain:

###### Proposition 3.4

There exists a unique embedding $\iota_{2}:Y_{2}\rightarrow Y$ satisfying
$\iota_{2}^{*}(\omega)=\omega$, $\iota_{2}^{*}(\omega^{1})=\omega^{1}$,
$\iota_{2}^{*}(\omega^{2})=\omega^{2}$, $\iota_{2}^{*}(\varphi)=\varphi$.

Proof. As $Y_{2}$ and $Y$ are both coframe bundles over the line bundle $E$ of
all contact forms, we can assume that the embedding projects to the identity
map on $E$. Over $E$, $Y$ is a coframe bundle with structure group

$\left\\{\left(\begin{array}[]{ccc}a&c&e\\\ 0&\frac{1}{a^{2}}&f\\\
0&0&a\par\end{array}\right)\right\\}.$

In order to determine the embedding we need to choose functions $c$, $e$ and
$f$. The diagonal matrix correspond to the fiber of $Y_{2}$ and does not need
to be fixed. Consider then a map from $M$ to the group above given by

$h=\left(\begin{array}[]{ccc}1&c&e\\\ 0&{1}&f\\\ 0&0&1\par\end{array}\right).$

Recall that

${R_{h}}^{*}\pi=h^{-1}d\,h+Ad_{h^{-1}}\pi.$

We obtain, neglecting the terms of the connection of $Y$ which are not
relevant, the following transformation formulae. Remark that the term
$h^{-1}d\,h$ does not appear in the transformation of these components.

$\displaystyle\tilde{\omega}$ $\displaystyle=\omega$
$\displaystyle\tilde{\omega}^{1}$ $\displaystyle=\omega^{1}-f\,\omega$
$\displaystyle\tilde{\omega}^{2}$ $\displaystyle=\omega^{2}+{c}\,\omega$
$\displaystyle\tilde{\varphi}$
$\displaystyle=\varphi-\frac{1}{2}c\,\omega^{1}-{f}\,\omega^{2}+(\frac{1}{2}cf-{e})\,\omega$
(30)

The forms $\omega^{1}$ and $\omega^{2}$ defined at each point of $Y_{2}$
define corresponding forms $\omega^{1}$ and $\omega^{2}$ in $Y$. We observe
then that the functions $f$ and $c$ must be zero in order that
$\iota_{2}^{*}(\omega^{1})=\omega^{1}$,
$\iota_{2}^{*}(\omega^{2})=\omega^{2}$. Finally the form $\varphi$ on $Y_{2}$
defines a corresponding form on $Y$ and we conclude that $e=0$ if we impose
that $\iota_{2}^{*}(\varphi)=\varphi$.

$\Box$

####  3.4.1 The curvature of $Y$ in terms of the curvature of $Y_{2}$

We obtain the following equations by pulling back to $Y_{2}$ the structure
equations on $Y$ through the embedding $\iota_{2}$:

$d\omega=2\varphi\wedge\omega+\omega^{1}\wedge\omega^{2}$
$d\omega^{1}=\varphi\wedge\omega^{1}+3{\tilde{w}}\wedge\omega^{1}+\omega\wedge\varphi^{1}$
(31)
$d\omega^{2}=\varphi\wedge\omega^{2}-3{\tilde{w}}\wedge\omega^{2}-\omega\wedge\varphi^{2}$
(32)
$d\varphi=\omega\wedge{\tilde{\psi}}-\frac{1}{2}(\varphi^{2}\wedge\omega^{1}+\varphi^{1}\wedge\omega^{2})$
(33)
$d{\tilde{w}}=-\frac{1}{2}\varphi^{2}\wedge\omega^{1}+\frac{1}{2}\varphi^{1}\wedge\omega^{2}$
(34)
$d\varphi^{1}+3\varphi^{1}\wedge{\tilde{w}}+\omega^{1}\wedge{\tilde{\psi}}+\varphi\wedge\varphi^{1}=Q^{1}\omega\wedge\omega^{2}$
(35)
$d\varphi^{2}-3\varphi^{2}\wedge{\tilde{w}}-\omega^{2}\wedge{\tilde{\psi}}+\varphi\wedge\varphi^{2}=Q^{2}\omega\wedge\omega^{1}$
$d{\tilde{\psi}}-\varphi^{1}\wedge\varphi^{2}-2\varphi\wedge{\tilde{\psi}}=(U_{1}\omega^{1}+U_{2}\omega^{2})\wedge\omega.$

In the formulae above we write the pull back of any form $\alpha$ defined on
$Y$ using the same notation $\alpha$ except for the pull backs
$\tilde{w}=\iota_{2}^{*}w$ and ${\tilde{\psi}}=\iota_{2}^{*}\psi$. We should
compare with the structure equations of $Y_{2}$ and obtain an expression for
$Q^{1}$ and $Q^{2}$:

$d\omega=2\varphi\wedge\omega+\omega^{1}\wedge\omega^{2}$
$d\omega^{1}=\varphi\wedge\omega^{1}+3w\wedge\omega^{1}+\omega\wedge\tau^{1}$
$d\omega^{2}=\varphi\wedge\omega^{2}-3w\wedge\omega^{2}-\omega\wedge\tau^{2}$

with $\tau^{1}\wedge\omega^{2}=\tau^{2}\wedge\omega^{1}=0$.

Recall also that $d\varphi=\omega\wedge\psi$ with
$\psi=A\omega^{1}+B\omega^{2}$ and
$dw=C\omega\wedge\omega^{1}+D\omega\wedge\omega^{2}+S\omega^{1}\wedge\omega^{2}$,
where $A,B,C,D$ and $S$ are functions on $Y_{2}$.

  * •

The differences between the pull back equations and the structure equations
for $d\omega^{1}$ and $d\omega^{2}$ give, respectively,

$3({\tilde{w}-w})\wedge\omega^{1}+\omega\wedge(\varphi^{1}-\tau^{1})=0$

and

$3({\tilde{w}-w})\wedge\omega^{2}+\omega\wedge(\varphi^{2}-\tau^{2})=0.$

Therefore, by Cartan’s lemma

${\tilde{w}-w}=m\omega$

for a function $m$ to be determined and

$\varphi^{1}-\tau^{1}=-3m\omega^{1}+n\omega\ \mbox{and}\
\varphi^{2}-\tau^{2}=-3m\omega^{2}+P\omega,$

where $n$ and $P$ are functions to be determined.

  * •

The difference between the equation $d\varphi=\omega\wedge\psi$ and the pull
back equation for $d\varphi$ above is

$\omega\wedge({\tilde{\psi}}-\psi)-\frac{1}{2}(\varphi^{2}\wedge\omega^{1}+\varphi^{1}\wedge\omega^{2})=0.$

Substituting the expressions for $\varphi^{1}$ and $\varphi^{2}$ obtained in
the item above we obtain

$\omega\wedge({\tilde{\psi}}-A\omega^{1}-B\omega^{2})-\frac{1}{2}((\tau^{2}-3m\omega^{2}+P\omega)\wedge\omega^{1}+(\tau^{1}-3m\omega^{1}+n\omega)\wedge\omega^{2})=0,$

which simplifies to

$\omega\wedge({\tilde{\psi}}-A\omega^{1}-B\omega^{2}-\frac{P}{2}\omega^{1}-\frac{n}{2}\omega^{2})=0.$

This implies that

${\tilde{\psi}}=(A+\frac{P}{2})\omega^{1}+(B+\frac{n}{2})\omega^{2}+q\omega,$

where $q$ is a function to be determined.

  * •

The difference between the equations for $d\tilde{w}$ and $dw$ is then

$d(m\omega)=d\tilde{w}-dw=-\frac{1}{2}\varphi^{2}\wedge\omega^{1}+\frac{1}{2}\varphi^{1}\wedge\omega^{2}-S\omega^{1}\wedge\omega^{2}-C\omega\wedge\omega^{1}-D\omega\wedge\omega^{2}.$

Substituting in this formula the expressions for $\varphi^{1}$ and
$\varphi^{2}$ in terms of the enriched structure we obtain

$dm\wedge\omega+m(2\varphi\wedge\omega+\omega^{1}\wedge\omega^{2})=(-S-3m)\omega^{1}\wedge\omega^{2}+(-\frac{P}{2}-C)\omega\wedge\omega^{1}+(\frac{n}{2}-D)\omega\wedge\omega^{2}.$

That is,

$(S+4m)\omega^{1}\wedge\omega^{2}+\omega\wedge\left(-dm-2m\varphi+(\frac{P}{2}+C)\omega^{1}-(\frac{n}{2}-D)\omega^{2}\right)=0.$

Therefore

$m=-S/4$

and so

$\frac{1}{4}dS+\frac{1}{2}S\varphi+(\frac{P}{2}+C)\omega^{1}-(\frac{n}{2}-D)\omega^{2}+E\omega=0$

where $E$ is a function determined by the derivative of $S$. Writing
$dS+2S\varphi=S_{0}\omega+S_{1}\omega^{1}+S_{2}\omega^{2}$ and comparing with
the above expression, we obtain

$S_{0}=-4E,\ \ S_{1}=-2(P+2C),\ \ S_{2}=-2(-n+2D).$ (36)

Therefore the functions $P$ and $n$ are determined. It remains to determine
the function $q$.

  * •

Computing $d\varphi^{1}=d(\frac{3S}{4}\omega^{1}+n\omega+\tau^{1})$ and
equating to the structure equation
$d\varphi^{1}=-3\varphi^{1}\wedge{\tilde{w}}-\omega^{1}\wedge{\tilde{\psi}}-\varphi\wedge\varphi^{1}+Q^{1}\omega\wedge\omega^{2}$
we obtain, after a computation writing

$dn+3n(\varphi-w)=n_{0}\omega+n_{1}\omega^{1}+n_{2}\omega^{2},$ (37)
$n_{1}=-\tau^{1}_{2}\tau^{2}_{1}-q+\frac{9}{16}S^{2}-3E\ \
,n_{2}=-Q^{1}+\frac{3}{2}S\tau^{1}_{2}+\tau^{1}_{20}.$ (38)

Recalling that $n=S_{2}/2+4D$, $S$ and $D$ are determined by $Y_{2}$, we
obtained an expression for $Q^{1}$ in terms of the enriched structure. Note
also that $q$ is determined by the first equation.

  * •

Analogously, computing
$d\varphi^{2}=-3\frac{3S}{4}\omega^{2}+p\omega+\tau^{2}$ and equating to the
structure equation
$d\varphi^{2}-3\varphi^{2}\wedge{\tilde{w}}-\omega^{2}\wedge{\tilde{\psi}}+\varphi\wedge\varphi^{2}=Q^{2}\omega\wedge\omega^{1}$
we obtain, after a computation, writing
$dP+3P(\varphi+w)=P_{0}\omega+P_{1}\omega^{1}+P_{2}\omega^{2}$),

$P_{1}=-Q^{2}-\frac{3}{2}S\tau^{2}_{1}+\tau^{2}_{10}\ \
,P_{2}=\tau^{1}_{2}\tau^{2}_{1}+q-\frac{9}{16}S^{2}-3E.$ (39)

Recalling that $P=-S_{1}/2-2C$, $S$ and $C$ are determined by $Y_{2}$, we
obtained an expression for $Q^{2}$ in terms of the enriched structure.

The following proposition follows directly from the computations above.

###### Proposition 3.5

Suppose $Y_{2}$ with its adapted Cartan connection has null torsion, that is,
satisfies $\tau^{1}=\tau^{2}=0$ and $dw=S\omega^{1}\wedge\omega^{2}$. Then

$Q^{2}=\frac{1}{2}S_{11}$

and

$Q^{1}=-\frac{1}{2}S_{22},$

where $S_{11}$ is the $\omega^{1}$ component of the form $dS_{1}$ and $S_{22}$
is the $\omega^{2}$ component of the form $dS_{2}$.

Proof. From Proposition 2.6, null torsion and the condition that
$dw=S\omega^{1}\wedge\omega^{2}$ (that is, $C=D=0$) implies that $P=-S_{1}/2$
and $n=S_{2}/2$. The result is therefore implied from the previous formulas.
$\Box$

###  3.5 The embedding $Y_{1}\to Y$

Recall that $Y_{1}$ is a coframe bundle of forms
$(\theta,\theta^{1},\theta^{2})$ over $M$. Choosing a local section, the
pullback forms over $M$ are also denoted by $(\theta,\theta^{1},\theta^{2})$.
We recall here an embedding $Y_{1}\rightarrow Y$ obtained in [FV].

A section $(\theta,\theta^{1},\theta^{2})$ of the coframe bundle $Y_{1}$
clearly defines a path geometry on $M$. We obtain then a line bundle $E$ and a
principal bundle $Y$ with its associated Cartan connection. Also,
$(\theta,\theta^{1},\theta^{2})$ defines, up to the action by the group of
matrices

$\left(\begin{array}[]{ccc}1&c&e\\\ 0&1&f\\\ 0&0&1\par\end{array}\right)$

sections of the tautological forms $(\omega,\omega^{1},\omega^{2})$ on $Y$. In
order to define a canonical section we use the following

###### Proposition 3.6

Let $\theta,\theta^{1},\theta^{2}$ be a coframe section of $Y_{1}$ and
consider the principal bundle $Y$ defined by this coframe. Then there exists a
unique section $s:M\to Y$ such that $s^{*}\omega=\theta$,
$s^{*}\omega^{1}=\theta^{1}$, $s^{*}\omega^{2}=\theta^{2}$ and
$s^{*}\varphi=0$.

It is easy to verify that this definition is equivariant with respect to the
action $G_{1}$, the one parameter group of the strict contact structure. This
defines then the embedding $Y_{1}\rightarrow Y$.

###  3.6 The equivalence problem for a second order differential equation

In this section we recall the treatment by Cartan of the point equivalence
between second order differential equations. It is included in order to fix
conventions and to compare the invariants defined in the next section.

Recall that for a second order differential equation we define

$\theta=dy-pdx,$

and

$L_{1}=\ker\\{dp-Fdx\\}\cap\ker\\{dy-pdx\\},\ \ \ L_{2}=\ker dx\cap\ker dy.$

For $Z^{1}=dx$ and $Z^{2}=dp-Fdx$, one has then $d\theta=Z^{1}\wedge Z^{2}$.
The general forms defining the lines at each tangent space may be described by

$\omega^{1}=a_{1}Z^{1},\ \ \omega^{2}=a_{2}Z^{2},\\\ \omega=a_{1}a_{2}\theta$

where $a_{1},a_{2}$ are non-vanishing positive functions on the manifold, so
that we have always

$2\varphi\wedge\omega+\omega^{1}\wedge\omega^{2}=d\omega=(\frac{da_{1}}{a_{1}}+\frac{da_{2}}{a_{2}})\wedge\omega+a_{1}Z^{1}\wedge
a_{2}Z^{2},$

and we obtain comparing with 4

$\varphi=\frac{1}{2}(\frac{da_{1}}{a_{1}}+\frac{da_{2}}{a_{2}})+r\omega.$

One computes

$(\varphi+3w)\wedge\omega^{1}+\omega\wedge\tau^{1}=d\omega^{1}=da_{1}\wedge
Z^{1}+a_{1}.0=\left(\frac{1}{2}(\frac{da_{1}}{a_{1}}+\frac{da_{2}}{a_{2}})+\frac{1}{2}(\frac{da_{1}}{a_{1}}-\frac{da_{2}}{a_{2}})\right)\wedge\omega^{1}$
(40)

and obtain

$3w=\frac{1}{2}(\frac{da_{1}}{a_{1}}-\frac{da_{2}}{a_{2}})-r\omega+s\omega^{1},\\\
\tau^{1}_{2}=0.$

Observe that, if $f(x,y,p)$ then

$df=\frac{1}{a_{1}}\frac{df}{dx}\omega^{1}+\frac{1}{a_{2}}f_{p}\omega^{2}+\frac{1}{a_{1}a_{2}}f_{y}\omega,$

where $\frac{df}{dx}=f_{x}+f_{y}p+f_{p}F$. Also

$\begin{array}[]{rcl}d\omega^{2}&=&\frac{da_{2}}{a_{2}}\wedge\omega^{2}+a_{2}(-\frac{1}{a_{2}}F_{p}\omega^{2}-\frac{1}{a_{1}a_{2}}F_{y}\omega)\wedge\frac{1}{a_{1}}\omega^{1}\\\
&=&(\frac{1}{2}(\frac{da_{1}}{a_{1}}+\frac{da_{2}}{a_{2}})-\frac{1}{2}(\frac{da_{1}}{a_{1}}-\frac{da_{2}}{a_{2}}))\wedge\omega^{2}+\frac{1}{a_{1}}F_{p}\omega^{1}\wedge\omega^{2}-\frac{1}{(a_{1})^{2}}F_{y}\omega\wedge\omega^{1}\\\
&=&(\varphi-3w-2r\omega+(s+\frac{1}{a_{1}}F_{p})\omega^{1})\wedge\omega^{2}-\omega\wedge\frac{F_{y}}{(a_{1})^{2}}\omega^{1}\end{array}$
(41)

Comparing with
$d\omega^{2}=(\varphi-3w)\wedge\omega^{2}-\omega^{\wedge}\tau^{2}$ we obtain
$r=0,s=-\frac{F_{p}}{a_{1}},\tau^{2}_{1}=\frac{F_{y}}{a_{1}^{2}}$,

$\varphi=\frac{1}{2}(\frac{da_{1}}{a_{1}}+\frac{da_{2}}{a_{2}}),\\\
3w=\frac{1}{2}(\frac{da_{1}}{a_{1}}-\frac{da_{2}}{a_{2}})-\frac{F_{p}}{a_{1}}\omega^{1}.$

From $d\varphi=0$, we obtain $\psi=0$, or $A=B=0$, and from $\tau^{1}_{2}=0$
it follows $D=\tau^{1}_{20}=\tau^{1}_{22}=0$.

From above we get

$3dw=-dF_{p}\wedge\frac{1}{a_{1}}\omega^{1}=-(\frac{F_{pp}}{a_{2}}\omega^{2}+\frac{F_{py}}{a_{1}a_{2}}\omega)\wedge\frac{1}{a_{1}}\omega^{1}$

and comparing with $dw=C\omega\wedge\omega^{1}+S\omega^{1}\wedge\omega^{2}$ we
obtain

$3C=-\frac{F_{py}}{a_{1}^{2}a_{2}},\ \ \ 3S=\frac{F_{pp}}{a_{1}a_{2}}.$

Also

$d\tau^{2}_{1}=d(\frac{F_{y}}{a_{1}^{2}})=-\frac{2}{a_{1}^{2}}(\varphi+3w+\frac{F_{p}}{a_{1}}\omega^{1})F_{y}+\frac{1}{a_{1}^{2}}(\frac{1}{a_{1}}\frac{dF_{y}}{dx}\omega^{1}+\frac{1}{a_{1}a_{2}}F_{yy}\omega+\frac{1}{a_{2}}F_{yp}\omega^{2}).$

Comparing with
$d\tau^{2}_{1}=-2\tau^{2}_{1}(\varphi+3w)-3C\omega^{2}+\tau^{2}_{10}\omega+\tau^{2}_{11}\omega^{1}$
we obtain

$\tau^{2}_{11}=\frac{1}{a_{1}^{3}}(-2F_{p}F_{y}+\frac{dF_{y}}{dx}),\ \ \
\tau^{2}_{10}=\frac{1}{a_{1}^{3}a_{2}}F_{yy}.$

Now

$3dS=-2(3S\varphi)+\frac{1}{a_{1}a_{2}}(\frac{1}{a_{1}}\frac{dF_{pp}}{dx}\omega^{1}+\frac{1}{a_{2}}F_{ppp}\omega^{2}+\frac{1}{a_{1}a_{2}}F_{ppy}\omega)$

and comparing with $dS=-2S\varphi+S_{0}\omega+S_{1}\omega^{1}+S_{2}\omega^{2}$
we obtain

$S_{1}=\frac{1}{3a_{1}^{2}a_{2}}\frac{dF_{pp}}{dx},\ \ \
S_{2}=\frac{1}{3a_{1}a_{2}^{2}}F_{ppp},\\\
S_{0}=\frac{1}{3a_{1}^{2}a_{2}^{2}}F_{ppy}.$

It follows from $S_{1}=-2P-4C$ that
$6P=\frac{1}{a_{1}^{2}a_{2}}(4F_{yp}-\frac{dF_{pp}}{dx}).$ Then

$6dP=-\frac{1}{a_{1}^{2}a_{2}}(3\varphi+3w+\frac{F_{p}}{a_{1}}\omega^{1})(4F_{yp}-\frac{dF_{pp}}{dx})+\frac{4}{a_{1}^{2}a_{2}}(\frac{1}{a_{1}}\frac{dF_{yp}}{dx}\omega^{1}+\frac{1}{a_{2}}F_{ypp}\omega^{2}+\frac{1}{a_{1}a_{2}}F_{yyp}\omega)$
$-\frac{1}{a_{1}^{2}a_{2}}(\frac{1}{a_{1}}\frac{d^{2}F_{pp}}{dx^{2}}\omega^{1}+\frac{1}{a_{2}}(\frac{dF_{ppp}}{dx}+F_{ypp}+F_{ppp}F_{p})\omega^{2}+\frac{1}{a_{1}a_{2}}(\frac{dF_{ypp}}{dx}+F_{ppp}F_{y})\omega)$

Comparing with
$dP=-(3\varphi+3w)P+P_{0}\omega+P_{1}\omega^{1}+P_{2}\omega^{2}$ we obtain

$P_{0}=\frac{1}{6a_{1}^{3}a_{2}^{2}}(4F_{yyp}-\frac{dF_{ypp}}{dx}-F_{ppp}F_{y}),\
\ \ \
P_{1}=\frac{1}{6a_{1}^{3}a_{2}}(-4F_{p}F_{yp}+F_{p}\frac{dF_{pp}}{dx}+4\frac{dF_{yp}}{dx}-\frac{d^{2}F_{pp}}{dx^{2}}),$
$P_{2}=\frac{1}{6a_{1}^{2}a_{2}^{2}}(4F_{ypp}-\frac{dF_{ppp}}{dx}-F_{ypp}-F_{ppp}F_{p}).$

From $Q^{2}=\tau^{2}_{10}-\frac{3}{2}S\tau^{2}_{1}-P_{1}$ it follows

$Q^{2}=\frac{1}{6a_{1}^{3}a_{2}}(6F_{yy}-3F_{y}F_{pp}+4F_{p}F_{yp}-F_{p}\frac{dF_{pp}}{dx}-4\frac{dF_{yp}}{dx}+\frac{d^{2}F_{pp}}{dx^{2}}).$
(42)

It follows from $S_{2}=2n-4D$ that $6n=\frac{1}{a_{1}a_{2}^{2}}F_{ppp}$. Then

$6dn=-\frac{1}{a_{1}a_{2}^{2}}(3\varphi-3w-\frac{F_{p}}{a_{1}}\omega^{1})F_{ppp}+\frac{1}{a_{1}a_{2}^{2}}(\frac{1}{a_{1}}\frac{dF_{ppp}}{dx}\omega^{1}+\frac{1}{a_{2}}F_{pppp}\omega^{2}+\frac{1}{a_{1}a_{2}}F_{pppy}\omega).$

Comparing with
$dn=-n(3\varphi-3w)+n_{0}\omega+n_{1}\omega^{1}+n_{2}\omega^{2}$ we obtain

$n_{0}=\frac{1}{6a_{1}^{2}a_{2}^{3}}F_{pppy},\ \ \
n_{1}=\frac{1}{6a_{1}^{2}a_{2}^{2}}(\frac{dF_{ppp}}{dx}+F_{p}F_{ppp})\ \ \
n_{2}=\frac{1}{6a_{1}a_{2}^{3}}F_{pppp}.$

From $Q^{1}=\tau^{1}_{20}+\frac{3}{2}S\tau^{1}_{2}-n_{2}$ it follows

$Q^{1}=-\frac{1}{6a_{1}a_{2}^{3}}F_{pppp}.$ (43)

Formulas 43 and 42 are in [Car].

##  4 A global invariant

In this section we define the global invariant for path structures. It has a
very similar definition with the global invariant obtained in [FV] in the
context of a structure defined on the complexified tangent space of a
3-manifold. But we make the definition explicit in the case of path structures
for the sake of clarity and to adapt differences of conventions with our
previous paper.

Define the second Chern class of the bundle $Y$ with connection form $\pi$ as

$c_{2}(Y,\pi)=\frac{1}{8\pi^{2}}\hbox{\rm tr}\,(\Pi\wedge\Pi).$

$\left(\begin{array}[]{ccc}0&\Phi^{2}&\Psi\\\ 0&0&\Phi^{1}\\\
0&0&0\end{array}\right)\wedge\left(\begin{array}[]{ccc}0&\Phi^{2}&\Psi\\\
0&0&\Phi^{1}\\\
0&0&0\end{array}\right)=\left(\begin{array}[]{ccc}0&0&\Phi^{1}\wedge\Phi^{2}\\\
0&0&0\\\ 0&0&0\end{array}\right).$

As $\Phi^{1}=Q^{1}\omega\wedge\omega^{2}$ and
$\Phi^{2}=Q^{2}\omega\wedge\omega^{1}$ we have $\Pi\wedge\Pi=0$ and therefore

$c_{2}(Y,\pi)=0.$

###### Definition 4.1

The transgression form is defined as

$TC_{2}(\pi)=\frac{1}{8\pi^{2}}\left(\hbox{\rm
tr}\,(\pi\wedge\Pi)+\frac{1}{3}\hbox{\rm
tr}\,(\pi\wedge\pi\wedge\pi)\right)=\frac{1}{24\pi^{2}}\hbox{\rm
tr}\,(\pi\wedge\pi\wedge\pi).$

###### Lemma 4.1

The transgression form is closed, that is, $d\,TC_{2}(\pi)=c_{2}(Y,\pi)=0$.

Proof. We compute first, using the expressions of $\Phi^{1}$, $\Phi^{2}$ and
$\Psi$, that

$\hbox{\rm
tr}\,(\Pi\wedge\pi)=\Phi^{2}\wedge\omega^{1}+\Phi^{1}\wedge\omega^{2}+\Psi\wedge\omega=0.$

Differentiating the curvature form we obtain
$d\,\Pi=\Pi\wedge\pi-\pi\wedge\Pi$ and therefore

$0=d\,\hbox{\rm tr}\,(\Pi\wedge\pi)=\hbox{\rm tr}\,(d\,\Pi\wedge\pi+\Pi\wedge
d\,\pi)=\hbox{\rm
tr}\,((\Pi\wedge\pi-\pi\wedge\Pi)\wedge\pi+\Pi\wedge(\Pi-\pi\wedge\pi))$
$=-\hbox{\rm tr}\,(\pi\wedge\Pi\wedge\pi).$

Note that $\hbox{\rm tr}\,(\alpha\wedge\beta)=(-1)^{kl}\hbox{\rm
tr}\,(\beta\wedge\alpha)$ if $\alpha$ and $\beta$ are two matrices of forms of
degree $k$ and $l$ respectively. Therefore, computing

$\frac{1}{3}d\,\hbox{\rm tr}\,(\pi\wedge\pi\wedge\pi)=\hbox{\rm
tr}\,(d\,\pi\wedge\pi\wedge\pi)=\hbox{\rm
tr}\,((\Pi-\pi\wedge\pi)\wedge\pi\wedge\pi)$ $=-\hbox{\rm
tr}\,(\pi\wedge\pi\wedge\pi\wedge\pi)=0.$

$\Box$

###### Definition 4.2

Suppose that the fiber bundle $Y\rightarrow M$ is trivial and let
$s:M\rightarrow Y$ be a section, we define then

$\mu=\int_{M}s^{*}TC_{2}(\pi)=\frac{1}{24\pi^{2}}\int_{M}s^{*}\hbox{\rm
tr}\,(\pi\wedge\pi\wedge\pi).$

In principle that integral depends on the section but the following
proposition shows that the integrand

$s^{*}TC_{2}(\pi)$

defines an element in $H^{3}(M,{\mathbb{R}})$ which does not depend on the
section.

###### Proposition 4.3

Suppose $s$ and $\tilde{s}$ are two sections. Then

$\tilde{s}^{*}TC_{2}(\pi)-s^{*}TC_{2}(\pi)=-\frac{1}{8\pi^{2}}d\,s^{*}\hbox{\rm
tr}\,(h^{-1}\pi\wedge d\,h).$

where $h:M\rightarrow H$ is a map such that $\tilde{s}=R_{h}\circ s$.

Proof. Fix the section $s$. Than there exists a map $h:M\rightarrow H$ such
that $\tilde{s}=R_{h}\circ s$. We have then

$\tilde{s}^{*}TC_{2}(\pi)=\frac{1}{24\pi^{2}}s^{*}\hbox{\rm
tr}\,(R_{h}^{*}\pi\wedge R_{h}^{*}\pi\wedge R_{h}^{*}\pi).$

From the formula

${R_{h}}^{*}\pi=h^{-1}d\,h+Ad_{h^{-1}}\pi,$

we obtain

$\hbox{\rm tr}\,(R_{h}^{*}\pi\wedge R_{h}^{*}\pi\wedge R_{h}^{*}\pi)=$
$\hbox{\rm tr}\,\left(h^{-1}d\,h\wedge h^{-1}d\,h\wedge
h^{-1}d\,h+3h^{-1}d\,h\wedge h^{-1}\pi\wedge d\,h+3h^{-1}\pi\wedge\pi\wedge
d\,h+\pi\wedge\pi\wedge\pi\right)$ $=\hbox{\rm tr}\,\left(-h^{-1}d\,h\wedge
d\,h^{-1}\wedge d\,h-3d\,h^{-1}\wedge\pi\wedge d\,h+3h^{-1}\pi\wedge\pi\wedge
d\,h+\pi\wedge\pi\wedge\pi\right).$

Observe that the first term in the right hand side vanishes. Indeed,
$d\,h^{-1}\wedge d\,h$ is upper triangular with null diagonal. Moreover
$h^{-1}d\,h$ is upper triangular and therefore the Lie algebra valued form
also has zero diagonal. Therefore

$\hbox{\rm tr}\,(h^{-1}d\,h\wedge d\,h^{-1}\wedge d\,h)=0.$

By the same argument $\hbox{\rm tr}\,\left(h^{-1}\Pi\wedge d\,h\right)=0$.

Now we show that

$d\,\hbox{\rm tr}\,(h^{-1}\pi\wedge d\,h)=\hbox{\rm
tr}\,\left(d\,h^{-1}\wedge\pi\wedge d\,h-h^{-1}\pi\wedge\pi\wedge
d\,h\right).$

Compute $d\hbox{\rm tr}\,(h^{-1}\pi\wedge d\,h)=\hbox{\rm
tr}\,\left(d\,h^{-1}\wedge\pi\wedge d\,h+h^{-1}d\pi\wedge d\,h\right)$

$=\hbox{\rm tr}\,\left(d\,h^{-1}\wedge\pi\wedge
d\,h+h^{-1}(\Pi-\pi\wedge\pi)\wedge d\,h\right),$

which gives, using that $\hbox{\rm tr}\,\left(h^{-1}\Pi\wedge d\,h\right)=0$,

$d\hbox{\rm tr}\,(h^{-1}\pi\wedge d\,h)=\hbox{\rm
tr}\,\left(d\,h^{-1}\wedge\pi\wedge d\,h-h^{-1}\pi\wedge\pi\wedge
d\,h\right).$

We obtained therefore that

$\tilde{e}s^{*}TC_{2}(\pi)=s^{*}TC_{2}(\pi)-\frac{1}{8\pi^{2}}d\,s^{*}\hbox{\rm
tr}\,(h^{-1}\pi\wedge d\,h)$

and this completes the proof of the proposition.

$\Box$

Let $\mu(t)$ be the invariant defined as a function of the a parameter
describing the deformation of the structure on a closed manifold $M$ and
define $\delta\mu=\frac{d}{dt}\mu(0)$. One can interpret the flat structures
as giving critical points of the global invariant $\mu$ through the first
variation formula which we refer to [FV] for a proof.

###### Proposition 4.4

$\delta\mu=-\frac{1}{4\pi^{2}}\int_{M}s^{*}\hbox{\rm
tr}\,(\dot{\pi}\wedge\Pi).$

The global invariant can be computed most easily for a path structure induced
by an enriched or strict path structure.

###### Proposition 4.5

Let $M$ be an enriched path structure and $Y_{2}\rightarrow Y$ be the
canonical embedding of the enriched geometry into the induced path geometry.
Then

$8\pi^{2}\mu(Y)=(n_{1}-\frac{3}{4}S_{0}+2\tau^{1}_{2}\tau^{2}_{1})\omega\wedge\omega^{1}\wedge\omega^{2}+\omega\wedge\omega^{1}\wedge(-2A\varphi-(\frac{3}{2}S_{1}+6C)w)$
$+\omega\wedge\omega^{2}(-2B\varphi-(\frac{3}{2}S_{2}+6D)w)-\frac{9}{2}Sw\wedge\omega^{1}\wedge\omega^{2}.$

Proof. One compute first the following formula.

$\frac{1}{3}\hbox{\rm
tr}\,(\pi\wedge\pi\wedge\pi)=(2\omega\wedge\varphi-\omega^{1}\wedge\omega^{2})\wedge\psi-\omega\wedge\varphi^{1}\wedge\varphi^{2}+\omega^{1}\wedge(\varphi+3w)\wedge\varphi^{2}+\omega^{2}\wedge(\varphi-3w)\wedge\varphi^{1}.$

Therefore using the embedding of $Y_{2}\rightarrow Y$ in the previous section
we obtain by a computation:

$\frac{1}{3}\hbox{\rm
tr}\,(\pi\wedge\pi\wedge\pi)=(n_{1}-\frac{3}{4}S_{0}+2\tau^{1}_{2}\tau^{2}_{1})\omega\wedge\omega^{1}\wedge\omega^{2}+\omega\wedge\omega^{1}\wedge(-2A\varphi-(\frac{3}{2}S_{1}+6C)w)$
$+\omega\wedge\omega^{2}(-2B\varphi-(\frac{3}{2}S_{2}+6D)w)-\frac{9}{2}Sw\wedge\omega^{1}\wedge\omega^{2}.$

$\Box$

Using the embedding of $Y_{1}\rightarrow Y$ in the previous section we obtain
by a similar computation:

###### Proposition 4.6

Let $M$ be a strict path structure and $Y_{1}\rightarrow Y$ be the canonical
embedding of the strict geometry into the induced path geometry. Then

$8\pi^{2}\mu(Y)=(n_{1}-\frac{3}{4}S_{0}+2\tau^{1}_{2}\tau^{2}_{1})\omega\wedge\omega^{1}\wedge\omega^{2}-w\wedge((\frac{3}{2}S_{1}+6C)\omega\wedge\omega^{1}+(\frac{3}{2}S_{2}+6D)\omega\wedge\omega^{2}+\frac{9}{2}S\omega^{1}\wedge\omega^{2})$
(44)

###  4.1 The global invariant for second order differential equations on the
torus

In this section we obtain formulas for the global invariant in the case of an
ordinary differential equation defined on the torus. Recall that the
projectivized cotangent bundle $\pi:PT^{*}S\to S$ of a surface $S$ is
described locally by $(x,y,[p,q])$ where $(x,y)$ are local coordinates on the
surface and $pdx+qdy$ is a form at $(x,y)$. The Liouville form $\theta$ on
$T^{*}S$ is defined to be the tautological form
$\theta(x,y,-pdx+qdy)=\pi^{*}(-pdx+qdy)$. It induces a contact distribution on
$PT^{*}S$, which in the chart $(x,y,p)\to(x,y,[p,1])$ is given by the kernel
of the form $dy-pdx$. On the chart $(x,y,q)\to(x,y,[1,q])$ the contact
distribution is the kernel of $dx-qdx$. One can also consider, fixing a metric
on the surface, the unit cotangent bundle $(T^{*})^{1}S$ which is a double
cover of $PT^{*}S$.

The fibers of the bundle $PT^{*}S$ give a canonical field of directions on the
Liouville distribution. Observe that, in local coordinates $(x,y,p)$, it is
described by $\ker dx\cap\ker dy=\ker dx\cap\ker(dy-pdx)$. Choosing another
direction on the contact distribution amounts to define a form, in local
coordiantes $(x,y,p)$, $dp-G(x,y,p)dx$, where $G(x,y,p)$ is a function. On the
chart $(x,y,q)$ one writes then

$d(\frac{1}{q})-G(x,y,\frac{1}{q})dx=-\frac{1}{q^{2}}dq-G(x,y,\frac{1}{q})dx.$

Therefore, the direction is determined by $dq+G(x,y,\frac{1}{q})q^{3}dy$ (the
contact distribution is $\ker(dx-qdy)$). In order to have a well defined
direction we need that the function $G(x,y,\frac{1}{q})q^{3}$ has a
differentiable extension for $q=0$.

###### Definition 4.7

A second order differential equation on a surface $S$ is a path structure on
the projective cotangent bundle with contact structure induced by the
Liouville form and such that one of the directions is given by the fibers.

It is convenient to introduce a new coordinate in the fiber
$\alpha\in]-\pi,\pi]$ through the formula $p=\tan{\alpha/2}$. The contact
distribution is defined by a globally defined form on the coordinates
$(x,y,\alpha)$:

$\theta=\cos{\alpha/2}dy-\sin{\alpha/2}dx.$

The fiber direction is defined by the equations $dx=dy=0$ which can also be
described by, defining $\theta^{1}=\sin{\alpha/2}dy+\cos{\alpha/2}dx$, as
$\ker\theta^{1}\cap\ker\theta$. The last form, which depends on a choice of a
function, is

$\theta^{2}=d\alpha-F(x,y,\alpha)\theta^{1}.$

Observe that

$d\theta=\frac{1}{2}\theta^{1}\wedge\theta^{2}.$

The relation with the differential equation given on the chart $(x,y,p)$ is
given writing

$dp=\frac{1}{2}(1+p^{2})d\alpha$

and therefore, as $dy=pdx$ in that chart,

$d\alpha-F(x,y,\alpha)\theta^{1}=\frac{2}{1+p^{2}}dp-F(x,y,2\arctan{p})(\sin{\alpha/2}dy+\cos{\alpha/2}dx)$
$=\frac{2}{1+p^{2}}dp-F(x,y,2\arctan{p})(\sin{\alpha/2}.pdx+\cos{\alpha/2}dx)$
$=\frac{2}{1+p^{2}}dp-F(x,y,2\arctan{p})(\sin{\alpha/2}.\tan{\alpha/2}+\cos{\alpha/2})dx.$

and recalling that $\cos{\alpha/2}=\frac{1}{\sqrt{1+p^{2}}}$,

$=\frac{2}{1+p^{2}}dp-F(x,y,2\arctan{p}){(1+p^{2})^{1/2}}dx.$

Therefore $2G(x,y,p)=F(x,y,2\arctan{p}){(1+p^{2})^{3/2}}$.

####  4.1.1 The strict and enriched structure of a differential equation on
the torus

Here we will work with a double cover of the projective cotangent bundle of
the torus. We define the path structure associated to a differential equation
on the torus through a strict path structure defined by

$\theta=\cos{\alpha}dy-\sin{\alpha}dx.$
$\theta^{1}=\sin{\alpha}dy+\cos{\alpha}dx$

and

$\theta^{2}=d\alpha-F(x,y,\alpha)\theta^{1}.$

Here $F(x,y,\alpha)$ is a function defined on the torus. Observe that

$d\theta=\theta^{1}\wedge\theta^{2}.$

In the following we will write, for a function $f:T^{3}\to{\mathbb{R}}$,
defined on the torus,

$df=f_{0}\theta+f_{1}\theta^{1}+f_{2}\theta^{2},$

so that $f_{x}=-f_{0}\sin{\alpha}+(f_{1}-f_{2}F)\cos{\alpha}$,
$f_{y}=f_{0}\cos{\alpha}+(f_{1}-f_{2}F)\sin{\alpha}$ and $f_{\alpha}=f_{2}$.
Compute

$d\theta^{1}=(\theta^{2}+F\theta^{1})\wedge\theta$

and

$d\theta^{2}=((F_{0}-F^{2})\theta^{1}-F\theta^{2})\wedge\theta+F_{2}\theta^{1}\wedge\theta^{2}.$

Consider now the enriched structure defined by $\theta,\theta^{1}$ and
$\theta^{2}$ and the tautological forms
$\omega=a_{1}a_{2}\theta,\omega^{1}=a_{1}\theta^{1}$ and
$\omega^{2}=a_{2}\theta^{2}$.

We first compute

$d\omega=2\varphi\wedge\omega+\omega^{1}\wedge\omega^{2},$

where $\varphi=\frac{1}{2}(\frac{da_{1}}{a_{1}}+\frac{da_{2}}{a_{2}})$. Next
we compute $d\omega^{1}$ and $d\omega^{2}$:

$d\omega^{1}=\frac{da_{1}}{a_{1}}\wedge
a_{1}\theta^{1}+a_{1}(\theta^{2}+F\theta^{1})\wedge\theta$
$d\omega^{2}=\frac{da_{2}}{a_{2}}\wedge
a_{2}\theta^{2}+a_{2}(((F_{0}-F^{2})\theta^{1}-F\theta^{2})\wedge\theta+F_{2}\theta^{1}\wedge\theta^{2})$

Comparing with the structure equations of the enriched structure in 5 we may
write

$d\omega^{1}=\varphi\wedge\omega^{1}+3w\wedge\omega^{1}+\omega\wedge\tau^{1}\
\ \
d\omega^{2}=\varphi\wedge\omega^{2}-3w\wedge\omega^{2}-\omega\wedge\tau^{2}.$
(45)

with

$3w=\frac{1}{2}(\frac{da_{1}}{a_{1}}-\frac{da_{2}}{a_{2}})-\frac{1}{a_{1}a_{2}}F\omega-\frac{1}{a_{1}}F_{2}\omega^{1},$
$\tau^{1}=-\frac{1}{a^{2}_{2}}\omega^{2}$

and

$\tau^{2}=\frac{1}{a^{2}_{1}}(F_{0}-F^{2})\omega^{1}.$

####  4.1.2 Curvatures

We compute now $d\varphi=\omega\wedge(A\omega^{1}+B\omega^{2})$ and
$dw=C\omega\wedge\omega^{1}+D\omega\wedge\omega^{2}+S\omega^{1}\wedge\omega^{2}$(see
Proposition 2.5) . From $d\varphi=0$ we obtain $A=B=0$. Computing $dw$ and
comparing to the formula above we obtain

$C=\frac{F_{1}-F_{20}+FF_{2}}{3a^{2}_{1}a_{2}},\ \
D=\frac{2F_{2}}{3a_{1}a^{2}_{2}},\ \ S=\frac{F_{22}-F}{3a_{1}a_{2}}.$

In order to compute the global invariant we need to compute the coefficients
$S_{0},S_{1}$ and $S_{2}$ in equation 17 (a Bianchi identity) :
$dS+2S\varphi=S_{0}\omega+S_{1}\omega^{1}+S_{2}\omega^{2}$. One obtains

$S_{0}=\frac{F_{220}-F_{0}}{3a^{2}_{1}a^{2}_{2}},\ \
S_{1}=\frac{F_{221}-F_{1}}{3a^{2}_{1}a_{2}},\ \
S_{2}=\frac{F_{222}-F_{2}}{3a_{1}a^{2}_{2}}.$

Now we use the expressions obtained in section 3.4.1 of the curvatures of $Y$
in terms of the curvature of $Y_{2}$. In order to compute $\mu(Y)$ we need to
compute $n$ and its derivatives (see 36 and 37).

We have $2n=S_{2}+4D=\frac{F_{222}+7F_{2}}{3a^{2}_{2}a_{1}}$ and compute the
left hand of $dn+3(\varphi-w)n=n_{0}\omega+n_{1}\omega^{1}+n_{2}\omega^{2}$
(formula 37) to obtain then

$n_{0}=\frac{1}{6a^{2}_{1}a^{3}_{2}}\left(F(F_{222}+F_{2})+F_{2220}+7F_{20}\right),$
$n_{1}=\frac{1}{6a^{2}_{1}a^{2}_{2}}\left(F_{2}(F_{222}+F_{2})+F_{2221}+7F_{21}\right)$
$n_{2}=\frac{1}{6a_{1}a^{3}_{2}}\left(F_{2222}+7F_{22}\right)$

We use equations 38 and 39 to compute the curvature functions $Q^{1}$ and
$Q^{2}$. We have $Q^{1}=-n_{2}+\frac{3}{2}S\tau^{1}_{2}+\tau^{1}_{20}$ and
$Q^{2}=-P_{1}-\frac{3}{2}S\tau^{2}_{1}+\tau^{2}_{10}$. For that sake, we
compute first the derivatives of the torsion, $\tau^{1}_{20}$ and
$\tau^{2}_{10}$, using formulas 11 and 12. Computing the left hand side of the
equation
$d\tau^{1}_{2}+2\tau^{1}_{2}(\varphi-3w)+(B+3D)\omega^{1}=\tau^{1}_{20}\omega+\tau^{1}_{22}\omega^{2}$
and comparing to the right hand side we obtain

$\tau^{1}_{20}=-\frac{2F}{a_{1}a^{3}_{2}}.$

Analogously, computing the left hand side of the equation
$d\tau^{2}_{1}+2\tau^{2}_{1}(\varphi+3w)-(A-3C)\omega^{2}=\tau^{2}_{10}\omega+\tau^{2}_{11}\omega^{1}$
we obtain

$\tau^{2}_{10}=\frac{F_{00}-4FF_{0}+2F^{3}}{a^{3}_{1}a_{2}}.$

###### Proposition 4.8

Given a (local) differential equation as a path structure induced by the forms
$\theta,\theta^{1}$ and $\theta^{2}$ as above one computes the curvature
functions in terms of the enriched structure:

$Q^{1}=-\frac{1}{6a_{1}a^{3}_{2}}(F_{2222}+10F_{22}+9F)$

and

$Q^{2}=\frac{1}{6a^{3}_{1}a_{2}}(F_{2}F_{221}+3F_{2}F^{2}_{1}-4F_{2}F_{20}+4FF^{2}_{2}+F_{2211}+6F_{1}F_{11}$
$-4F_{201}+4F_{2}F_{1}+4FF_{21}-3F_{22}F_{0}+3F_{22}F^{2}-21FF_{0}+9F^{3}+6F_{00}).$

Proof. Recall from formula 36 that $P=-S_{1}/2-2C$ and we write
$dP+3P(\varphi+w)=P_{0}\omega+P_{1}\omega^{1}+P_{2}\omega^{2}$. Computing the
left side and comparing the right side we obtain the expression of $P_{1}$
which we use in the formulas above.  $\Box$

The following proposition describes locally differential equations satisfying
$Q^{1}=0$.

###### Proposition 4.9

Differential equations on an open subset with coordinates $(x,y)$ given by
$\theta=\cos{\alpha}dy-\sin{\alpha}dx,$
$\theta^{1}=\sin{\alpha}dy+\cos{\alpha}dx$ and
$\theta^{2}=d\alpha-F(x,y,\alpha)\theta^{1}$ satisfy

$Q^{1}=0$

if and only if

$F(x,y,\alpha)=A(x,y)\cos{\alpha}+B(x,y)\sin{\alpha}+C(x,y)\cos{3\alpha}+D(x,y)\sin{3\alpha}$

where $A,B,C$ and $D$ are functions on $x$ and $y$.

Proof. Observe that $Q^{1}=0$ is equivalent to
$F_{2222}+10F_{22}+9F=F_{\alpha\alpha\alpha\alpha}+10F_{\alpha\alpha}+9F=0$.
The only solutions to this linear equation are of the form above.  $\Box$

Using the coordinates $(x,y,p)$ as above where the differential equation is
described as $dp-G(x,y,p)dx=0$ the condition $Q^{1}=0$ implies that $G(x,y,p)$
is at most a third order polynomial in $p$ with coefficients functions of $x$
and $y$ (see [A]).

####  4.1.3 The global invariant

We are ready now to use Proposition 4.5 to detemine the global invariant:

###### Proposition 4.10

Let $M$ be an enriched path structure defined by an ordinary differential
equation of second order on the torus with strict structure defined by the
forms $\theta,\theta^{1}$ and $\theta^{2}$ as above. Let $Y_{2}\rightarrow Y$
be the canonical embedding of the enriched geometry into the induced path
geometry whose connection is $\pi$. Then

$8\pi^{2}s^{*}(TC_{2}(\pi))=\frac{1}{12}(-12F^{2}_{\alpha}+2(F_{\alpha\alpha\alpha
x}\cos{\alpha}+F_{\alpha\alpha\alpha
y}\sin{\alpha}+F_{\alpha\alpha\alpha\alpha}F)+14(F_{\alpha
x}\cos{\alpha}+F_{\alpha y}\sin{\alpha}$
$+F_{\alpha\alpha}F)-3(-F_{\alpha\alpha x}\sin{\alpha}+F_{\alpha\alpha
y}\cos{\alpha})+3F_{\alpha}-24(-F_{x}\sin\alpha+F_{y}\cos\alpha)+18F^{2}+6FF_{\alpha\alpha})\theta\wedge\theta^{1}\wedge\theta^{2}.$

and

$8\pi^{2}\mu(Y)=8\pi^{2}\int_{M}s^{*}(TC_{2}(\pi))=\frac{1}{12}\int_{M}(-32F^{2}_{\alpha}+2F^{2}_{\alpha\alpha}+18F^{2})\theta\wedge\theta^{1}\wedge\theta^{2}.$

Proof. The terms in Proposition 4.5 were all computed before. A substitution
of these terms in the formula gives the first formula. The second formula is
obtained by integration by parts.

$\Box$

###### Corollary 4.11

Let $M$ be equipped with a path structure defined by an ordinary differential
equation of second order on the torus with strict structure defined by the
forms $\theta,\theta^{1}$ and $\theta^{2}$. Let $Y$ be the canonical Cartan
bundle with its associated Cartan connection.

  1. 1.

If $\theta^{2}=d\alpha-F(x,y)\theta^{1}$ (the function $F$ does not depend on
$\alpha$). Then $\mu(Y)=0$ if and only if $F=0$.

  2. 2.

$Q^{1}(Y)=0$ and $\mu(Y)=0$ if and only if $F=0$.

Proof. Clearly, if $F=0$ then $\mu(Y)=0$ and $Q^{1}=0$. If $F$ does not depend
on $\alpha$ then the invariant becomes

$\frac{4}{3}\int_{M}F^{2}\theta\wedge\theta^{1}\wedge\theta^{2},$

which is zero only if $F=0$. Suppose now that $Q^{1}(Y)=0$ and $\mu(Y)=0$.
Observe that the integral formula for the invariant, by an integration by part
and a slight rearrangement, may be written as

$\frac{1}{12}\int_{M}(-12F^{2}_{\alpha}+2F_{\alpha\alpha\alpha\alpha}F+20F_{\alpha\alpha}F+18F^{2})\theta\wedge\theta^{1}\wedge\theta^{2}.$

Using the expression of $Q^{1}$ given in 4.8 and the hypothesis $Q^{1}=0$, we
obtain $F_{2222}+10F_{22}+9F=0$, and therefore

$2F_{\alpha\alpha\alpha\alpha}F+20F_{\alpha\alpha}F+18F^{2}=0.$

Therefore

$8\pi^{2}\mu(Y)=-\int_{M}F^{2}_{\alpha}\theta\wedge\theta^{1}\wedge\theta^{2}.$

We observe therefore that if $\mu(Y)=0$ then $F_{\alpha}$ should be null. But
if $F$ does not depend on $\alpha$ it should be null by the first part.
$\Box$

##  5 Path structures on a torus

We recall example II which is the torus $T^{3}$ with coordinates $(x,y,t)$
($\mod 1$) and the global contact form, for a fixed $n\in{{\mathbb{Z}}}^{*}$,

$\theta=\cos(2\pi nt)dx-\sin(2\pi nt)dy.$

It was proven independently by E. Giroux and Y. Kanda that the contact
structures defined by these contact forms classify all tight structures on
$T^{3}$ (see [Y]). We will show here that for each of these contact structures
one can define a flat path structure.

There are two canonical global vector fields on the distribution given by
$X_{1}=\frac{\partial}{\partial t}$ and $X_{2}=\sin(2\pi
nt)\frac{\partial}{\partial x}+\cos(2\pi nt)\frac{\partial}{\partial y}$.

We define

$\theta^{1}=-2\pi ndt,\ \ \theta^{2}=\sin(2\pi nt)dx+\cos(2\pi nt)dy,$

so that $d\theta=\theta^{1}\wedge\theta^{2}$ and we define the strict path
structure defined by these forms. We compute

$d\theta^{1}=0,\ \ d\theta^{2}=-\theta^{1}\wedge\theta.$

Comparing now with the enriched path connection we obtain

$\varphi=\frac{1}{2}(\frac{da_{1}}{a_{1}}+\frac{da_{2}}{a_{2}}),\ \
3w=\frac{1}{2}(\frac{da_{1}}{a_{1}}-\frac{da_{2}}{a_{2}}),\ \ \tau^{1}_{2}=0,\
\ \tau^{2}_{1}=-\frac{1}{a_{1}^{2}}$

and therefore
$d\varphi=dw=d\tau^{1}_{2}=d\tau_{1}^{2}+2\tau^{2}_{1}(\varphi+3w)=0$. It
follows that $A=B=C=D=S=\tau^{1}_{20}=\tau^{2}_{10}=0$, and it follows from
formulas 38 and 39 that $Q^{1}=Q^{2}=0$. We proved:

###### Lemma 5.1

The path structures defined by the forms $\theta^{1},\theta^{2},\theta$ on
$T^{3}$ are flat.

We now define a new strict path structure by fixing the contact form $\theta$
and changing $\theta^{1}$ and $\theta^{2}$ by a constant matrix:

$\left(\begin{array}[]{c}{\theta^{1}}^{\prime}\\\
{\theta^{2}}^{\prime}\par\end{array}\right)=\left(\begin{array}[]{cc}a&b\\\
c&f\\\ \end{array}\right)\left(\begin{array}[]{c}\theta^{1}\\\
\theta^{2}\par\end{array}\right)\ \ \ \ \mbox{or}\ \
\left(\begin{array}[]{c}{\theta^{1}}\\\
{\theta^{2}\par}\end{array}\right)=\left(\begin{array}[]{cc}f&-b\\\ -c&a\\\
\end{array}\right)\left(\begin{array}[]{c}{\theta^{1}}^{\prime}\\\
{\theta^{2}}^{\prime}\par\end{array}\right),$

where

$\det\left(\begin{array}[]{cc}a&b\\\ c&f\\\ \end{array}\right)=1.$

We compute
$d{\theta^{1}}^{\prime}=ad{\theta^{1}}+bd{\theta^{2}}=-b\theta^{1}\wedge\theta=-bf{\theta^{1}}^{\prime}\wedge\theta+b^{2}{\theta^{2}}^{\prime}\wedge\theta$
and
$d{\theta^{2}}^{\prime}=cd{\theta^{1}}+fd{\theta^{2}}=-f\theta^{1}\wedge\theta=-f^{2}{\theta^{1}}^{\prime}\wedge\theta+bf{\theta^{2}}^{\prime}\wedge\theta$.
In order to compute the enriched connection we need to find
$\varphi^{\prime},w^{\prime},{\tau^{1}}^{\prime},{\tau^{2}}^{\prime}$
satisfying

$d{\omega^{1}}^{\prime}=(\varphi^{\prime}+3w^{\prime})\wedge{\omega^{1}}^{\prime}+\omega^{\prime}\wedge{\tau^{1}_{2}}^{\prime}{\omega^{2}}^{\prime},\
\ \
d{\omega^{2}}^{\prime}=(\varphi^{\prime}-3w^{\prime})\wedge{\omega^{2}}^{\prime}-\omega^{\prime}\wedge{\tau^{2}_{1}}^{\prime}{\omega^{1}}^{\prime}.$

Comparing with the structure equations and observing
${\omega^{i}}^{\prime}=a_{i}{\theta^{i}}^{\prime}$ we obtain

$\varphi^{\prime}=\frac{1}{2}(\frac{da_{1}}{a_{1}}+\frac{da_{2}}{a_{2}}),\ \
3w^{\prime}=\frac{1}{2}(\frac{da_{1}}{a_{1}}-\frac{da_{2}}{a_{2}})+\frac{bf}{a_{1}a_{2}}\omega,\
\ $ $\ {\tau^{1}_{2}}^{\prime}=-\frac{b^{2}}{a_{2}^{2}}{\theta^{2}}^{\prime}\
\ \mbox{and}\ \
{\tau^{2}_{1}}^{\prime}=-\frac{f^{2}}{a_{1}^{2}}{\theta^{1}}^{\prime}.$

Then $d\varphi=0,\ \
3dw=\frac{bf}{a_{1}a_{2}}{\omega^{1}}^{\prime}\wedge{\omega^{2}}^{\prime}$,
and it follows $A=B=C=D=0$ and $3S=\frac{bf}{a_{1}a_{2}}$. Also

$d{\tau^{1}_{2}}^{\prime}=-2{\tau^{1}_{2}}^{\prime}(\varphi-3w)+\frac{2b^{3}f}{a_{2}^{3}a_{1}}\omega,\
\
d{\tau^{2}_{1}}^{\prime}=-2{\tau^{2}_{1}}^{\prime}(\varphi+3w)-\frac{2bf^{3}}{a_{1}^{3}a_{2}}\omega,$

and ${\tau^{1}_{20}}^{\prime}=\frac{2b^{3}f}{a_{2}^{3}a_{1}},\ \
{\tau^{2}_{10}}^{\prime}=-\frac{2bf^{3}}{a_{1}^{3}a_{2}}$. At last
$dS=-2S\varphi$ and we get $S_{0}=S_{1}=S_{2}=0$. Then $P=n=0$, and we obtain
from formulas 38 and 39 that

$Q^{1}=\frac{3}{2}\frac{b^{3}f}{a_{1}a_{2}^{3}},\ \
Q^{2}=-\frac{3}{2}\frac{bf^{3}}{a_{1}^{3}a_{2}}.$

We proved

###### Lemma 5.2

The path structures defined by the forms $\theta^{\prime 1},\theta^{\prime
2},\theta$ on $T^{3}$ have curvatures $Q^{1}=\frac{3}{2}b^{3}f,\ \ \
Q^{2}=-\frac{3}{2}bf^{3}$ (computed through a section on the torus).

Note that the path structure is flat if and only if the one of the torsions
${\tau^{1}}^{\prime}$ or ${\tau^{2}}^{\prime}$ are zero and this happens if
the direction defined by $\frac{\partial}{\partial t}$ is one of the line
bundles contained in the contact bundle of the path structure. The couple
$(b,f)$ is determined up to a sign by the curvatures $Q^{1}$ and $Q^{2}$.

The global invariant is given in the next Proposition.

###### Proposition 5.3

Let ${T^{3}}_{n}(a,b,c,d,f)$ as the path structure on the torus defined as
above. Then the global invariant is

$\mu({T^{3}}_{n}(a,b,c,d,f))=\frac{3n}{8\pi}(bf)^{2}.$

Proof. This is a direct computation using the formula for the global invariant
(see formula 44):

$\int_{T^{3}}s^{*}TC_{2}(\pi)=\int_{T^{3}}\frac{1}{8\pi^{2}}(2{\tau^{1}_{2}}^{\prime}{\tau^{2}_{1}}^{\prime}\theta\wedge\theta^{1}\wedge\theta^{2}-\frac{9}{2}w\wedge\theta^{1}\wedge\theta^{2})=\int_{T^{3}}\frac{1}{8\pi^{2}}\frac{3}{2}b^{2}f^{2}\theta\wedge\theta^{1}\wedge\theta^{2}.$

Therefore

$\mu({T^{3}}_{n}(a,b,c,d,f))=\int_{T^{3}}\frac{3}{16\pi^{2}}(bf)^{2}\theta
d\theta=\frac{3n}{8\pi}(bf)^{2}.$

$\Box$

Note that the global invariant is null if and only if the path structure is
flat.

##  6 Invariant path structures on ${\hbox{\bf SU}}(2)$

Tight contact structures on $S^{3}$ are all contactomorphic (see [E]). In this
section we explicit homogeneous strict path structures on ${\hbox{\bf SU}}(2)$
which are carried by a fixed left invariant tight contact structure.

Let $\alpha,\beta,\gamma$ be a basis of left invariant 1-forms defined on
$\hbox{\bf SU}(2)$ with

$d\alpha=-\beta\wedge\gamma,\ \ \ d\beta=-\gamma\wedge\alpha,\ \ \
d\gamma=-\alpha\wedge\beta\ \ \ $

A strict path structure on ${\hbox{\bf SU}}(2)$ is given by fixing the contact
form $\theta=\gamma$ and the line fields $E^{1}=\ker{\alpha}\cap\ker{\theta}$
and $E^{2}=\ker{\beta}\cap\ker{\theta}$.

We define strict path structures by choosing a map from $\hbox{\bf SU}(2)$ to
$\hbox{\bf SL}(2,{\mathbb{R}})$:

$\theta=\gamma,\ \ \ Z^{1}=r_{1}\beta+r_{2}\alpha,\ \ \
Z^{2}=s_{1}\beta+s_{2}\alpha,$

with $r_{1}s_{2}-r_{2}s_{1}=1$. Then

$d\theta=Z^{1}\wedge Z^{2}.$

In the case the map $\hbox{\bf SU}(2)\rightarrow\hbox{\bf SL}(2,{\mathbb{R}})$
is constant, from $\beta=s_{2}Z^{1}-r_{2}Z^{2}$ and
$\alpha=-s_{1}Z^{1}+r_{1}Z^{2}$, we obtain

$dZ^{1}=r_{1}d\beta+r_{2}d\alpha=\theta\wedge\left(xZ^{1}+yZ^{2}\right)$

and analogously,

$dZ^{2}=\theta\wedge\left(zZ^{1}-xZ^{2}\right),$

where

$x=r_{1}s_{1}+r_{2}s_{2},\ \ \ y=-(r_{1}^{2}+r_{2}^{2}),\ \ \
z=s_{1}^{2}+s_{2}^{2}.$

Observe that $x^{2}+yz=-1$. Then for a enriched path structure with coframes
obtained from t!he tautological forms $\omega=a_{1}a_{2}\theta$,
$\omega^{1}=a_{1}Z^{1}$ and $\omega^{2}=a_{2}Z^{2}$ we obtain

$d\omega^{1}=(\frac{da_{1}}{a_{1}}+x\theta)\wedge\omega^{1}+a_{1}y\theta\wedge
Z^{2}.$
$d\omega^{2}=(\frac{da_{2}}{a_{2}}-x\theta)\wedge\omega^{2}+a_{2}z\theta\wedge
Z^{1}$

From Proposition 2.3 we have

$\varphi=\frac{1}{2}(\frac{da_{1}}{a_{1}}+\frac{da_{2}}{a_{2}}),\ \
3w=\frac{1}{2}(\frac{da_{1}}{a_{1}}-\frac{da_{2}}{a_{2}})+\frac{x}{a_{1}a_{2}}\omega,\
\ $ $\tau^{1}_{2}=\frac{y}{a_{2}^{2}},\ \ \
\tau^{2}_{1}=-\frac{z}{a_{1}^{2}}.$

and therefore

$d\varphi=0,\ \ 3dw=d(x\theta)=\frac{x}{a_{1}a_{2}}\omega^{1}\wedge\omega^{2}$

so that $S=\frac{x}{3a_{1}a_{2}},A=B=C=D=0$.

From

$d\tau_{2}^{1}=-2\frac{da_{2}}{a_{2}}\frac{y}{a_{2}^{2}}=-2\tau^{1}_{2}(\varphi-3w)-2\frac{xy}{a_{1}a_{2}^{3}}\omega$
$d\tau_{1}^{2}=2\frac{da_{1}}{a_{1}}\frac{z}{a_{1}^{2}}=-2\tau^{2}_{1}(\varphi+3w)-2\frac{xz}{a_{2}a_{1}^{3}}\omega$

we obtain

$\tau_{20}^{1}=-2\frac{xy}{a_{1}a_{2}^{3}},\ \
\tau_{10}^{2}=-2\frac{xz}{a_{2}a_{1}^{3}}$

From

$dS=-\frac{x}{3a_{1}a_{2}}(\frac{da_{1}}{a_{1}}+\frac{da_{2}}{a_{2}})=-2\varphi
S$

we obtain $S_{0}=S_{1}=S_{2}=0$, ans $P=n=0$.

It follows from formulas 38 and 39 that
$Q^{1}=\tau^{1}_{20}+\frac{3}{2}S\tau^{1}_{2}$ and
$Q^{2}=\tau^{2}_{10}-\frac{3}{2}S\tau^{2}_{1},$ therefore

$Q^{1}=-\frac{xy}{a_{1}a_{2}^{3}}$

and

$Q^{2}=-\frac{xz}{a_{1}^{3}a_{2}}.$

Observe that $y$ and $z$ never vanish. We conclude that the invariant strict
structure on $\hbox{\bf SU}(2)$ is a flat path structure if and only if $x=0$.
This can be interpreted, because $x=r_{1}s_{1}+r_{2}s_{2}$, as the strict
structures such that the directions $E^{1}$ and $E^{2}$ are perpendicular for
the canonical metric defined by the forms $\alpha$ and $\beta$.

###### Proposition 6.1

Define strict path structures on $\hbox{\bf SU}(2)$ by choosing a constant map
from $\hbox{\bf SU}(2)$ to $\hbox{\bf SL}(2,{\mathbb{R}})$:

$\theta=\gamma,\ \ \ Z^{1}=r_{1}\beta+r_{2}\alpha,\ \ \
Z^{2}=s_{1}\beta+s_{2}\alpha,$

with $r_{1}s_{2}-r_{2}s_{1}=1$. Let $x=r_{1}s_{1}+r_{2}s_{2}$. Then the global
invariant of the induced path structure is

$\mu(\hbox{\bf
SU}(2)(r_{1},r_{2},s_{1},s_{2}))=-\frac{1}{2}-\frac{3}{8}x^{2}.$

Proof. We compute, using formula 44, the global invariant for the family of
structures defined on $\hbox{\bf SU}(2)$. We have from above that
$x=r_{1}s_{1}+r_{2}s_{2},y=-(r_{1}^{2}+r_{2}^{2}),z=s_{1}^{2}+s_{2}^{2}$ and
that $x^{2}+yz=-1$. Then it follows

$\int_{\hbox{\bf SU}(2)}s^{*}TC_{2}(\pi)=\int_{\hbox{\bf
SU}(2)}\frac{1}{8\pi^{2}}(2{\tau^{1}_{2}}{\tau^{2}_{1}}\theta-\frac{9}{2}Sw)\wedge\theta^{1}\wedge\theta^{2}=-\int_{\hbox{\bf
SU}(2)}\frac{1}{8\pi^{2}}(2yz+\frac{1}{2}x^{2})\gamma\wedge\beta\wedge\alpha$
$=\int_{\hbox{\bf
SU}(2)}\frac{1}{8\pi^{2}}(-2-\frac{3}{2}x^{2})\gamma\wedge\beta\wedge\alpha.$

We use then that $\int_{\hbox{\bf
SU}(2)}\gamma\wedge\beta\wedge\alpha=2\pi^{2}$.  $\Box$

Observe that the invariant is never null for this family even in the case of a
flat path structure (which happens when $x=0$). Also the critical point of the
invariant along this family is a maximal at $x=0$, at a flat structure, and it
is equal to $-\frac{1}{2}$.

## References

  * [A] Arnold, V. : Chapitres supplémentaires de la théorie des équations différentielles ordinaires. Mir, Moscow, 1980.
  * [B] Bryant, R. : Élie Cartan and geometric duality. Preprint 1998.
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E. Falbel
Institut de Mathématiques
de Jussieu-Paris Rive Gauche
CNRS UMR 7586 and INRIA EPI-OURAGAN
Sorbonne Université, Faculté des Sciences
4, place Jussieu 75252 Paris Cedex 05, France
<EMAIL_ADDRESS>
J. M. Veloso
Faculdade de Matemática - ICEN
Universidade Federal do Pará
66059 - Belém- PA - Brazil
<EMAIL_ADDRESS>