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%    On differential equations and Cartan's projective connections. 
%    In: Geometry in Partial Differential Equations. World Scientific, 
%    Singapore, 1994, 329-344.
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%                      The title of the paper:
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\centerline{\bf ON DIFFERENTIAL EQUATIONS}
\centerline{\bf AND CARTAN'S PROJECTIVE CONNECTIONS}
\vskip 70pt
 {
\baselineskip=12pt

\centerline{YOURY R. ROMANOVSKY}
\centerline{\tenit Program Systems Institute of the Russian 
                Academy of Sciences}
\centerline{\tenit Pereslavl-Zalessky 152140, Russia}
\vskip 48pt
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                       \centerline{ABSTRACT}
\parindent=3pc
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We describe a method for constructing a normal projective connection
associated with a second order ordinary differential equation. Our construction
shows that differential equations generate Cartan's connections in the same way
as Riemannian metrics generate Levi-Civita's connections. We use this
construction to give a classification, in terms of projective curvature, of
differential equations admitting Lie groups of point symmetries.
 \par}
 }
% 
%                   The main part of the article: 
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\rm
\vskip 10mm
\leftline{\bf 1. Introduction}
\vskip 5mm\nobreak

This work is devoted to Elie Cartan's theory of {\it
normal projective connections}$^{1}$ and its applications to differential
equations.  Projective connections themselves provide a geometric theory of
ordinary differential equations. However, this theory gave rise to the notion
of {\it Cartan connection}$^{2}$ which has a wider range of applications. For
example, the Cartan connection associated with a second order partial
differential equation of Monge-Amp\`ere type was discovered in the paper by
T.~Morimoto$^{3}$.

In \S 2, we begin with a second order ordinary differential equation and
describe a natural way leading to the notion of normal projective connection
associated with a {\it cubic differential equation} (see Eq.~8). An
important intermediate step is the construction of a {\it projective
structure}$^{2}$. A projective structure is a principal bundle in which the
normal projective connection appears in an intrisic manner. Cubic differential
equations and normal projective connections are in one-to-one
correspondence. The differential 1-form of a normal projective connection is an
absolute parallelism on a projective structure. Hence, the equivalence problem
for cubic differential equations under the pseudo-group of coordinate
transformations reduces to the equivalence problem for absolute
parallelisms. The latter problem has a well-known solution$^{4}$.

In \S 3, we explain how one can use the connection form to solve
the equivalence problem for a pair of cubic differential equations. Here we
discuss the notion of {\it projective curvature} and prove an
elegant Cartan's criterion for an equation to be locally equivalent to the
differential equation of straight lines (see Eq.~1). The other interesting 
result is Cartan's {\it theorema egregium} stating that the curves on a surface
in Euclidean space that are orthogonal to the level lines of Gaussian
curvature can be computed in terms of the differential equation of geodesics
(see~Eq.~3).

In \S 4, we use the connection form to investigate the
{\it symmetry group} of a cubic differential equation. Here we deduce the
generalized {\it Bianchi identities}. As a consequence, we
give a necessary and sufficient condition, in terms of projective curvature,
for an equation to admit a symmetry group of dimension $n\ge3$. It should be
emphasized that such an equation is locally equivalent either to the
differential equation of straight lines or to the differential equation of
quadrics (see Eq.~2).

Finally, we note that an analogous classification can be obtained for cubic
differential equations with a symmetry group of dimension $n=2$. Moreover, in
this case the machinery of normal projective connections allows us to find
{\it by quadratures} the infinitesimal generators of a symmetry group,
{\it i.e.} the generators of its Lie algebra. Thus, if an equation has a
symmetry algebra of dimension two, then the generators of this algebra can be
found by quadratures. This means that the solutions of the equation can be
found by quadratures too$^5$. We will study the problem of integration
by quadratures for cubic differential equations in a forthcoming paper.

\vskip 10mm
\leftline{\bf 2. Projective Structures}
\vskip 5mm\nobreak

We begin by recalling that the general solution of a second order ordinary
differential equation $y''=f(x,y,y')$ is, by definition, the two-parameter
family of its integral curves on the plane $\P.$ 
\medskip
{\bf Example 1.} The differential equation of straight lines:
$$
y''=0.\eqno(1)
$$
The general solution: $x\cos t_1+y\sin t_1+t_2=0.$
\medskip
{\bf Example 2.} The differential equation of quadrics:
$$
y''=\pm (xy'-y)^3.\eqno(2)
$$
The general solution: $\;XY(X^2\pm Y^2-1)=0\;$ where 
$\;X=(x\cos t_1+y\sin t_1)t_2,\break Y=(y\cos t_1-x\sin t_1)/t_2.$
\medskip
{\bf Example 3.} The differential equation of geodesics:
$$
y''=a_0+a_1y'+a_2y'^2+a_3y'^3,\eqno(3)
$$
where
$$
\eqalign{a_0&=\,\,\,\,\,((EE_y+FE_x)/2-EF_x)/(EG-F^2),\cr
         a_1&=\,\,\,\,\,(3FE_y/2-FF_x-EG_x+E_xG/2)/(EG-F^2),\cr
         a_2&=         -(3FG_x/2-FF_y-GE_y+G_yE/2)/(EG-F^2),\cr
         a_3&=         -((GG_x+FG_y)/2-GF_y)/(EG-F^2).\cr}
$$
The general solution: geodesics of the Riemannian metric
$Edx^2+2Fdx\,dy+Gdy^2.$
Here we note that Eq.~1  and Eq.~2 are 
particular cases of Eq.~3. 
\medskip
Let us consider a special equivalence problem: determine all local
diffeomorphisms $\varphi\colon\P \rightarrow \P$ which take the general
solution of Eq.~1 into the general solution of the target equation
$y''=f(x,y,y').$ We say that the target equation is locally equivalent to the
differential equation of straight lines, if the set of such diffeomorphisms is
not empty.  The first obstacle to construct $\varphi$ arises at the
level of a one-parameter pencil of integral curves:
$$ 
y-y_0=p(x-x_0)+{f(x_0,y_0,p)\over 2}(x-x_0)^2+O((x-x_0)^3),
                                                    \quad p=y'(x_0).\eqno(4) 
$$
\medskip
{\bf Example$^6$.} The pencil of circles
$$
y=px+a(p)(x^2+y^2)
$$
can be rectified by a diffeomorphism in a neigbourhood of the point $(0,0)$, if
and only if the function $a(p)$ is a linear function. In other words, the
circles should have one more common point. Otherwise the rectifying
transformations are not smooth at the origin of $\P.$ This is an immediate
consequence of the following proposition.
\medskip
{\bf Proposition 1.} {\it Pencil (4) transforms by a diffeomorphism 
$$
\eqalign{x&=\varphi^1(z^1,z^2)\cr
         y&=\varphi^2(z^1,z^2)\cr}
$$
into the form 
$$
z^2=qz^1+O(|z|^3),\quad p={u^2_2q+u^2_1 \over u^1_2q+u^1_1},\quad 
u^i_j={\partial\varphi^i \over \partial z^j}(0),                  \eqno(5)
$$
if and only if $\;f(p)\;$ is a cubic polynomial: 
$$
f(p)=A+3Bp+3Cp^2+Dp^3.                                            \eqno(6)
$$
Suppose $\;\varphi,\,\psi\;$ is a pair of diffeomorphisms transforming pencil
(4) to pencil (5). Then there is a unique projective transformation
$$
h\colon\Pz\to \Pz,\quad h^i={a^i_jz^j \over 1+a_jz^j},              \eqno(7)
$$
such that
$$
j^2_0(\varphi^{-1}\circ\psi)=j^2_0(h),
$$
where $\;j^2_0(\varphi)\;$ is the 2-jet of $\;\varphi\;$ at $\;z=0.$}
\medskip
{\it Proof.\/} Follows from a straightforward calculation.
\medskip
Proposition 1 has three important corollaries.
\medskip
{\bf Corollary 1.} {\it The condition $\;\partial ^4f/\partial y'^4=0\;$ is 
a necessary condition for the equation $\;y''=f(x,y,y')\;$ to be locally 
equivalent to the differential equation of straight lines. The set of
``suspicious equations''
$$
y''=A(x,y)+3B(x,y)y'+3C(x,y)y'^2+D(x,y)y'^3                       \eqno(8)
$$
is invariant under the pseudo-group of coordinate transformations.}
\medskip
We call Eq.~8 a {\it cubic differentual equation}. From now on we shall be only
interested in cubic differentual equations. For the reader whose interest is
in geometry of general second order ordinary differential equations, we would
recommend the paper by E.~Cartan$^7$.
\medskip
{\bf Corollary 2.} {\it A cubic differential equation generates a principal 
fibre bundle
$$
\pi\colon P\to \P                                                \eqno(9)
$$
whose fibre $\;\pi ^{-1}(x_0,y_0)\;$ consists of 2-jets of diffeomorphisms
$$
\varphi\colon \Pz\to \P,\quad \varphi (0)=(x_0,y_0),             \eqno(10)
$$
which take pencil (5) to pencil (4) where $\;f(p)\;$ is cubic polynomial (6).
The structure group $\;H\;$ of bundle (9) coincides with the group of 
projective transformations~(7).}
\medskip
We recall$^2$ that the $r$-jet $j^r_0(\varphi)$ of diffeomorphism (10) at $0$
is, by definition, an {\it r-frame} of $\P$ at $(x_0,y_0).$ The set of
$r$-frames of $\P$ will be denoted by $P^r.$ It is a smooth manifold. For
example, each 3-frame $u\in P^3$ has a unique polynomial representation
$u=j^3_0(\varphi)$ of the form
$$
\varphi ^i(z)=u^i+u^i_jz^j+u^i_{jk}z^jz^k+u^i_{jkl}z^jz^kz^l.
$$
The coefficients $u^i_{jk},\;u^i_{jkl}$ are symmetric in the lower indices.
We take $(u^i,u^i_{j},u^i_{jk},u^i_{jkl})$ as the natural coordinate system in
$P^3.$ The bundle
$$
\pi_r\colon P^r\to\P,\quad \pi _r(j^r_0(\varphi ))=\varphi (0),
$$
is a principal fibre bundle. Its structure group $G^r$ consists of $r$-jets of 
the diffeomorphisms $g\colon \Pz\to\Pz,\;g(0)=0.$ The group $G^r$ acts on $P^r$
by $j^r_0(\varphi)\cdot j^r_0(g)=j^r_0(\varphi\circ g)$. For example, if
$(a^i_{j},a^i_{jk})$ are the natural coordinates of $j^2_0(g)\in G^2,$ then we
have
$$
(u^i,u^i_{j},u^i_{jk})\cdot (a^i_{j},a^i_{jk})=
(u^i,u^i_pa^p_{j},u^i_pa^p_{jk}+u^i_{qr}a^q_ja^r_k).
$$
Let $H^2=\{j^2_0(h)\in G^2\mid h\in H\}.$ Then $H^2$ is isomorphfic to $H.$ 
This follows from the formula
$$j^2_0(h)=(a^i_j,-(a^i_ja_k+a^i_ka_j)/2).$$
\medskip
{\bf Definition$^2$.} A subbundle $P$ of $P^2$ with structure group $H^2$ is
called a\break {\it projective structure} on $\P.$
\medskip
{\bf Corollary 3.} {\it Bundle (9) is a projective structure on $\P.$
Conversely, any projective structure on $\P$ is generated by a cubic
differential equation. Cubic differential equations and projective structures
are in one-to-one correspondence.}
\medskip
{\it Proof.} Let $P$ be a projective structure. Then there is a unique cross
section
$$
s\colon \P\to P,\quad s(x,y)=(x,y,u^i_j(x,y),u^i_{jk}(x,y)),
$$
such that
$$
\eqalign{&u^i_j=\delta ^i_j,\quad u^i_{jk}=-{1 \over 2}\Pi^i_{jk},\cr
         &\Pi ^1_{11}+\Pi ^2_{12}=\Pi ^2_{22}+\Pi ^1_{21}=0,\cr}
$$
where $\delta ^i_j$ is Kronecker's delta. This cross section is called the
{\it natural frame} ({\it le rep\`ere naturel} in Cartan$^1$). Thus, each
element of $P$ has a unique representation of the form 
$$ 
s\cdot j^2_0(h)=(x,y,a^i_j,-(a^i_ja_k+a^i_ka_j+\Pi ^i_{qr}a^q_ja^r_k)/2).  
                                                                    \eqno(11)
$$ 
In the sequel, we shall use 
$$ 
(x,y,a),\quad a=\left(\matrix{1 & a_1   & a_2   \cr 
                              0 & a^1_1 & a^1_2 \cr 
                              0 & a^2_1 & a^2_2 \cr}\right),        \eqno(12) 
$$ 
as the natural coordinate system in $P.$ A simple calculation shows
that diffeomorphisms (11) take pencil (5) to pencil (4) where
$f(p)$ is polynomial (6) with the coefficients
$$
\left\{
\eqalign{A&=-\Pi^2_{11},\cr
         B&=\;\;\> \Pi^1_{11}=-\Pi^2_{12}=-\Pi^2_{21},\cr
         C&=-\Pi^2_{22}=\;\;\> \Pi^1_{12}=\;\;\> \Pi^1_{21},\cr
         D&=\;\;\> \Pi^1_{22}.\cr}
\right.                                                             \eqno(13)
$$
The proof is complete.
\medskip
In order to solve the special equivalence problem for Eq.~8, we should give now
the answer to the following question.
\medskip
{\it Does there exist the prolongation of the projective structure $P$
$$
P_{\infty}=\{\varphi\colon \Pz\to \P\mid j^2_0(\varphi)\in P\}
$$
such that all composite maps
$$
\varphi^{-1}\circ\psi : \Pz\to\Pz\qquad (\varphi ,\psi\in P_{\infty})
$$
are collineations of $\;\Pz\;$ ?}
\medskip
If the answer to this question is affirmative, then $P_{\infty}$ is 
the solution of the problem. It is clear that $P_{\infty}$ is a principal fibre
bundle over $\P$ with structure group $H.$
Let $\varphi$ be an arbitrary element of 
$P_{\infty}$ and $\varphi_t \in P_{\infty}$ be a one-parameter family such
that $\varphi_0=\varphi.$ Then the composite map 
$\;\chi_t=\varphi^{-1}\circ\varphi_t\;$ is a collineation
$$
\chi^i_t={b^i(t)+b^i_j(t)z^j \over 1+b_j(t)z^j},                    \eqno(14)
$$
where $b^i(0)=b_j(0)=0,\;\;b^i_j(0)=\delta^i_j.$ Hence we have
$$
d\chi^i_t|_{t=0}=db^i+db^i_j\,z^j
-{1 \over 2}(\delta^i_j\,db_k+\delta^i_k\,db_j)z^jz^k+O(|z|^3).     \eqno(15)
$$
On the other hand, we obtain
$$
d\chi^i_t|_{t=0}=\theta^i+\theta^i_j\,z^j
+\theta^i_{jk}\,z^jz^k+O(|z|^3),                                    \eqno(16)
$$
where $\;\theta=(\theta^i,\,\theta^i_{j},\,\theta^i_{jk})\;$ is the so-called
{\it canonical (differential) 1-form$^8$} on $P^3$ at the point
$j^3_0(\varphi).$ The values of $\theta$ can be viewed as the 2-jets of
infinitesimal composite maps. A straightforward computation gives the following
formul\ae:
$$
\eqalign{\theta^i\,\,  &=v^i_p\,du^p,                                    \cr
         \theta^i_j\,\,&=v^i_p\,du^p_j\,\,-2v^i_pu^p_{qj}\,\theta^q,     \cr
         \theta^i_{jk} &=v^i_p\,du^p_{jk} -2v^i_pu^p_{qj}\,\theta^q_k
                                          -3v^i_pu^p_{qjk}\,\theta^q,    \cr}
$$
where $(v^i_j)$ is the inverse matrix of $(u^i_j)$;
$(u^i,u^i_{j},u^i_{jk},u^i_{jkl})=j^3_0(\varphi)$. Now comparing expressions 
(15) and (16), we see that the restriction of the canonical form
$\theta$ to
$$
P_3=\{j^3_0(\varphi)\mid \varphi\in P_{\infty}\}
$$
satisfies the condition
$$
\varepsilon^i_{jk}\buildrel \rm def \over = 
\theta^i_{jk}-{1 \over 2}(\delta^i_j\,\theta^k_{kk}+\delta^i_k\,\theta^j_{jj})
\;=\;0.
$$
In other words, $P_3$ is a subbundle of $P^3$
annihilating the Pfaffian system $\{\varepsilon^i_{jk}\}.$

Now, in answer to the preceding question, we have the following fundamental
observation: the prolongation $P_{\infty}$ does not generally exist, while the
prolongation $P_3$ can be constructed for any projective structure $P$.
\medskip
{\bf Proposition 2.} {\it For any projective structure $P$ there exists
a unique map
$$
\sigma\colon P\to P^3\qquad (\,j^2_0\circ\sigma=id\,)
$$
such that 
$$
\sigma^{\ast}(\varepsilon^i_{jk})\;=\;0.
$$
The bundle
$$
\pi_3|_{P_3}:\; P_3\buildrel \rm def \over = \sigma (P)\;\to\; \P
$$
is a principal subbundle of $\,P^3$ with structure group 
$H^3=\{j^3_0(h)\mid h\in H\}\simeq H^2\simeq H.$}
\medskip
{\it Proof.} Follows from a straightforward computation in the natural
coordinates.
\medskip
Proposition 2 enables us to take the canonical form $\theta$ on $P^3$ back to
a form on the projective structure $P.$ We set
$$
\omega=\left(\matrix{0        & \omega_1   & \omega_2   \cr 
                     \omega^1 & \omega^1_1 & \omega^1_2 \cr 
                     \omega^2 & \omega^2_1 & \omega^2_2 \cr}\right),
$$
where
$$
\omega^i=\sigma^{\ast}(\theta^i),\quad
\omega^i_j=\sigma^{\ast}(\theta^i_j),\quad
\omega_j=-\sigma^{\ast}(\theta^j_{jj}).                            \eqno(17)
$$
\medskip
{\bf Definition.} The differential 1-form $\omega$ on $P$ is called the {\it
normal projective connection} associated with the projective structure $P.$
\medskip
{\bf Remark.} The motivation behind this definition is the following simple
observation. Each tangent vector $\dot u (t) \in T(P)$ generates an
infinitesimal collineation of the form (14) where
$$
\left(\matrix{1   &b_1   &b_2   \cr 
              b^1 &b^1_1 &b^1_2 \cr 
              b^2 &b^2_1 &b^2_2 \cr}\right)=I\,-\,\omega(\dot u)\,dt,
$$ 
$I$ is the unit matrix.
This infinitesimal shift is interpreted as the infinitesimal
``{\it parallel displacement}'' of the {\it tangent projective plane} to
$\P$ at the point $\pi(u)$ in the direction $\pi_{\ast}(\dot u).$
As usual the infinitesimal picture can be integrated up to give the global
picture of parallel displacement along curves.
\medskip
The correspondence between normal projective connections and cubic
differential equations satisfies the following condition.
\medskip
{\bf Proposition 3.} {\it Let $\;(P,\omega)\;$ and $\;(\overline
P,\overline\omega)\;$ be normal projective connections associated with
Eq.~8 and Eq.~\=8$,$ respectively. Then the map $\;\Psi\colon P\to\overline P$
takes the form $\;\overline\omega\;$ back into the form $\;\omega,\;$i.e.
$\;\Psi^{\ast}(\overline\omega)=\omega,\;$ if and only if $\;\Psi =\Phi |_P\;$
where $\;\Phi\colon P^2\to P^2\;$ is the natural prolongation of the map
$\;\varphi\colon\P\to\P\;$ which takes the general solution of Eq.~\=8
back into the general solution of Eq.~8.}
\medskip
{\it Proof.} Follows from Corollary 3 and the properties of the canonical 
form$^7\;\theta.$
\medskip
Finally, to conclude this section, we shall express the connection form
$\omega$ in terms of natural coordinate system (12) on a projective structure
$P.$ For this goal, one should construct the prolongation $S\colon\P\to P_3$ of
the natural frame $s$ such that $S^{\ast}(\varepsilon^i_{jk})=0.$ Then the map
$\sigma$ is given by 
$$
\sigma\left(\,s\cdot j^2_0(h)\,\right)=S\cdot j^3_0(h).
$$
Now, from definition (17) it follows that $\omega$ is given by the
``gauge transformation''
$$
\omega=a^{-1}wa+a^{-1}da+I\,a_i\omega^i,\quad
w=\left(\matrix{0   &w_1   &w_2   \cr
                w^1 &w^1_1 &w^1_2 \cr
                w^2 &w^2_1 &w^2_2 \cr}\right),                      \eqno(18)
$$
where
$$
\eqalign{w^i&=\;\;\>s^{\ast}(\theta^i)\;\;  =\>\>\,\delta^i_1\,dx
                                              +\;\;\delta^i_2\,dy,         \cr
       w^i_j&=\;\;\,s^{\ast}(\theta^i_j)\;\;=\Pi^i_{j1}\,dx+\Pi^i_{j2}\,dy,\cr
         w_j&=     -S^{\ast}(\theta^j_{jj}) =\Pi_{j1}\,dx+\Pi_{j2}\,dy,    \cr}
$$
$$
\eqalign{\Pi_{12}=\Pi_{21}&=\;\;\;\Pi^1_{22}\Pi^2_{11}-\Pi^1_{11}\Pi^2_{22}\;\,
                            +{\partial\Pi^2_{22}\over\partial x}
                            +{\partial\Pi^1_{11}\over\partial y},        \cr
                  \Pi_{11}&=2(\Pi^1_{11}\Pi^1_{11}-\Pi^2_{11}\Pi^2_{22}) 
                            -{\partial\Pi^1_{11}\over\partial x}
                            -{\partial\Pi^2_{11}\over\partial y},        \cr
                  \Pi_{22}&=2(\Pi^2_{22}\Pi^2_{22}-\Pi^1_{11}\Pi^1_{22})
                            -{\partial\Pi^1_{22}\over\partial x}
                            -{\partial\Pi^2_{22}\over\partial y}.        \cr}
$$
The coefficients $\Pi^i_{jk}$ are given by formula (13).

\vskip 10mm
\leftline{\bf 3. The Equivalence Problem}
\vskip 5mm\nobreak

The equivalence problem is the following question. Given a pair of cubic
differential equations: Eq.~8 and Eq.~$\overline 8$, one should determine all
diffeomorphisms $\varphi\colon\P\to\P$ which take the general solution of Eq.~8
into the general solution of Eq.~$\overline 8$. If the set of such
diffeomorphisms is not empty, then the equations are said to be locally
equivalent.

From proposition 3 it follows that the normal projective connection is a good
tool to solve the equivalence problem. Actually, the components
$\omega^i,\,\omega^i_j,\,\omega_j$ of Cartan's connection (18) form a basis of
the cotangent space $T^{\ast}_u(E)$ at each point $u=(x,y,a)$ of the direct
product
$$
E=\P\times H.
$$
This means that $\omega$ is an absolute parallelism on $E.$ Proposition~3 
asserts that the equivalences $\varphi$ of Eq.~8 and Eq.~$\overline 8$ are in
one-to-one correspondence  with diffeomorphisms
$$
\Psi\colon E\to E                                                   \eqno(19)
$$
such that
$$
\Psi^{\ast}(\overline\omega)=\omega.                                \eqno(20)
$$
This correspondence is given by
$$
\Psi(x,y,a)=(\,\varphi^1,\,\varphi^2,\,\varphi_{\star}\cdot a\,),   \eqno(21)
$$
where $\varphi_{\star}$ is the {\it projective differential} of $\varphi$:
$$
\varphi_{\star}=\left(\matrix{1 &J_x         &J_y         \cr
                              0 &\varphi^1_x &\varphi^1_y \cr
                              0 &\varphi^2_x &\varphi^2_y \cr}\right),\qquad
J=-{1 \over 3}\log(\varphi^1_x\varphi^2_y-\varphi^1_y\varphi^2_x\,),
$$
which maps the tangent projective plane to $\P$ at $p$ to the
tangent projective plane to $\P$ at $\varphi(p)$.

Thus, the equivalence problem for cubic differential equations reduces to
the equivalence problem for absolute parallelisms which can be solved by the
standard method$^4.$ Following this method, we describe the finite system of
invariants recognizing a cubic differential equation up to local
equivalence. The algorithm that will find such invariants rests on the notion
of {\it structure equations}. For the problem under consideration the structure
equations are nothing else but the structure equations of Cartan's connection
$\omega$:
$$
d\omega =-\omega\wedge\omega+I\,\omega_i\wedge\omega^i
         +\Omega\,\omega^1\wedge\omega^2,                           \eqno(22)
$$
where
$$
\Omega=\left(\matrix{0 &\Omega_1 &\Omega_2 \cr
                     0 &0        &0        \cr
                     0 &0        &0        \cr}\right),\quad
\Omega_j=L_ia^i_j|a|,\quad |a|=a^1_1a^2_2-a^1_2a^2_1,
$$
$$
\eqalign{
L_1={\partial\Pi_{12}\over\partial x}-{\partial\Pi_{11}\over\partial y}
  +\Pi_{11}\Pi^1_{12}-\Pi_{12}\Pi^1_{11}+\Pi_{12}\Pi^2_{12}-\Pi_{22}\Pi^2_{11},
\cr
L_2={\partial\Pi_{22}\over\partial x}-{\partial\Pi_{12}\over\partial y}
  +\Pi_{11}\Pi^1_{22}-\Pi_{12}\Pi^1_{12}+\Pi_{12}\Pi^2_{22}-\Pi_{22}\Pi^2_{12}.
\cr}
$$
The differential 2-form $\;\Omega\,\omega^1\wedge\omega^2\;$ is called the {\it
projective curvature} of $\omega$. It follows from structure equations (22)
that the projective curvature vanishes, {\it i.e.} $\Omega=0$, if and only if 
the ``parallel displacement'' along each contractible closed path is the 
identical transformation.

We are now in position to describe the method for solving a special equivalence
problem.
\medskip
{\bf Theorem 1 (Elie Cartan).} {\it A cubic differential equation is locally
equivalent to the differential equation of straight lines, if and only if its
projective curvature vanishes.}
\medskip
{\it Proof.} Let $\omega$ and $\overline\omega$ be the normal projective
connections associated with Eq.~8 and Eq.~1 respectively. It is clear that
$$
d\overline\omega =-\overline\omega\wedge\overline\omega
                  +I\,\overline\omega_i\wedge\overline\omega^i.
                                                                    \eqno(23)
$$
Since the exterior derivative is an intrinsic operation, from (20) and (22) we
obtain
$$
\Psi^{\ast}(\overline\Omega)=\Omega.                                \eqno(24)
$$
Hence we have
$$
d\omega =-\omega\wedge\omega+I\,\omega_i\wedge\omega^i              \eqno(25)
$$
provided Eq.~8 is locally equivalent to Eq.~1. Now subtracting Eq.~(25) from
Eq.~(23), we find
$$
d(\omega-\overline\omega)=-(\omega-\overline\omega)\wedge\omega
                     -\overline\omega\wedge(\omega-\overline\omega)
                     +I\,(\omega_i-\overline\omega_i)\wedge\overline\omega^i
                     +I\,\overline\omega_i\wedge(\omega_i-\overline\omega^i).
$$
Thus, the differential system $\omega-\overline\omega=0$ is a Frobenius system
on the direct product $E\times E$. The $8$-parameter family of its
8-dimensional integral surfaces is nothing else but the family of graphs of
diffeomorphisms (19) satisfying condition (20). The proof is complete.
\medskip
In the general case ($\,\Omega\not=0,\;\overline\Omega\not=0\,$), condition
(24) is a first (but not a unique!) obstacle to construct an equivalence
$\varphi$.  Let us study the geometric meaning of this obstacle. For this aim,
we set
$$
\gamma=\Omega_1\omega^1+\Omega_2\omega^2=(L_1\,dx+L_2\,dy)|a|.
$$
The general solution of the first order equation 
$$
\gamma=0                                                            \eqno(26)
$$ 
is a one-parameter family of curves on the plane $\P$. Since condition (24) is
equivalent to $\Psi^{\ast}(\overline\gamma)=\gamma$, we see that any
equivalence $\varphi$ of Eq.~8 and Eq.~$\overline 8$ must take the
general solution of Eq.~26 into the general solution of Eq.~$\overline{26}$.
\medskip
{\bf Theorem 2 (Elie Cartan).} {\it Suppose $\omega$ is the normal projective
connection associated with differential equation of geodesics {\sl (3)}. Then
the general solution of Eq.~26 consists of the curves orthogonal to the
level lines of Gaussian curvature.}
\medskip
{\it Proof.} See Chapitre V in the book by Cartan$^1$.
\medskip
{\bf Remark.} In other words, the theorem states that the curves on a surface 
in Euclidean space that are orthogonal to the level lines of Gaussian
curvature can be computed in terms of the differential equation of geodesics.
In this sense, Cartan's theorem is analogous to Gauss's {\it theorema egregium}
stating that Gaussian curvature of a surface can be computed in terms of the
Riemannian metric.
\medskip
Now, we shall describe the total system of obstacles to construct an
equivalence $\varphi$. Consider the family
$$
{\cal F}_r=\{\,\Omega,\;\nabla(\Omega),\,\ldots,\;
             \underbrace{\nabla\cdots\nabla}_{r\rm\;times}(\Omega)\,\},\quad
\nabla=(\nabla_i,\nabla^j_i,\nabla^j),
$$
where $\;\nabla_i,\nabla^j_i,\nabla^j\;$
are the {\it covariant derivatives} with respect to the absolute parallelism
$\omega$, {\it i.e.} the vector fields on $E$ such that
$$
df=\nabla_i(f)\,\omega^i+\nabla^j_i(f)\,\omega^i_j+\nabla^j(f)\,\omega_j
$$
for any smooth function $f\in C^{\infty}(E)$; $r=\ord(\omega)$ is the {\it
order} of the parallelism $\omega$, that is the smallest $r\ge 0$ such that
$\rank({\cal F}_r)=\rank({\cal F}_{r+1})$; $\rank({\cal F})$ equals the number
of functionally independent elements in $\cal F$. Recall that the
$\rank(\omega)$ is, by definition, the number $\rank({\cal F}_r)$. Let us fix
the system
$$
\{\,x_1=\Omega_1,\;x_2=\Omega_2,\,\ldots,\;x_k\,\}\subset {\cal F}_r,
\quad r=\ord(\omega),\quad k=\rank(\omega),
$$
of functionally independent elements. Then any other element
$f\in {\cal F}_{r+1}$ generates a unique function $F_f(t_1,\ldots,t_k)$
defined on an open subset $\cal D$ of ${\bf R}^k$
such that
$$
f=F_f(x_1,\ldots,x_k).
$$
It is clear that the domain $\cal D$ does not depend on $f$.
\medskip
{\bf Theorem 3.} {\it Let $\omega$ and $\overline{\omega}$ be normal projective
connections associated with Eq.~8 and Eq.~\=8, respectively. Then the equations
are locally equivalent, if and only if the following conditions hold:

\item{{\rm (a)}} $\ord(\omega)= \ord(\overline{\omega}),\quad 
    \rank(\omega)=\rank(\overline{\omega})$;

\item{{\rm (b)}}
$F_f(t_1,\ldots,t_k)=\overline{F}_{\overline f}(t_1,\ldots,t_k)\;$
for each $\;(t_1,\ldots,t_k)\in {\cal D}\cap\overline{\cal D}\not=\emptyset$
and any pair
$\;(f,\overline{f})\in {\cal F}_{r+1}\times \overline{\cal F}_{r+1}\;$
of covariant derivatives of the same names.}
\medskip
{\it Proof.} The necessity is evident. To prove sufficiency, we observe that
the restriction of the differential system
$\omega-\overline\omega=0$ to the submanifold
$$
x_i-\overline{x}_i=0\quad (1\le i\le k)
$$
of the direct product $E\times E$ is a Frobenius system whenever conditions
(a) and (b) hold. The ($8-k$)-parameter
family of its 8-dimensional integral surfaces is nothing else but the family of
graphs of diffeomorphisms (19) satisfying condition (20). See the book by
S.~Sternberg$^4$ for details.
\medskip
See also the paper by C.~Grissom {\it et al.}$^9$ for an alternative
presentation of Cartan's method for solving the equivalence problem for second
order equations.

\vskip 10mm
\leftline{\bf 4. Group Analysis}
\vskip 5mm\nobreak

There exists an extensive literature on the symmetry groups of differential
equation. Here, we just mention the paper by L.~Hsu and N.~Kamran$^{10}$ which
is sufficiently close to the material of the present section.

By a (point) symmetry of a differential equation, we understand a local
diffeomorphism $\varphi\colon\P\to\P$ which takes the general solution of the
equation into itself. From proposition 3 it follows that the symmetry group of
a cubic differential equation is isomorphic to the group
$$
\Sym(\omega)=\{\,\Psi\colon E\to E\mid \Psi^{\ast}(\omega)=\omega\,\}
$$
of symmetries of the normal projective connection associated with the equation.
The isomorphism is given by relation (21). In order to investigate the group
$\Sym(\omega)$, we shall use the machinery developed in \S 3. In particular, we
shall determine the dimension of the group by the formula
$$
\dim\,\Sym(\omega)=8-\rank(\omega)
$$
which follows from Theorem 3. Now, we supply this formula with three lemmas 
that allow one to simplify the computation of the number $\rank(\omega)$.
\medskip
{\bf Lemma 1.} {\it Let ${\cal F}^s$ be the following family of functions
$$
{\cal F}^s=\{\,\Omega,\;\partial(\Omega),\,\ldots,\;
             \underbrace{\partial\cdots\partial}_{s\rm\;times}(\Omega)\,\},
\quad\partial=(\nabla_1,\nabla_2),
$$
and $\;\langle{\cal F}^s\rangle\;$ be the linear space spanned by
$\;{\cal F}^s.\;$ Then
$$
\nabla^i({\cal F}^s)\subset\langle{\cal F}^s\rangle,\quad
\nabla^i_j({\cal F}^s)\subset\langle{\cal F}^s\rangle
$$
for all $s\ge 0$. In particular,}
$$
\rank({\cal F}_s)=\rank({\cal F}^s).
$$
\medskip
{\it Proof.} Applying the exterior derivative $d$ to the structure equation
(22) and using a result from exterior algebra known as Cartan's lemma$^4$,
we find
$$
\nabla^i(\Omega_j)=\nabla^2_1(\Omega_1)=\nabla^1_2(\Omega_2)=0,
$$
$$
         \nabla^1_1(\Omega_1)=2\Omega_1,\;\nabla^1_2(\Omega_1)=\Omega_2,\;
         \nabla^2_2(\Omega_1)=\Omega_1,
$$
$$
         \nabla^2_2(\Omega_2)=2\Omega_2,\;\nabla^2_1(\Omega_2)=\Omega_1,\;
         \nabla^1_1(\Omega_2)=\Omega_2.
$$
This proves the statement of the lemma in the case $s=0$. To prove the lemma
in general, we proceed by induction using the commutation relations
$$
\vcenter{\openup2\jot\halign{\indent#\hfil&\quad\quad\qquad#\hfil\cr
$[\nabla^1,\nabla_1]=-2\nabla^1_1-\nabla^2_2$&
$[\nabla^1,\nabla_2]=-\nabla^1_2$\cr
$[\nabla^2,\nabla_1]=-\nabla^2_1$&
$[\nabla^2,\nabla_2]=-\nabla^1_1-2\nabla^2_2$\cr
$[\nabla^1_1,\nabla_1]=\nabla_1$&
$[\nabla^1_1,\nabla_2]=0$\cr
$[\nabla^2_2,\nabla_1]=0$&
$[\nabla^2_2,\nabla_2]=\nabla_2$\cr
$[\nabla^1_2,\nabla_1]=\nabla_2$&
$[\nabla^1_2,\nabla_2]=0$\cr
$[\nabla^2_1,\nabla_1]=0$&
$[\nabla^2_1,\nabla_2]=\nabla_1$\cr}}
$$
which are nothing else but the same structure equations (22)
written in terms of covariant derivatives.
\medskip
{\bf Corollary.} {\it Theorem {\sl 3} will remain valid, if we replace
(everywhere) the families $\;{\cal F}_s\;$ by the families $\;{\cal F}^s$.}
\medskip
{\bf Lemma 2.} {\it If $\;\Omega\not=0,\;$ then the functions
$$
x_1=\Omega_1,\quad x_2=\Omega_2,\quad
y_1=\nabla_1(\Omega_1),\quad y_2=\nabla_2(\Omega_2)                \eqno(27)
$$
are functionally independent. In particular,}
$$
\rank({\cal F}^1)\ge 4.
$$
\medskip
{\it Proof.} Follows from the relation
$$
\displaystyle\pmatrix{
\displaystyle{\partial y_1 \over \partial a_1}&
\displaystyle{\partial y_1 \over \partial a_2}\cr
\displaystyle{\partial y_2 \over \partial a_1}&
\displaystyle{\partial y_2 \over \partial a_2}\cr}=-5
\pmatrix{x_1&0\cr
         0&x_2\cr},
$$
which one can readily verify using the explicit formula
$$
\nabla_i=X_i-(a^p_qw_p(X_i)+a_ia_q){\partial \over \partial a_q}
        -(a^p_qw^r_p(X_i)+a_qa^r_i+a_ia^r_q){\partial \over \partial a^r_q},
$$
where $X_i=a^1_i\displaystyle{\partial \over \partial x}
          +a^2_i\displaystyle{\partial \over \partial y}$.
\bigskip
{\bf Lemma 3.} {\it If $\;\Omega\not=0,\;$ then
$$
\rank(\omega)\ge 5,                                                  \eqno(28)
$$
i.e. the dimension of the symmetry group is not greater than three.}
\medskip
{\it Proof.} Lemmas 1 and 2 imply that
$\rank({\cal F}_1)=\rank({\cal F}^1)\ge 4$. The equality is attained at the
equation
$$
y''=y^2.                                                            \eqno(29)
$$
Hence we need to look at the family ${\cal F}^2$. Let us denote
$$
\Omega'_{i,j}=\nabla_j(\Omega_i),\quad
\Omega''_{i,jk}=\nabla_k\nabla_j(\Omega_i)
$$
and show that
$$
\rank({\cal F}^2)\ge 5.
$$
\medskip\noindent
Suppose, on the contrary, that $\rank({\cal F}^2)<5$, {\it i.e.}
$$
\Omega'_{i,j}=\Sigma_{i,j}(x_1,x_2,y_1,y_2),\quad
\Omega''_{i,jk}=\Sigma_{i,jk}(x_1,x_2,y_1,y_2).
$$
Then we immediately find
$$
\vcenter{\openup2\jot
\halign{
$\hfil#$&${}#\hfil$&
\qquad$\hfil#$&${}#\hfil$\cr
\Sigma_{i,j} &=\displaystyle{1\over 5}\displaystyle{y_ix_j\over x_i}+
               \displaystyle{4\over 5}\displaystyle{x_iy_j\over x_j},&
\Sigma_{i,ii}&=\displaystyle{6\over 5}\displaystyle{y^2_i\over x_i},    \cr
\Sigma_{i,ij}&=\displaystyle{1\over 5}\displaystyle{y^2_ix_j\over x^2_i}
              +\displaystyle{y_iy_j\over x_j},&
\Sigma_{i,jj}&=\displaystyle{4\over 5}\displaystyle{x_iy^2_j\over x^2_j}
              +\displaystyle{2\over 5}{y_iy_j\over x_i}.                \cr}}
$$
Indeed, the sought functions $\Sigma_{i,j}$ must satisfy the
generalized {\it Bianchi identities}
$$
\vcenter{\openup2\jot
\halign{
$\hfil#$&${}#\hfil$&
\qquad$\hfil#$&${}#\hfil$&
\qquad$\hfil#$&${}#\hfil$\cr
\nabla^1(\Omega'_{i,j})&=-(\delta^1_i+4\delta^2_i)x_2,
 &\nabla^1_1(\Omega'_{i,j})&=2\Omega'_{i,j},
 &\nabla^1_2(\Omega'_{i,j})&=y_2,\cr
\nabla^2(\Omega'_{i,j})&=-(\delta^1_j+4\delta^2_j)x_1,
 &\nabla^2_2(\Omega'_{i,j})&=2\Omega'_{i,j},
 &\nabla^2_1(\Omega'_{i,j})&=y_1,\cr}}
$$
which can be derived from the commutation relations of Lemma~1. We integrate
these identities and determine the functions $\Sigma_{i,j}$ explicitly,
using the fact that the differential operators
$\nabla^i$ and $\nabla^i_j$ take the following form
$$
\eqalign{\nabla^i  &=-5x_i{\partial \over\partial y_i},\cr
         \nabla^1_1&=2x_1{\partial \over\partial x_1}
                    +x_2{\partial \over\partial x_2}
                    +3y_1{\partial \over\partial y_1}
                    +y_2{\partial \over\partial y_2},  \cr
         \nabla^2_2&=2x_2{\partial \over\partial x_2}
                    +x_1{\partial \over\partial x_1}
                    +3y_2{\partial \over\partial y_2}
                    +y_1{\partial \over\partial y_1},  \cr
         \nabla^i_j&=x_j{\partial \over\partial x_i}
                    +(\Omega'_{1,2}+\Omega'_{2,1})
                     {\partial \over\partial y_i}      \cr}
$$
within the coordinate system (27).
The sought functions $\Sigma_{i,jk}$ must satisfy the similar ``Bianchi
identities'' which also can be derived from the commutation relations of
Lemma~1. Therefore, these functions can be found explicitly too.

Using now the commutation relation
$$
[\nabla_2,\nabla_1]=\Omega_1\nabla^1+\Omega_2\nabla^2,
$$
we derive two more ``Bianchi identities''
$$
\nabla_j(\Omega''_{i,ii})-\nabla_i(\Omega''_{i,ij})=(-1)^i5x^2_i.\eqno(30)
$$
Substituting the functions $\Sigma_{i,j}$ and $\Sigma_{i,jk}$ into Eq.~30 and
taking into account the fact that the differential operators $\nabla_1$ and
$\nabla_2$ take the form
$$
\displaylines{
\nabla_1=y_1{\partial \over\partial x_1}
        +\Omega'_{2,1}{\partial \over\partial x_2}
        +\Omega''_{1,11}{\partial \over\partial y_1}
        +\Omega''_{2,21}{\partial \over\partial y_2},\cr
\nabla_2=y_2{\partial \over\partial x_2}
        +\Omega'_{1,2}{\partial \over\partial x_1}
        +\Omega''_{2,22}{\partial \over\partial y_2}
        +\Omega''_{1,12}{\partial \over\partial y_1} \cr}
$$
within the coordinate system (27),
we see that the left-hand side of Eq.~30 equals zero. Hence the assumption
$\rank({\cal F}^2)<5$ contradicts the condition $\Omega\not= 0$, and the lemma
is proven.
\bigskip
Note that the equality in (28) is attained at the differential equation of
quadrics (2). The symmetry group of Eq.~2 is isomorphic to $SL(2,{\bf R})$.
It consists of linear unimodular transformations of the plane $\P$.

Set
$$
\Omega_{i,i}=\Omega'_{i,i},\quad
\Omega_{i,j}=\Omega''_{i,j}-\Sigma_{i,j},\quad 
\Omega_{i,jk}=\Omega''_{i,jk}-\Sigma_{i,jk},
$$
where
$\Omega'_{i,j},\;\Omega''_{i,jk},\;\Sigma_{i,j},\;\Sigma_{i,jk}$
are defined in the proof of Lemma 3.
\bigskip
{\bf Theorem.} {\it The following conditions are equivalent:

\item{{\rm (i)}} Eq.~8 admits a three-dimensional group of symmetries,

\item{{\rm (ii)}} Eq.~8 is locally equivalent to Eq.~2,

\item{{\rm (iii)}} the projective curvature of Eq.~8 satisfies the relations}
$$
\eqalignno{
\Omega_{1,2}+\Omega_{2,1}&=0,&(31)\cr
3\Omega_{i,ii}\Omega_{1,2}+5\Omega^3_i&=0,\cr
\Omega_{i,ij}\Omega_i+\Omega_{j,i}\Omega_{i,i}-\Omega_{i,ii}\Omega_j&=0,\cr
\Omega_{i,jj}\Omega_j+2\Omega_{j,i}\Omega_{j,j}-\Omega_{j,jj}\Omega_i&=0.\cr}
$$
\medskip
{\bf Remark.} Relaton (31) is satisfied identically for an equation of the 
form
$$
y''=A(x,y).                                                          \eqno(32)
$$
This means that relation (31) is a necessary condition for a cubic
differential equation to be locally equivalent to Eq.~32. Relation (31) is
also a necessary and sufficient condition for a normal projective connection
to coincide with the induced connection on a surface in three dimensional
projective space$^{11}$.
\medskip
{\it Proof.} Suppose Eq.~8 admits a three-dimensional group of symmetries.
Then Lemmas~1--3 give $\;\rank({\cal F}^1)\ge 4,\;\rank({\cal F}^2)=5$. Using
the technique of Lemma 1, one can readily verify that the equality
$\rank({\cal F}^1)=4$ contradicts the condition (i). For example, in the case 
of Eq.~29 we get
$\>\rank({\cal F}^1)=4,\>\rank({\cal F}^2)=5,\>\rank({\cal F}^3)=6.$
Therefore, the condition (i) is equivalent to the constraint
$$
\rank({\cal F}^1)=\rank({\cal F}^2)=5.
$$
In other words, the covariant derivatives $\Omega_{1,2},\;\Omega_{i,jk}$ are
functions of $x_1,x_2,y_1,y_2$ and $z=\Omega_{2,1}$. Using the
fact that these functions are subjected to the generalized Bianchi identities,
we determine them explicitly and, thereby, obtain the condition (iii) which
defines (Theorem 3) a unique differential equation up to local
equivalence. This differential equation coincides with Eq.~2.

\vskip 10mm
\leftline{\bf Acknowledgements}
\vskip 5mm\nobreak

It is a pleasure to thank V.V.~Lychagin for many valuable discussions on this
research. Also, I thank S.V.~Duzhin for having the time to read the 
manuscript, and for many helpful criticisms and corrections.

\vskip 10mm
\leftline{\bf References}
\vskip 5mm\nobreak
{\parindent=18pt
\frenchspacing

\item{1.} E.~Cartan, {\it Le\c cons sur la Th\'eorie des Espaces \`a Connexion
Projective}, (Gauthier-Villars, Paris, 1937).

\item{2.} S.~Kobayashi and T.~Nagano, On Projective Connections,
{\it J. Math. Mech.} {\bf 13} (1964) 215-235.

\item{3.} T.~Morimoto, La g\'eom\'etrie des \'equations de Monge-Amp\`ere,
{\it C.~R.~Acad.~Sc. Paris (S\'er.~A)} {\bf 289} (1979) 25-28.

\item{4.} S.~Sternberg, {\it Lectures on Differential Geometry},
(Prentice-Hall, Englwood Cliffs, NJ, 1964).

\item{5.} S.V.~Duzhin and V.V.~Lychagin, Symmetries of Distributions and
Quadrature of Ordinary Differential Equations,
{\it Acta Appl. Math.} {\bf 24} (1991) 29-57.

\item{6.} A.G.~Khovansky, On Rectification of Circles,
{\it Siberian Mathematical Journal} {\bf XXI} (1980) 221-226 [Russian].

\item{7.} E.~Cartan, Sur les Vari\'et\'es \`a Connexion Projective,
{\it Bull. Soc. Math. France} {\bf 52} (1924) 205-241.

\item{8.} S.~Kobayashi, Canonical Forms on Frame Bundles of Higher Order
Contact, {\it Proc. Symp. Pure Math.} {\bf III} (1961) 186-193.

\item{9.} C.~Grissom, G.~Thompson, and G.~Wilkens, Linearization of Second
Order Ordinafy Differential Equations via Cartan's Equivalence Method,
{\it J. Differential Equations} {\bf 77} (1989) 1-15.

\item{10.} L.~Hsu and N.~Kamran, Classification of Second Order Ordinary
Differential Equations Admitting Lie Groups of Fibre-Preserving Point
Symmetries, {\it Proc. London Math. Soc.} {\bf 58} (1989) 387-416.

\item{11.} V.~Prokofiev, Embedding of Two-Dimensional Spaces of Normal
Projective Connection in Three-Dimensional Projective Space,
{\it Dokl. Acad. Nauk SSSR} {\bf XXXVI} (1942) 95-97 [Russian].

}
\bye