# Chains of path geometries on surfaces:
theory and examples

Gil Bor111 CIMAT, A.P. 402, Guanajuato, Gto. 36000, Mexico<EMAIL_ADDRESS>Travis Willse222 Guilford College, 5800 W Friendly Ave, Greensboro, NC 27410,
USA<EMAIL_ADDRESS>
###### Abstract

We derive the equations of chains for path geometries on surfaces by solving
the equivalence problem of a related structure: sub-Riemannian geometry of
signature $(1,1)$ on a contact 3-manifold. This approach is significantly
simpler than the standard method of solving the full equivalence problem for
path geometry. We then use these equations to give a characterization of
projective path geometries in terms of their chains (the chains projected to
the surface coincide with the paths) and study the chains of four examples of
homogeneous path geometries. In one of these examples (horocycles in the
hyperbolic planes) the projected chains are bicircular quartics.

###### Contents

  1. 1 Introduction
    1. 1.1 A quick reminder about path geometries on surfaces
    2. 1.2 An abstract reformulation of path geometry
    3. 1.3 Chains of path geometries via the Fefferman metric
    4. 1.4 Contents of the article
  2. 2 The Fefferman metric for CR 3-manifolds (revisited)
    1. 2.1  Example: left-invariant CR structures on $\mathrm{SU}_{2}$
  3. 3 The Fefferman metric for path geometries
    1. 3.1 Chains of $y^{\prime\prime}=f(x,y,y^{\prime})$
    2. 3.2 Chains of projective path geometries
  4. 4 Examples of path geometries and their chains
    1. 4.1  The flat path geometry
      1. 4.1.1 First proof of Proposition 4.1: via the Heisenberg group
      2. 4.1.2 Second proof of Proposition 4.1: via the Euclidean group
    2. 4.2 Circles of fixed radius
    3. 4.3 Hooke ellipses of fixed area
    4. 4.4 Horocycles in the hyperbolic plane

##  1 Introduction

###  1.1 A quick reminder about path geometries on surfaces

A path geometry on a surface consists of a surface $\Sigma$ (a 2-dimensional
differentiable manifold) together with a non-degenerate 2-parameter family of
unparametrized curves in $\Sigma.$333This definition will be reformulated
below more abstractly and precisely; in particular, the non-degeneracy
condition will be spelled out.An equivalence of path geometries on two
surfaces is a diffeomorphism of the surfaces which maps the paths of one
surface onto those of the other. A symmetry of a path geometry on a surface is
a self-equivalence.

The basic example is $\Sigma=\mathbb{R}P^{2}$ (the $2$-dimensional real
projective plane) equipped with the family of straight lines in it. A path
geometry444We shall henceforth usually drop the qualifier “on a surface” since
that is the only situation this article considers. which is locally equivalent
to this example is called flat. A less obvious flat example is given by all
parabolas whose focus is at the origin (‘Kepler parabolas’; here
$\Sigma:=\mathbb{R}^{2}\setminus\\{0\\}$). It is doubly covered by straight
lines via the (complex) quadratic map $z\mapsto z^{2}$.

Figure 1:  A gallery of 2D path geometries: (a) Kepler ellipses of fixed major
axis. (b) Kepler parabolas. (c) Straight lines. (d) Circles of fixed radius.
(e) Hooke ellipses of fixed area. (f) Kepler ellipses of fixed minor axis. (g)
Kepler ellipses tangent to a fixed Kepler ellipse. (h) Circles tangent to a
fixed circle (horocycles). Can you find the equivalent and dual geometries?
(Answer: $a=e=f,b=c=g$ (flat); $a^{*}=h,b^{*}=b,d^{*}=d.$)

Every path geometry is given locally by the graphs of solutions of a second-
order ODE $y^{\prime\prime}=f(x,y,y^{\prime})$. Conversely, a path geometry
determines the ODE up to so-called point transformations, that is, changes of
coordinate $(x,y)\mapsto(\tilde{x},\tilde{y})$. The flat example of straight
lines in $\mathbb{R}P^{2}$ corresponds to $y^{\prime\prime}=0$. A path
geometry is projective if its paths are the (unparametrized) geodesics of a
torsion-free affine connection on $\Sigma$. Such path geometries correspond to
ODEs $y^{\prime\prime}=f(x,y,y^{\prime})$ where $f$ is at most cubic in
$y^{\prime}$. Note that, somewhat surprisingly, this condition is independent
of the coordinates $x,y$ used on $\Sigma.$ Thus a ‘generic’ path geometry is
not projective, and in particular, non-flat. A non-projective example is the
path geometry in $\mathbb{R}^{2}$ whose paths are all circles of a fixed
radius.

A path geometry on a surface $\Sigma$ defines a dual path geometry on the path
space $\Sigma^{*}$, whose paths are parametrized by $\Sigma$: for each point
$x\in\Sigma$ the corresponding path in $\Sigma^{*}$ consists of all paths in
$\Sigma$ passing through $x$. Clearly, the dual of a flat path geometry is
flat as well, an example of a self-dual path geometry. The path geometry of
circles of fixed radius in $\mathbb{R}^{2}$ is an example of a self-dual non-
projective path geometry. A projective path geometry is flat if and only if
its dual is projective as well.

A flat path geometry admits an 8-dimensional (local) group of symmetries (the
projective group $\mathrm{PSL}_{3}(\mathbb{R})$). Conversely, a path geometry
admitting an 8-dimensional local group of symmetries is necessarily flat (a
theorem of Sophus Lie). The sub-maximal symmetry dimension, i.e. the maximum
dimension of the local symmetry group of a non-flat path geometry, is 3. The
path geometry of circles with a fixed radius is sub-maximal. Its symmetry
group is the Euclidean group. Another sub-maximal example is given by central
ellipses (‘Hooke ellipses’) of fixed area. The symmetry group
${\mathrm{SL}_{2}(\mathbb{R})}$, acting by its standard linear action on
$\mathbb{R}^{2}$ (here $\Sigma=\mathbb{R}^{2}\setminus\\{0\\}$). In contrast
to the previous example of circles with fixed radius, this example is
projective and non–self-dual: its dual is the hyperbolic plane with the set of
horocycles (in the Poincaré disk or upper half-plane model horocycles are
precisely the circles tangent to the boundary). The set of Kepler conics of
fixed major or minor axis (either hyperbolas or ellipses, with one of their
foci at the origin) defines an interesting path geometry which is locally
equivalent to that of Kepler ellipses of fixed area, see [4].

The subject was studied extensively in the second half of the 19th century by
Roger Liouville (a relative of the more famous Joseph Liouville), Sophus Lie
and his student Arthur Tresse, who produced a local classification, over the
complex numbers, of sub-maximal path geometries (i.e. those admitting a
3-dimensional group of symmetries) [24]. This classification has been since
refined over the real numbers [13]. The only non-flat projective items on the
list is the above mentioned case of central ellipses of fixed area
(equivalently, Kepler ellipses of fixed major or minor axis) and central
hyperbolas of fixed discriminant (equivalently, Kepler hyperbolas of fixed
minor axis; see Table 2 in the Appendix of [3]).

###  1.2 An abstract reformulation of path geometry

We describe here briefly a more abstract and rigorous reformulation of path
geometries on surfaces, useful also for introducing chains. For further
details we recommend V. I. Arnol’d’s book [1, Chapter 1, Section 6].

Given a surface $\Sigma$, let $\mathbb{P}T\Sigma$ be the ($3$-dimensional)
total space of its projectivized tangent bundle. That is, a point in
$\mathbb{P}T\Sigma$ corresponds to a point in $\Sigma$ together with a tangent
line at the point (a 1-dimensional linear subspace of the tangent space at the
point). There is a standard contact distribution $D$ on $\mathbb{P}T\Sigma$,
given by the ‘skating’ condition: “the point moves along the line”, or “the
line rotates about the point.” The fibers of the base point projection
$\mathbb{P}T\Sigma\to\Sigma$ are integral curves of $D$. Their tangents form
the vertical line field $L_{1}\subset D$. A path $\gamma\subset\Sigma$ is
lifted to $\mathbb{P}T\Sigma$ by mapping a point on $\gamma$ to the tangent
line to $\gamma$ at this point. The lifted curve is clearly an integral curve
of $D$, as it satisfies the skating condition. The non-degeneracy assumption
on a path geometry on $\Sigma$ is that the lifted paths form a smooth
1-dimensional foliation of $\mathbb{P}T\Sigma$, transverse to $L_{1}$ (in
$D$); equivalently, the tangent lines to the lifted curves form a smooth line
field $L_{2}\subset D$, complementary to $L_{1}$, so that $D=L_{1}\oplus
L_{2}$.

We thus arrive at an abstract reformulation of a path geometry:

######  Definition 1.1.

A (2-dimensional) path geometry is a smooth 3-manifold $M$ together with an
(ordered) pair of smooth line fields $L_{1},L_{2}\subset TM$, spanning a
contact distribution $D=L_{1}\oplus L_{2}$. The path geometry dual to
$(M,L_{1},L_{2})$ is $(M,L_{2},L_{1})$.

######  Remark 1.2.

Another common name for $(M,L_{1},L_{2})$ is a para-CR structure, due to the
formal similarity with a (Levi–non-degenerate) CR structure $(M,D,J)$. The
latter is a contact distribution $D$ on a 3-manifold $M$ together with a
complex structure $J\in\operatorname{End}(D)$, i.e.
$J^{2}=-\operatorname{id}_{D}$; equivalently, it is a splitting
$D\otimes\mathbb{C}=D^{1,0}\oplus D^{0,1}$, the direct sum of a conjugate pair
of complex line bundles (the $\pm i$-eigenbundles of $J\otimes\mathbb{C}$).

In the real-analytic setting, CR and para-CR structures have a common
complexification: a complex 3-manifold together with a pair of (complex) line
fields spanning a (complex) contact distribution.

######  Remark 1.3.

Some authors define a path geometry as a 2-parameter family of curves on a
surface $\Sigma$, a unique curve through any given point of $\Sigma$ in any
given direction (see, e.g. the first paragraph of [14], or the “fancy
formulation” of Section 8.6 of [18]). Definition 1.1 is more precise and
general: first, the surface $\Sigma$ is recovered from $(M,L_{1},L_{2})$ as
the space of integral curves of $L_{1}$, which may exist as a smooth surface
only locally. Second, even if $\Sigma$ exists, the set of directions at a
given $x\in\Sigma$ for which a curve exists may be only an open subset in
$\mathbb{P}T_{x}\Sigma$. For example, for the path geometry of central
ellipses in $\mathbb{R}^{2}\setminus\\{0\\}$ a curve exists only in non-radial
directions. Third, there may be more then one curve in a given direction. For
example, for circles of fixed radius in $\mathbb{R}^{2}$, there are two
circles passing through each point in a given direction. This can be remedied
by considering instead oriented circles of fixed radius and the spherized
tangent bundle $\mathbb{S}T\mathbb{R}^{2}$ ($T\mathbb{R}^{2}$, with the zero
section removed, mod $\mathbb{R}^{+}$) instead of $\mathbb{P}T\mathbb{R}^{2}$.
An analogous remedy applies to the aforementioned path geometry of horocycles
in the hyperbolic plane.

We shall not dwell here further on these details and refer the interested
reader to Sections 4.2.3 and 4.4.3 of [6], where our notion of a path geometry
on a surface is called both a generalized path geometry and a Lagrangean
contact structure on a $3$-manifold; the two notions differ in higher
dimension.

###  1.3 Chains of path geometries via the Fefferman metric

In the CR case there is a well-known, naturally associated 4-parameter family
of curves on $M$, called chains, one chain for each given point in $M$ in a
given direction transverse to the contact distribution. They are considered
the CR analog of geodesics in Riemannian geometry (see the recent article [12]
for a variational formulation). Chains were introduced by É. Cartan while
solving the equivalence problem of CR geometry [10, 19] and were studied
extensively by many authors, such as Chern-Moser [11] and C. Fefferman [16],
who showed that they arise from a natural construction, considerably simpler
than Cartan’s, nowadays called the Fefferman metric: a conformal Lorentzian
metric, i.e. of signature $(3,1)$, defined on the total space of a certain
circle bundle over $M$. The chains of the CR structure are then the
projections onto $M$ of the null geodesics of the Fefferman metric.

Similarly, to each path geometry $(M,L_{1},L_{2})$ one can associate a natural
4-parameter family of curves on $M$, a unique curve through any given point in
$M$ in any given direction transverse to the contact distribution
$D:=L_{1}\oplus L_{2}$. The study of this natural class of curves is quite
recent. The earliest reference we know of is a 2005 article of A. Čap and V.
Žádník [7] (path geometries on surfaces appear there in Section 2 as
$3$-dimensional Lagrangean contact structures). See also Sections 5.3.7–8 and
5.3.13–14 of [6]. Both references define chains using the associated Cartan
geometry. However, as in CR case, there is a significant shortcut via the
Fefferman metric. This is a conformal metric of signature $(2,2)$ on the total
space of an $\mathbb{R}^{*}$-bundle over $M$, and the chains are the
projections onto $M$ of non-vertical null geodesics of the Fefferman metric.
In this article we explain this construction and use it to give several
concrete examples.

######  Remark 1.4.

As mentioned in Remark 1.3, path geometries on surfaces generalize in higher
dimension to either (generalized) path geometries or Lagrangean contact
geometries. The Fefferman-type construction of a conformal structure described
here generalizes in higher dimensions to Lagrangean contact structures but not
to path geometries.

###  1.4 Contents of the article

In the next section we re-derive, as a warm-up and a reminder, the Fefferman
metric for a CR structure $(M,D,J)$. The construction appeared first in
Fefferman’s article [16] for a CR manifold embedded as a real hypersurface in
a complex manifold, followed by intrinsic constructions, first direct ones in
[15, 21], then more advanced constructions that use the full solution of the
equivalence problem for CR structures (Cartan bundle and connection), such as
[5, 7, 22]. We view instead a CR structure as a conformal class of sub-
Riemannian geometries of contact type, solve the equivalence problem of sub-
Riemannian geometries of contact type following [17]—which is much simpler
than that for CR geometry, use a sub-Riemannian metric on $D$ to define a
Lorentzian metric on $\mathbb{S}D$ (the spherization of $D$), then show that
conformally equivalent sub-Riemannian metrics on $D$ induce conformally
equivalent Lorentzian metrics on $\mathbb{S}D$. In retrospect, our
construction can be regarded as a shorter version of [15, 21], using [17]. It
is still too complicated conceptually for our taste, and the below formula
(24) for the metric appears a bit like magic, but this method is the best we
have so far and is quite easy to work with.

Once the construction of Fefferman metric for CR geometry is understood, we
construct in Section 3 in a similar fashion the Fefferman metric for a path
geometry. As far as we know, our derivation is new, and before this article
the only available construction of the Fefferman metric for path geometry has
been via the solution of the full equivalence problem for such a structure
(see, e.g., [7, 22]), which is considerably more involved than our derivation.

In Section 3.2 we prove the following theorem, apparently new:

######  Theorem 1.

A path geometry on a surface $\Sigma$ is projective if and only if the chains
on $\mathbb{P}T\Sigma$ project to the paths in $\Sigma$.

In the last section we study in some detail the chains of four homogeneous
path geometries mentioned above: straight lines, circles of fixed radius,
central ellipses of fixed area and horocycles in the hyperbolic plane.

##### Acknowledgments.

GB acknowledges support from CONACYT Grant A1-S-4588. TW is grateful for
support and hospitality from CIMAT during an extended visit in the 2019–20
academic year and for support from Guilford College.

##  2 The Fefferman metric for CR 3-manifolds (revisited)

Let $(M,D,J)$ be a CR 3-manifold, i.e. $D\subset TM$ is a contact
2-distribution (that is, $[D,D]=TM$) and $J\in{\rm End}(D)$ satisfies
$J^{2}=-\operatorname{id}_{D}$. Canonically associated to the CR structure is
the circle bundle $\mathbb{S}D\to M$ ($D$ with the zero section removes, mod
$\mathbb{R}^{+}$) and a conformal class of metrics of signature $(3,1)$ on
$\mathbb{S}D$, the Fefferman metric. It depends on the second-order jet of the
CR structure, so is not so easy to see. The fibers of $\mathbb{S}D\to M$ are
null geodesics, and the projections of the non-vertical null geodesics to $M$
are the chains of the CR structure, forming a 4 parameter family of curves on
$M$.

##### The construction.

Fix a positive contact form $\eta^{3}$ on $M$, i.e. a 1-form satisfying

$\displaystyle D={\rm Ker}(\eta^{3}),$ (1)
$\displaystyle\mathrm{d}\eta^{3}(X,JX)>0\mbox{ for every }X\in D,\ X\neq 0.$
(2)

######  Remark 2.1.

A general contact manifold does not admit necessarily a global contact form (a
1-form whose kernel is $D$) but the contact structure of a CR manifold does,
using the orientation of $D$ induced by $J$. If $M$ is connected then any
global contact form is either positive or negative.

Recall that the coframe bundle $\pi:F^{*}\to M$ is the principal
$\mathrm{GL}_{3}(\mathbb{R})$-bundle whose fiber at a point $x\in M$ consists
of all linear isomorphisms $u:T_{x}M\to\mathbb{R}^{3}$. The tautological
1-form on $F^{*}$ is the $\mathbb{R}^{3}$-valued 1-form $\omega$ whose value
at $u\in F^{*}$ is $u\circ(\mathrm{d}\pi)_{u}.$

Now a positive contact form $\eta^{3}$ on $M$ defines a positive-definite
inner product on $D$, $\langle X,Y\rangle:=\mathrm{d}\eta^{3}(X,JY)$. An
adapted coframe is an extension of $\eta^{3}$ to a coframe
$\eta=(\eta^{1},\eta^{2},\eta^{3})^{t}$ (we view elements of $\mathbb{R}^{3}$
as column vectors), satisfying

$\displaystyle\ \mathrm{d}\eta^{3}=\eta^{1}\wedge\eta^{2}$ (3)
$\displaystyle\langle\cdot,\cdot\rangle=\left.\left[(\eta^{1})^{2}+(\eta^{2})^{2}\right]\right|_{D}.$
(4)

It is easy to show that for a fixed $\eta^{3}$ these 2 equations define a
circle’s worth of coframes at each $x\in M$. Thus, let
$S^{1}\subset\mathrm{GL}_{3}(\mathbb{R})$ be the set of matrices of the form

$\left(\begin{array}[]{ccc}\cos\varphi&\sin\varphi&0\\\
\sin\varphi&\cos\varphi&0\\\ 0&0&1\end{array}\right),$

and $B\subset F^{*}$ the set of coframes adapted to $\eta^{3}$. Then $B\to M$
is a principal $S^{1}$-subbundle, an $S^{1}$-reduction of $F^{*}$, whose local
sections consist of adapted coframes.

We continue to denote by $\omega=(\omega^{1},\omega^{2},\omega^{3})^{t}$ the
restriction of the tautological 1-form on $F^{*}$ to $B$. Then there are
unique 1-form $\alpha$ and functions $a_{1},a_{2}$ on $B$ such that

$\displaystyle\mathrm{d}\left(\begin{array}[]{c}\omega^{1}\\\ \omega^{2}\\\
\omega^{3}\end{array}\right)=-\left(\begin{array}[]{ccc}0&\alpha&0\\\
-\alpha&0&0\\\
0&0&0\end{array}\right)\wedge\left(\begin{array}[]{c}\omega^{1}\\\
\omega^{2}\\\
\omega^{3}\end{array}\right)+\left(\begin{array}[]{ccc}a_{1}&a_{2}&0\\\
a_{2}&-a_{1}&0\\\
0&0&1\end{array}\right)\left(\begin{array}[]{c}\omega^{2}\wedge\omega^{3}\\\
\omega^{3}\wedge\omega^{1}\\\ \omega^{1}\wedge\omega^{2}\end{array}\right).$
(20)

(See equation (1) of [17]). Furthermore, there are unique functions
$b_{1},b_{2},K$ on $B$ such that

$\mathrm{d}\alpha=b_{1}\omega^{2}\wedge\omega^{3}+b_{2}\omega^{3}\wedge\omega^{1}+K\omega^{1}\wedge\omega^{2}.$
(21)

(See equation (4) of [17]; in fact, $K$ descends to $M$. Also, $\alpha$ is
essentially the Webster connection form [25], $a_{1},a_{2}$ its torsion, and
$K$ the Webster scalar curvature).

Define a Lorentzian metric on $B$ by

$\mathrm{g}:=\omega^{1}\cdot\omega^{1}+\omega^{2}\cdot\omega^{2}+\omega^{3}\cdot\sigma,$
(22)

where $\sigma$ is a $1$-form, to be determined later, and $\cdot$ is the
symmetric product of 1-forms.

Let $\mathbb{S}D$ be the ‘spherization’ (or ‘ray projectivization’) of $D$,
the quotient of $D$, with the zero section removed, by the dilation action of
$\mathbb{R}^{+}$. There is an obvious $S^{1}$-action on $D$, commuting with
the $\mathbb{R}^{+}$ action, thus making $\mathbb{S}D$ a principal
$S^{1}$-bundle. Note that $\mathbb{S}D$, unlike $B$, is canonically associated
to $(M,D,J)$: to define $B$ we needed to choose the positive contact form
$\eta^{3}$. Define an isomorphism of principal $S^{1}$-bundles

$h:B\to\mathbb{S}D,\quad u\mapsto[u^{-1}{\mathbf{e}}_{1}],$ (23)

where ${\mathbf{e}}_{1}=(1,0,0)^{t}.$ That is, $h(u)=[X]\in\mathbb{S}D,$ where
$X\in D$ is the unique vector in $T_{x}M$, $x=\pi(u)$, satisfying
$u^{1}(X)=1,u^{2}(X)=u^{3}(X)=0$. We then use $h$ to map the Lorentzian metric
on $B$ of equation (22) to a Lorentzian metric on $\mathbb{S}D$. In general,
for arbitrary $\sigma$ in formula (22), the resulting metric on $\mathbb{S}D$
depends on the choice of $\eta^{3}$ in a complicated way, but for a careful
choice of $\sigma$ the conformal class of the Lorentzian metric on
$\mathbb{S}D$ is independent of the choice of $\eta^{3}$.

######  Remark 2.2.

There are other models for the underlying space of the Fefferman metric
instead of $\mathbb{S}D$ (a matter of taste). For example, one can take the
spherization of the dual bundle $D^{\vee}$, in which case the formula for the
identification $B\to\mathbb{S}D^{\vee}$ is a little simpler:
$u\mapsto[u^{1}|_{D}].$ Another model is the spherization of the canonical
bundle $\Lambda^{2,0}D\subset\Lambda^{2}T^{*}D\otimes\mathbb{C}$, as in [21];
the identification with $B$ in this case is
$u\mapsto[u^{3}\wedge(u^{1}+iu^{2})|_{D}].$ Also, the metric on $\mathbb{S}D$
is invariant under the antipodal map in each fiber (a circle), and so it
descends to the (full) projectivization $\mathbb{P}D$.

######  Theorem 2.

Let $(M,D,J)$ be a CR 3-manifold, $\mathbb{S}D\to M$ the spherization of $D$
and $\eta^{3}$ any positive contact $1$-form, as in equations (1) and (2).
Define a $1$-form $\sigma$ on the total space of the associated circle bundle
$B\to M$,

$\sigma={4\over 3}\alpha-{1\over 3}K\omega^{3},$ (24)

where $\alpha,K$ are defined via equations (20) and (21). Then the conformal
class of the Lorentzian metric induced on $\mathbb{S}D$ by equation (22), via
the isomorphism (23), is independent of the choice of $\eta^{3}$. In fact,
multiplying $\eta^{3}$ by a positive function rescales the induced metric on
$\mathbb{S}D$ by the same factor.

###### Proof.

If $\eta^{3}$ is a positive contact form on $M$, then any other positive
contact form is of the form $\tilde{\eta}^{3}=\lambda^{2}\eta^{3}$, for some
positive function $\lambda:M\to\mathbb{R}^{+}$. Changing $\eta^{3}$ to
$\tilde{\eta}^{3}$ changes $B$ to $\tilde{B}$, another $S^{1}$-reduction of
the coframe bundle of $M$, with corresponding metric $\tilde{\mathrm{g}}$ and
isomorphism $\tilde{h}:\tilde{B}\to S$. We thus need to show that the
composition $f:=\tilde{h}^{-1}\circ h:B\to\tilde{B}$ satisfies
$f^{*}(\tilde{g})=\lambda^{2}g.$

Let us pull-back $\lambda$ to $B$ by the projection $B\to M$, denoting the
result by $\lambda$ as well. Then

$\mathrm{d}\lambda=\lambda_{i}\omega^{i},\quad\mathrm{d}\lambda_{i}=\lambda_{i0}\alpha+\lambda_{ij}\omega^{j},$
(25)

for some functions $\lambda_{i},\lambda_{ij},\lambda_{i0}$ on $B$, $1\leq
i,j\leq 3$. (Note that by definition $\lambda$ descends to $M$; in general the
$\lambda_{i}$ do not, but $\lambda_{3}$ does.)

######  Lemma 2.3.

$\lambda_{10}=-\lambda_{2},\qquad\lambda_{20}=\lambda_{1},\qquad\lambda_{12}-\lambda_{21}=\lambda_{3}.$

###### Proof.

These identities follow immediately from expanding
$\mathrm{d}(\mathrm{d}\lambda)=0.$ $\square$

Now a section $\eta=(\eta^{1},\eta^{2},\eta^{3}):M\to B$ of $B\to M$ is a
coframe adapted to $\eta^{3}$, so $f\circ\eta:M\to\tilde{B}$ is a section of
$\tilde{B}\to M$, a coframe adapted to $\tilde{\eta}^{3}=\lambda^{2}\eta^{3}.$

######  Lemma 2.4.

$f\circ\eta=\Lambda\eta$, where

$\Lambda=\left(\begin{array}[]{ccr}\lambda&0&-2\lambda_{2}\\\
0&\lambda&2\lambda_{1}\\\ 0&0&\lambda^{2}\end{array}\right).$

###### Proof.

It is enough to check that $\tilde{\eta}:=\Lambda\eta$ satisfies equations
(1)-(4) above, with $\tilde{\eta}^{3}=\lambda^{2}\eta^{3}$ instead of
$\eta^{3}$, as well as
$\tilde{\eta}^{1}(\tilde{X})=1,\tilde{\eta}^{2}(\tilde{X})=\tilde{\eta}^{3}(\tilde{X})=0$
for $\tilde{X}=X/\lambda.$ $\square$

######  Lemma 2.5.

$f^{*}\omega_{\tilde{B}}=\Lambda\omega_{B}.$

###### Proof.

Let $\eta\in B$, $\tilde{\eta}=f(\eta)$. By Lemma 2.4,
$\tilde{\eta}=\Lambda\eta,$ hence
$(f^{*}\omega_{\tilde{B}})_{\eta}=(\omega_{\tilde{B}})_{\tilde{\eta}}\circ(\mathrm{d}f)_{\eta}=\tilde{\eta}\circ(\mathrm{d}\tilde{\pi})_{\tilde{\eta}}\circ(\mathrm{d}f)_{\eta}=\tilde{\eta}\circ\mathrm{d}(\tilde{\pi}\circ
f)_{\eta}=\tilde{\eta}\circ(\mathrm{d}\pi)_{\eta}=\Lambda\eta\circ\mathrm{d}\pi_{\eta}=\Lambda\omega_{\eta}.$
$\square$

Notation. For sake of readability, we adopt henceforth the following
abbreviated notation:

$\omega:=\omega_{B},\ \mathrm{g}:=\mathrm{g}_{B},\ \ldots,\
\tilde{\omega}:=f^{*}\omega_{\tilde{B}},\
\tilde{\mathrm{g}}:=f^{*}\mathrm{g}_{\tilde{B}},\ \ldots,\ \textrm{etc.}$

Thus, for example, Lemma 2.5 reads $\tilde{\omega}=\Lambda\omega$.

We proceed with the proof of Theorem 2. It is clearly enough to show an
infinitesimal version of the claimed conformal invariance. Suppose
$\lambda=\lambda(t)$ is differentiable and that it satisfies $\lambda(0)=1$.
Denote by a dot the derivative with respect to $t$ at $t=0$ of objects on
$\tilde{B}$ pulled back to $B$ by $f$, e.g.,
$\dot{\lambda}=\lambda^{\prime}(0),\
\dot{\lambda}_{i}=\lambda_{i}^{\prime}(0),\
\dot{\lambda}_{ij}=\lambda_{ij}^{\prime}(0)$,
$\dot{\mathrm{g}}=\left.{\mathrm{d}\over\mathrm{d}t}\right|_{t=0}\tilde{\mathrm{g}},$
etc. Then $\tilde{\mathrm{g}}=\lambda^{2}\mathrm{g}$ if and only if
$\dot{\mathrm{g}}=2\dot{\lambda}\mathrm{g}$ (for all $\eta^{3}$ and
$\lambda(t)$ satisfying $\lambda(0)=1$). Now calculate using the previous
lemmas:

$\displaystyle\dot{\omega}^{1}=\dot{\lambda}\omega^{1}-2\dot{\lambda}_{1}\omega^{3},\
\ \dot{\omega}^{2}=\dot{\lambda}\omega^{2}+2\dot{\lambda}_{2}\omega^{3},\ \
\dot{\omega}^{3}=2\dot{\lambda}\omega^{3},$
$\displaystyle\dot{\mathrm{g}}=2\dot{\lambda}\mathrm{g}+(\dot{\sigma}-4\dot{\lambda}_{2}\omega^{1}+4\dot{\lambda}_{1}\omega^{2})\cdot\omega^{3}.$

Thus $\dot{\mathrm{g}}=2\dot{\lambda}\mathrm{g}$ if and only if

$\dot{\sigma}=4(\dot{\lambda}_{2}\omega^{1}-\dot{\lambda}_{1}\omega^{2}).$
(26)

To calculate $\dot{\sigma}$, using formula (24), we need formulas for
$\dot{\alpha}$ and $\dot{K}.$ To find $\dot{\alpha}$ we find first a formula
for $\tilde{\alpha}.$ Write the structure equations (20) for $\tilde{\omega}$,
substitute $\tilde{\omega}=\Lambda\omega$, and equate coefficients. The result
is

$\tilde{\alpha}=\alpha+3{\lambda_{2}\over\lambda}\omega^{1}-3{\lambda_{1}\over\lambda}\omega^{2}-\left[3{(\lambda_{1})^{2}+(\lambda_{2})^{2}\over\lambda^{2}}+{\lambda_{11}+\lambda_{22}\over\lambda}\right]\omega^{3}.$

Taking derivative with respect to $t$ at $t=0$ of the last formula, we get

$\dot{\alpha}=3\dot{\lambda}_{2}\omega^{1}-3\dot{\lambda}_{1}\omega^{2}-(\dot{\lambda}_{11}+\dot{\lambda}_{22})\omega^{3}.$

To find $\dot{K}$, there is a shortcut, avoiding an explicit formula for
$\tilde{K}$, by noting first that $K$ is defined by $\mathrm{d}\alpha\equiv
K\omega^{1}\wedge\omega^{2}$ (mod $\alpha,\omega^{3}$). Taking $\mathrm{d}$ of
the above formula for $\dot{\alpha}$, we get, using equations (25),
$\mathrm{d}\dot{\alpha}\equiv-4(\dot{\lambda}_{11}+\dot{\lambda}_{22})\omega^{1}\wedge\omega^{2}$
(mod $\alpha,\omega^{3}$). Taking derivative with respect to $t$ of
$\mathrm{d}\tilde{\alpha}\equiv\tilde{K}\tilde{\omega}^{1}\wedge\tilde{\omega}^{2}$
(mod $\tilde{\alpha},\tilde{\omega}^{3}$), we get
$\mathrm{d}\dot{\alpha}\equiv(\dot{K}+2\dot{\lambda}K)\omega^{1}\wedge\omega^{2}$
(mod $\alpha,\omega^{3}$), hence

$\dot{K}=-2\dot{\lambda}K-4(\dot{\lambda}_{11}+\dot{\lambda}_{22}).$

Now let $\tilde{\sigma}=c_{1}\tilde{\alpha}+c_{2}\tilde{K}\tilde{\omega}^{3}$,
with some constants $c_{1},c_{2}$. Then

$\dot{\sigma}=c_{1}\dot{\alpha}+c_{2}(\dot{K}\omega^{3}+K\dot{\omega}^{3})=3c_{1}(\dot{\lambda}_{2}\omega^{1}-\dot{\lambda}_{1}\omega^{2})-(c_{1}+4c_{2})(\dot{\lambda}_{11}+\dot{\lambda}_{22})\omega^{3}.$

Thus equation (26) is satisfied if $c_{1}=4/3,c_{2}=-1/3$. $\square$

###  2.1  Example: left-invariant CR structures on $\mathrm{SU}_{2}$

The left-invariant $\mathfrak{su}_{2}$-valued Maurer–Cartan form on
${\mathrm{SU}_{2}}$ is

$\Theta=g^{-1}\mathrm{d}g=\left(\begin{array}[]{cc}i\theta^{1}&\theta^{2}+i\theta^{3}\\\
-\theta^{2}+i\theta^{3}&-i\theta^{1}\end{array}\right).$ (27)

The Maurer–Cartan equation $\mathrm{d}\Theta=-\Theta\wedge\Theta$ gives

$\mathrm{d}\theta^{1}=-2\theta^{2}\wedge\theta^{3},\qquad\mathrm{d}\theta^{2}=-2\theta^{3}\wedge\theta^{1},\qquad\mathrm{d}\theta^{3}=-2\theta^{1}\wedge\theta^{2}.$
(28)

For each $t\in[1,\infty)$ let

$\eta^{1}=\sqrt{t}\,\theta^{1},\qquad\eta^{2}=\theta^{2}/\sqrt{t},\qquad\eta^{3}=-\theta^{3}/2.$

One can show that every left-invariant CR structure $D^{0,1}\subset
T{\mathrm{SU}_{2}}\otimes\mathbb{C}$ is equivalent (via right translation),
for a unique $t\in[1,\infty)$, to $\\{\eta^{1}+i\eta^{2},\eta^{3}\\}^{\perp}.$
For $t=1$ we obtain the standard ‘spherical’ CR structure on
$\mathrm{SU}_{2}\simeq S^{3}$, for $t>1$ these are non-spherical CR
structures, distinct $t$ determine inequivalent structures (see [3], Prop.
5.1). We use (28) to find

$\mathrm{d}\eta^{1}=4t\,\eta^{2}\wedge\eta^{3},\qquad\mathrm{d}\eta^{2}=(4/t)\eta^{3}\wedge\eta^{1},\qquad\mathrm{d}\eta^{3}=\eta^{1}\wedge\eta^{2}.$

Using this coframe we identify $B\simeq{\mathrm{SU}_{2}}\times S^{1}$ and
$\omega=\bar{u}\cdot\eta$, where $u=e^{i\varphi}.$ Explicitly,

$\omega^{1}=\sqrt{t}(\cos\theta)\theta^{1}+{1\over\sqrt{t}}(\sin\theta)\theta^{2},\
\omega^{2}=-\sqrt{t}(\sin\theta)\theta^{1}+{1\over\sqrt{t}}(\cos\theta)\theta^{2},\
\omega^{3}=-{1\over 2}\theta^{3}.$

Inserting these into equations (20)–(21), we obtain

$\alpha=\theta^{4}-\left(t+{1\over t}\right)\theta^{3},\quad
K=2\left(t+{1\over t}\right),$

where $\theta^{4}:=\mathrm{d}\varphi$. Inserting all this into equations
(22)–(24), we get

$\mathrm{g}=t\left(\theta^{1}\right)^{2}+{1\over
t}\left(\theta^{2}\right)^{2}+{1\over 2}\left(t+{1\over
t}\right)\left(\theta^{3}\right)^{2}-{2\over 3}\theta^{3}\cdot\theta^{4}.$

This is essentially formula (15) of [11]; the coefficient of our
$\theta^{3}\cdot\theta^{4}$ term can be made to agree with the cited formula
by rescaling the $\varphi$ coordinate by a constant. See also [11] for a study
of the chains of this example via null geodesics of the Fefferman metric.

##  3 The Fefferman metric for path geometries

Let $(M,L_{1},L_{2})$ be a path geometry, i,e, $L_{1},L_{2}$ is a pair of line
fields on a 3-manifold $M$, spanning a contact distribution $D:=L_{1}\oplus
L_{2}$. Let us fix a contact form $\eta^{3}$, that is, $D={\rm Ker}(\eta^{3})$
(possibly defined only locally, see Remark 3.1 below). An adapted coframe
(with respect to $\eta^{3}$) is an extension of $\eta^{3}$ to a (local)
coframe $(\eta^{1},\eta^{2},\eta^{3})$ satisfying

$\displaystyle\mathrm{d}\eta^{3}=\eta^{1}\wedge\eta^{2},$ (29)
$\displaystyle\eta^{1}|_{L_{2}}=\eta^{2}|_{L_{1}}=0.$ (30)

These equations define an $\mathbb{R}^{*}$-structure, i.e. an
$\mathbb{R}^{*}$-principal subbundle $B\subset F^{*}$, whose local sections
are the coframes adapted to $\eta^{3}$, where $s\in\mathbb{R}^{*}$ acts by
$(\eta^{1},\eta^{2},\eta^{3})\mapsto(\eta^{1}/s,s\eta^{2},\eta^{3}).$

Let $D^{*}=D\setminus(L_{1}\cup L_{2})\subset D$, with spherization
$\mathbb{S}D^{*}\subset\mathbb{S}D$. The Fefferman metric associated to the
path geometry is a conformal pseudo-Riemannian metric of signature $(2,2)$ on
$\mathbb{S}D^{*}$. We shall define it in a manner similar to the CR case. The
splitting $D=L_{1}\oplus L_{2}$ defines an involution $J\in{\rm End}(D),$
$J^{2}=\mathrm{id},$ by

$J(X_{1}+X_{2})=X_{1}-X_{2},\quad X_{i}\in L_{i},\ i=1,2.$ (31)

The contact form $\eta^{3}$ defines on $D$ an area form,
$\left.\mathrm{d}\eta^{3}\right|_{D}$, and an indefinite metric of signature
$(1,1)$, $\langle X,Y\rangle:=\mathrm{d}\eta^{3}(X,JY).$ Now $D^{*}=D^{+}\cup
D^{-}$, where $D^{\pm}$ are the positive (resp. negative) vectors with respect
to $\langle\cdot,\cdot\rangle$, and corresponding decomposition
$\mathbb{S}D^{*}=\mathbb{S}D^{+}\cup\mathbb{S}D^{-}$. Both $\mathbb{S}D^{\pm}$
are $\mathbb{R}^{*}$-principal bundles over $M$, where $s\in\mathbb{R}^{*}$
acts by $[X_{1}+X_{2}]\mapsto[sX_{1}+X_{2}/s],$ $X_{i}\in L_{i}$. Note that
$D^{\pm}$ are interchanged by $J$ or by taking $-\eta^{3}$ instead of
$\eta^{3}$. There is an identification of $\mathbb{R}^{*}$-principal bundles,

$h:B\to\mathbb{S}D^{+},\ u\mapsto[X],\ \mbox{where}\
u^{1}(X)=u^{2}(X)=1,u^{3}(X)=0.$ (32)

We shall define a pseudo-Riemannian metric of signature $(2,2)$ on $B$, map it
by $h$ to $\mathbb{S}D^{+}$, then by $J$ to $\mathbb{S}D^{-}$. As in the CR
case, we show that the associated conformal class of metrics on
$\mathbb{S}D^{*}$ is independent of the chosen contact form $\eta^{3}$.

######  Remark 3.1.

A general contact 3-manifold is naturally oriented. (Proof: choose a local
contact form $\eta^{3}$, then $\eta^{3}\wedge\mathrm{d}\eta^{3}$ is a volume
form; multiplying $\eta^{3}$ by a non-vanishing $\lambda$ multiplies this
volume form by $\lambda^{2}$, so does not change the associated orientation.)
The Lie bracket of sections of $D$ defines an isomorphism $\Lambda^{2}(D)\to
TM/D$, but these isomorphic line bundles need not be trivial, i.e. there might
not exist on $M$ a global contact form (a non-vanishing section of
$D^{\perp}\simeq(TM/D)^{*}).$ In the CR case, $J$ defines an orientation on
$D\subset TM$, hence of $TM/D$ (since $TM$ is oriented), and a dual
orientation of $D^{\perp}=(TM/D)^{*}$, so there is always a global contact
form. This is not the case for a path geometry (e.g.,
$M=\mathbb{P}T\mathbb{R}^{2}$, equipped with the standard flat path geometry).
But this topological difficulty is minor, we can still define $B$ locally,
then show that the conformal structures defined on $\mathbb{S}D^{*}$
restricted to open subsets of $M$ agree on intersections. We shall not dwell
on the details.

We shall now proceed with the plan outlined above, in the paragraph before
Remark 3.1.

The structure equations for any $\mathbb{R}^{*}$-connection form $\alpha$ on
$B\to M$ are

$\displaystyle\mathrm{d}\left(\begin{array}[]{c}\omega^{1}\\\ \omega^{2}\\\
\omega^{3}\end{array}\right)=-\left(\begin{array}[]{ccc}\alpha&0&0\\\
0&-\alpha&0\\\
0&0&0\end{array}\right)\wedge\left(\begin{array}[]{c}\omega^{1}\\\
\omega^{2}\\\
\omega^{3}\end{array}\right)+\left(\begin{array}[]{ccc}T^{1}_{23}&T^{1}_{31}&T^{1}_{12}\\\
T^{2}_{23}&T^{2}_{31}&T^{2}_{12}\\\
0&0&1\end{array}\right)\left(\begin{array}[]{c}\omega^{2}\wedge\omega^{3}\\\
\omega^{3}\wedge\omega^{1}\\\ \omega^{1}\wedge\omega^{2}\end{array}\right),$

where $T^{i}_{jk}$ are some real functions on $B$ (the coefficients of the
torsion tensor of the connection). Starting from any such connection it is
easy to show that it can be modified, in a unique way, by adding to $\alpha$
multiples of the $\omega^{i}$, so as to render
$T^{1}_{31}=T^{1}_{12}=T^{2}_{12}=0$ (in fact doing so also solves the
equivalence problem for path geometry equipped with a fixed contact form).
Taking the exterior derivative of
$\mathrm{d}\omega^{3}=\omega^{1}\wedge\omega^{2}$ shows that $T^{2}_{23}=0$ as
well. The structure equations now become

$\displaystyle\begin{split}\mathrm{d}\omega^{1}&=-\alpha\wedge\omega^{1}+a_{1}\omega^{2}\wedge\omega^{3}\\\
\mathrm{d}\omega^{2}&=\phantom{-}\alpha\wedge\omega^{2}+a_{2}\omega^{3}\wedge\omega^{1}\\\
\mathrm{d}\omega^{3}&=\phantom{-}\omega^{1}\wedge\omega^{2}\end{split}$ (33)

for some functions $a_{1},a_{2}$ on $B$. Taking exterior derivative of these
equations we get

$\displaystyle\mathrm{d}\alpha=b_{1}\omega^{2}\wedge\omega^{3}+b_{2}\omega^{3}\wedge\omega^{1}+K\omega^{1}\wedge\omega^{2}$
(34)

for some functions $b_{1},b_{2},K$ on $B$ (i.e. $\mathrm{d}\alpha$ is semi-
basic, containing no $\alpha\wedge\omega^{i}$ terms.)

######  Theorem 3.

Let $(M,L_{1},L_{2})$ be a path geometry and
$\mathbb{S}D^{*}\subset\mathbb{S}D$ the set of rays in $D=L_{1}\oplus L_{2}$
not contained in $L_{1}\cup L_{2}$. Then there is a canonically associated
conformal class of metrics of signature $(2,2)$ on $\mathbb{S}D^{*}$, called
the Fefferman metric, defined as follows. Associated with each contact 1-form
$\eta^{3}$ on $M$ is an $\mathbb{R}^{*}$-reduction $B\to M$ of the coframe
bundle of $M$, given by equations (29)-(30), a unique
$\mathbb{R}^{*}$-connection form $\alpha$ on $B$ satisfying equations (33) and
the 1-form

$\sigma:=-{2\over 3}\alpha+{1\over 6}K\omega^{3},$ (35)

where $K$ is defined via equations (34), and where
$\omega^{1},\omega^{2},\omega^{3}$ are the tautological 1-forms on the coframe
bundle of $M$ restricted to $B$. Then

$\mathrm{g}:=\omega^{1}\cdot\omega^{2}+\omega^{3}\cdot\sigma$ (36)

is a pseudo-Riemannian metric on $B$ of signature $(2,2)$. There is also
associated with $\eta^{3}$ a decomposition
$\mathbb{S}D^{*}=\mathbb{S}D^{+}\cup\,\mathbb{S}D^{-}$ and
$\mathbb{R}^{*}$-isomorphisms $h:B\to\mathbb{S}D^{+}$, $J\circ h:B\to D^{-}$,
where $h$ is given by equation (32) and $J$ by equation (31), such that the
conformal class of the induced metric on $\mathbb{S}D^{*}$ is independent of
the choice of $\eta^{3}$; in fact, multiplying $\eta^{3}$ by a smooth non-
vanishing function rescales the induced metric on $\mathbb{S}D^{*}$ by the
same factor.

###### Proof.

The proof is very similar to the CR case. Here are the formulas that differ:

$\displaystyle\lambda_{10}$
$\displaystyle=\lambda_{1},\qquad\lambda_{20}=-\lambda_{2},\qquad\lambda_{12}-\lambda_{21}=\lambda_{3},$
$\displaystyle\tilde{\alpha}$
$\displaystyle=\alpha+{3\lambda_{1}\over\lambda}\omega^{1}-{3\lambda_{2}\over\lambda}\omega^{2}-\left({\lambda_{12}+\lambda_{21}\over\lambda}+{6\lambda_{1}\lambda_{2}\over\lambda^{2}}\right)\omega^{3},$
$\displaystyle\dot{\alpha}$
$\displaystyle=3\dot{\lambda}_{1}\omega^{1}-3\dot{\lambda}_{2}\omega^{2}-\left(\dot{\lambda}_{12}+\dot{\lambda}_{21}\right)\omega^{3},$
$\displaystyle\mathrm{d}\dot{\alpha}$
$\displaystyle\equiv-4\left(\dot{\lambda}_{12}+\dot{\lambda}_{21}\right)\omega^{1}\wedge\omega^{2}\equiv\left(\dot{K}+2\dot{\lambda}K\right)\omega^{1}\wedge\omega^{2}\
\pmod{\alpha,\omega^{3}},$ $\displaystyle\dot{K}$
$\displaystyle=-2\dot{\lambda}K-4\left(\dot{\lambda}_{12}+\dot{\lambda}_{21}\right),$
$\displaystyle\dot{\mathrm{g}}$
$\displaystyle=2\dot{\lambda}\mathrm{g}+(\dot{\sigma}+2\dot{\lambda}_{1}\omega^{1}-2\dot{\lambda}_{2}\omega^{2}).$

$\square$

######  Definition 3.2.

A chain of a path geometry $(M,L_{1},L_{2})$ is the projection to $M$ of an
unparametrized non-vertical null geodesic of the associated Fefferman
conformal metric on $\mathbb{S}D^{*}$.

######  Proposition 3.3.

  1. 1.

The $\mathbb{R}^{*}$-action on $\mathbb{S}D^{*}$ is by conformal isometries.

  2. 2.

For every point in $M$, in every given direction transverse to $D$, there is a
unique chain passing through this point in the given direction.

  3. 3.

The fibers of $\mathbb{S}D^{*}\to M$ are null geodesics and project to
constant curves on $M$.

###### Proof.

(1) For every contact form $\eta^{3}$, the map $h:B\to\mathbb{S}D^{+}$ (by
definition, a conformal isometry) is $\mathbb{R}^{*}$-equivariant, hence it is
enough to verify that the pseudo-Riemannian metric on $B$ given by equations
(35)–(36) is $\mathbb{R}^{*}$-invariant. This follows from the
$\mathbb{R}^{*}$-invariance of $\alpha,\omega^{3},K$ and the
$\mathbb{R}^{*}$-equivariance $R_{s}^{*}\omega^{1}=\omega^{1}/s,$
$R_{s}^{*}\omega^{2}=s\omega^{2}$.

(2) Let $x\in M$ and $v\in T_{x}M,$ $v\not\in D_{x}$. Pick a contact form
$\eta^{3}$ and work on the associated bundle $B$. The fiber $B_{x}$ consists
of coframes $u=(u^{1},u^{2},u^{3})$ on $T_{x}M$ adapted to $\eta^{3}$, as in
equations (29)–(30). We show that for every $u\in B_{x}$ there is a unique
lift $\tilde{v}\in T_{u}B$ of $v$ which is null. Now $\tilde{v}$ is a lift of
$v$ if and only if $\omega^{i}(\tilde{v})=u^{i}(v)$, $i=1,2,3$. It remains to
determine $\sigma(\tilde{v})$. Now $\omega^{3}(\tilde{v})=u^{3}(v)\neq 0$ and,
by formula (36), $\tilde{v}$ is null if and only if
$\omega^{1}(\tilde{v})\omega^{2}(\tilde{v})+\omega^{3}(\tilde{v})\sigma(\tilde{v})=0,$
i.e. $\sigma(\tilde{v})=-u^{1}(v)u^{2}(v)/u^{3}(v).$ This shows that $v\in
T_{x}M$ has a unique null lift at $u\in B_{x}.$ The null geodesic through $u$
in the direction of $\tilde{v}$ projects to a chain through $x$ in the
direction of $v$. This proves existence of the required chain. As for
uniqueness, we need to show that if we repeat the above at another point of
$B_{x}$, say $s\cdot u\in B_{x}$, we obtain the same chain. We use the fact
that $s$ acts on $B$ by isometries $R_{s}$, mapping $\tilde{v}$ to the unique
null-lift of $v$ at $s\cdot u$, and the null geodesic through $u$ tangent to
$\tilde{v}$ to the null geodesic through $s\cdot u$ in the direction of
$(R_{s})_{*}\tilde{v}$. Since $R_{s}$ commutes with the projection $B\to M$,
the two null geodesics project to the same chain in $M$.

(3) The vertical distribution of $B\to M$ is given by
$\omega^{1}=\omega^{2}=\omega^{3}=0$, thus
$\mathrm{g}=\omega^{1}\cdot\omega^{2}+\omega^{3}\cdot\sigma$ restricted to the
fibers vanishes, so these fibers are null curves. We proceed to show that they
are null geodesics.

As shown in part (1) above, the principal $\mathbb{R}^{*}$-action on $B$ is
isometric. Let $\zeta$ denote an infinitesimal generator of this action (i.e.
a nonzero vertical null Killing vector field on $B$). The fibers of $B\to M$
are the integral curves of $\zeta$, hence to show that these fibers are null
geodesics it is enough to show that $\nabla_{\zeta}\zeta=0$, or in index
notation,

$\zeta^{b}\nabla_{b}\zeta^{a}=0.$

Lowering an index of $\nabla_{b}\zeta^{a}$ (using $\mathrm{g}$), splitting
$\nabla_{b}\zeta_{a}$ into its symmetric and antisymmetric parts, and
contracting with $\zeta^{b}$ gives

$\zeta^{b}\nabla_{b}\zeta_{a}=\zeta^{b}\cdot\frac{1}{2}(\nabla_{a}\zeta_{b}+\nabla_{b}\zeta_{a})+\zeta^{b}\cdot\frac{1}{2}(\nabla_{a}\zeta_{b}-\nabla_{b}\zeta_{a}).$
(37)

The quantity $\frac{1}{2}(\nabla_{a}\zeta_{b}+\nabla_{b}\zeta_{a})$ in the
first term is $(\mathcal{L}_{\zeta}g)_{ab}$, but, per part (1), $\mathrm{g}$
is $\zeta$-invariant—that is, $\mathcal{L}_{\xi}g=0$—and so the first term
vanishes.

The quantity $\frac{1}{2}(\nabla_{a}\zeta_{b}-\nabla_{b}\zeta_{a})$ in the
second term is $(d\zeta^{\flat})_{ab}$, so the second term is
$-\iota_{\zeta}(d\zeta^{\flat})$, where $\iota_{\zeta}$ denotes interior
multiplication by $\zeta$. Since $\zeta$ generates the $\mathbb{R}^{*}$-action
on $B\to M$ and $\alpha$ is a connection form thereon, $\alpha(\zeta)$ is a
nonzero constant, and by rescaling $\zeta$ by a nonzero constant we may as
well assume $\alpha(\zeta)=3$. Lowering an index with $\mathrm{g}$ (equations
(35)–(36)) then gives $\zeta^{\flat}=-\omega^{3}$, so the third equation of
(33) yields
$-\iota_{\zeta}(\mathrm{d}\zeta^{\flat})=\iota_{\zeta}\mathrm{d}\omega^{3}=\iota_{\zeta}(\omega^{1}\wedge\omega^{2})=0$.
$\square$

######  Remark 3.4.

In fact, chains come equipped with a preferred projective structure (see,
e.g., [6, Theorem 5.3.7], which applies to all so-called parabolic contact
structures), but we do not need that structure here.

###  3.1 Chains of $y^{\prime\prime}=f(x,y,y^{\prime})$

Here $\Sigma=J^{0}(\mathbb{R},\mathbb{R})=\mathbb{R}^{2}$, with coordinates
$(x,y)$, and $M=J^{1}(\mathbb{R},\mathbb{R})=\mathbb{R}^{3}$, with coordinates
$(x,y,p)$ and contact distribution $D={\rm Ker}(\mathrm{d}y-p\,\mathrm{d}x).$
The paths in $\Sigma$ are graphs of solutions $y(x)$ to
$y^{\prime\prime}=f(x,y,y^{\prime})$, and their lifts to $M$ are graphs of
their first jets, $(x,y(x))\mapsto(x,y(x),y^{\prime}(x))$. So here

$L_{1}=\mathrm{Span}\left(\partial_{p}\right),\quad
L_{2}=\mathrm{Span}\left[\partial_{x}+p\partial_{y}+f(x,y,p)\partial_{p}\right].$

We fix the contact form $\eta^{3}:=\mathrm{d}y-p\,\mathrm{d}x$. An adapted
coframe on $M$, satisfying equations (29)–(30), is

$\eta^{1}:=\mathrm{d}p-f\mathrm{d}x,\quad\eta^{2}:=-\mathrm{d}x,\quad\eta^{3}:=\mathrm{d}y-p\,\mathrm{d}x.$
(38)

Any other adapted coframe is of the form
$s\cdot\eta=(\eta^{1}/s,s\eta^{2},\eta^{3})^{t}$, $s:M\to\mathbb{R}^{*}$. This
defines an identification $\mathbb{R}^{3}\times\mathbb{R}^{*}\to B$,
$(x,y,p,s)\mapsto s\cdot\eta(x,y,p).$ Under this identification,

$\omega^{1}=\eta^{1}/s,\quad\omega^{2}=s\eta^{2},\quad\omega^{3}=\eta^{3}.$
(39)

The following proposition was proved in [22, equation (31)] by solving the
full equivalence problem for path geometry.

######  Proposition 3.5.

The Fefferman metric on $B=J^{1}(\mathbb{R},\mathbb{R})\times\mathbb{R}^{*}$
is

$\mathrm{g}=-\mathrm{d}x\cdot(\mathrm{d}p-f\mathrm{d}x)+{1\over
6}(\mathrm{d}y-p\,\mathrm{d}x)\cdot\left[4f_{p}\mathrm{d}x+f_{pp}(\mathrm{d}y-p\,\mathrm{d}x)-4\mathrm{d}\tau\right],$
(40)

where $\mathrm{d}\tau=\mathrm{d}s/s.$

###### Proof.

Solving equations (33)–(34), with $\omega^{i}$ given by equations (38)–(39),
we obtain

$\alpha=-f_{p}\mathrm{d}x+\mathrm{d}\tau,\ K=f_{pp}\ \Longrightarrow\
\sigma={2\over 3}f_{p}\mathrm{d}x+{1\over
6}f_{pp}(\mathrm{d}y-p\mathrm{d}x)-{2\over 3}{\mathrm{d}\tau}.$

Using this in equations (36)-(36) gives the claimed formula. $\square$

######  Proposition 3.6.

The chains of the path geometry corresponding to a 2nd order ODE
$y^{\prime\prime}=f(x,y,y^{\prime})$ are the curves in
$J^{1}(\mathbb{R},\mathbb{R})$ which are the graphs of solutions $(y(x),p(x))$
of the system

$\displaystyle y^{\prime\prime}$
$\displaystyle=f+f_{p}\Delta+\frac{1}{2}f_{pp}\Delta^{2}+{1\over
6}f_{ppp}\Delta^{3}$ (41)
$\displaystyle\begin{split}p^{\prime\prime}&=-\frac{2(p^{\prime}-f)^{2}}{\Delta}+f_{p}(3p^{\prime}-2f)+f_{x}+pf_{y}+\left[f_{pp}(p^{\prime}-f)+2f_{y}\right]\Delta\\\
&+\frac{1}{6}\left[f_{ppp}(p^{\prime}-2f)-f_{xpp}+4f_{yp}-pf_{ypp})\right]\Delta^{2},\end{split}$
(42)

where $\Delta=y^{\prime}-p.$

###### Proof.

Using the metric (40), we write the geodesic equations on $B$,

$\displaystyle\ddot{x}$
$\displaystyle=\frac{1}{6}\left[\left(p\dot{x}-\dot{y}\right)\left(f_{{ppp}}\left(\dot{y}-p\dot{x}\right)+2\dot{x}f_{{pp}}\right)-2\dot{x}^{2}f_{p}-4\dot{\tau}\dot{x}\right],$
$\displaystyle\ddot{y}$
$\displaystyle=\frac{1}{6}\left[2\dot{x}\left(pf_{{pp}}\left(p\dot{x}-\dot{y}\right)-p\dot{x}f_{p}-2p\dot{\tau}+3\dot{p}\right)-pf_{{ppp}}\left(\dot{y}-p\dot{x}\right)^{2}\right],$
$\displaystyle\ddot{p}$
$\displaystyle=\frac{1}{6}\left[-p^{3}\dot{x}^{2}f_{{ypp}}+2p^{2}\dot{x}\dot{y}f_{{ypp}}+4p^{2}\dot{x}^{2}f_{{yp}}\right.$
$\displaystyle\left.\qquad\quad-2f\left(\left(\dot{y}-p\dot{x}\right)\left(f_{{ppp}}\left(\dot{y}-p\dot{x}\right)+2\dot{x}f_{{pp}}\right)+2\dot{x}^{2}f_{p}+4\dot{\tau}\dot{x}\right)\right.$
$\displaystyle\qquad\quad\left.+2\dot{p}f_{{pp}}(\dot{y}-p\dot{x})-f_{{xpp}}\left(\dot{y}-p\dot{x}\right)^{2}-8p\dot{x}\dot{y}f_{{yp}}-6p\dot{x}^{2}f_{y}\right.$
$\displaystyle\qquad\quad\left.+8\dot{p}\dot{x}f_{p}-p\dot{y}^{2}f_{{ypp}}+12\dot{x}\dot{y}f_{y}+6\dot{x}^{2}f_{x}+4\dot{y}^{2}f_{{yp}}+4\dot{p}\dot{\tau}\right].$

(We do not need the $\tau$ equation.) Next use formula (40) and the nullity
condition to eliminate $\dot{\tau}$ from the equations,

$\dot{\tau}={1\over
4}\left(f_{{pp}}\left(\dot{y}-p\dot{x}\right)+4\dot{x}f_{p}\right)-{3\over
2}{\dot{x}(\dot{p}-f\dot{x})\over(\dot{y}-p\dot{x})},$

($\tau$ itself does not appear explicitly, because of the
$\mathbb{R}^{*}$-invariance of the metric). Then substitute into
$y^{\prime\prime}=(\ddot{y}\dot{x}-\ddot{x}\dot{y})/\dot{x}^{3},p^{\prime\prime}=(\ddot{p}\dot{x}-\ddot{x}\dot{p})/\dot{x}^{3}$
the expressions for $\ddot{x},\ddot{y},\ddot{p}$ from the geodesic equations,
and finally make the substitutions
$\dot{y}=\dot{x}y^{\prime},\dot{p}=\dot{x}p^{\prime}$ to obtain the desired
equations (all instances of $\dot{x}$ cancel out because the geodesic equation
is homogeneously quadratic in velocities). $\square$

###  3.2 Chains of projective path geometries

Here we prove Theorem 1, which was announced in the introduction. Recall that,
by definition, a path geometry is projective if the paths are the
(unparametrized) geodesics of a torsion-free affine connection.

Theorem 1. A path geometry on a 2-dimensional manifold $\Sigma$ is projective
if and only if all chains on $\mathbb{P}T\Sigma$ project to the paths in
$\Sigma$.

###### Proof.

This is a local statement so we can assume without loss of generality the
situation studied in the previous subsection, i.e. the paths are given in the
$xy$ plane by graphs of solutions $y(x)$ of
$y^{\prime\prime}=f(x,y,y^{\prime})$ for some smooth $f$, and the associated
chains in $xyp$-space are the graphs of solutions $(y(x),p(x))$ to the chain
equations (41)–(42) of Proposition 3.6. As is well known, such a path geometry
is projective if and only if $f(x,y,p)$ is a polynomial in $p$ of degree at
most 3 (see [9], also Section 4 of [13]). The statement we are to prove
therefore reduces to the following lemma:

######  Lemma 3.7.

Every solution $(y(x),p(x))$ of equations (41)–(42) satisfies
$y^{\prime\prime}=f(x,y,y^{\prime})$ if and only if $f(x,y,p)$ is polynomial
in $p$ of degree at most $3$.

We proceed with the proof of the lemma. Assume $f(x,y,p)$ is polynomial in $p$
of degree $\leq 3$. Then $f(x,y,y^{\prime})$ is given by the cubic Taylor
polynomial of $f$ with respect to $p$:

$f(x,y,y^{\prime})=f+f_{p}(y^{\prime}-p)+\frac{1}{2}f_{pp}(y^{\prime}-p)^{2}+\frac{1}{6}f_{ppp}(y^{\prime}-p)^{3},$
(43)

where $f$ and its derivatives on the right hand side are evaluated at
$(x,y,p)$. Now the right hand side of the last equation, evaluated at
$y=y(x),y^{\prime}=y^{\prime}(x),p=p(x)$, is the right hand side of the chain
equation (41). It follows that if $(y(x),p(x))$ satisfy equations (41)–(42)
then $y(x)$ satisfies $y^{\prime\prime}(x)=f(x,y(x),y^{\prime}(x)),$ as
needed.

Conversely, suppose $f(x,y,p)$ is not polynomial in $p$ of degree $\leq 3$.
Then there is a neighborhood $U\subset\mathbb{R}^{3}$ such that for all
$(x,y,p),(x,y,y^{\prime})\in U$, with $y^{\prime}\neq p$, equation (43) does
not hold. It follows that the chains in this neighborhood do not project to
solutions of $y^{\prime\prime}=f(x,y,y^{\prime})$. $\square$

######  Remark 3.8.

One should also be able to prove Theorem 1 using the general machinery of
parabolic geometry concerning correspondence spaces [6, Section 4.4] and
canonical curves [6, Section 5.3] in a way that may be readily generalizable
to other types of parabolic geometries and families of curves. Such a proof
would take us too far afield here, so we will take up this approach elsewhere.

##  4 Examples of path geometries and their chains

In this section we illustrate the general theory of the previous section by
determining explicitly the chains of some homogeneous path geometries. First,
the flat path geometry on $\mathbb{R}P^{2}$, admitting an 8-dimensional
symmetry group, then 3 of the items of Tresse’s classification [24] of
‘submaximal’ path geometries, i.e. those admitting a 3-dimensional group of
symmetries. In each case we exploit the symmetry to reduce the chain equations
to determining null geodesics on a group with respect to a left-invariant sub-
Riemannian metric. Then a well-known procedure reduces the equation to the
Euler equations on the dual of the Lie algebra of the group and are
integrable.

###  4.1  The flat path geometry

Here $M\subset\mathbb{R}P^{2}\times(\mathbb{R}P^{2})^{*}$ is the set of
incident point-line pairs $(q,\ell)$ (equivalently, the manifold $F_{1,2}$ of
full flags in $\mathbb{R}^{3}$) and $L_{1},L_{2}\subset TM$ are the tangents
to the fibers of the projections onto the first and second factor
(respectively).

######  Proposition 4.1.

For each non-incident pair
$(q_{*},\ell_{*})\in\mathbb{R}P^{2}\times(\mathbb{R}P^{2})^{*}\setminus M$
consider the set of incident pairs $(q,\ell)\in M$ such that
$q\in\ell_{*},q_{*}\in\ell.$ This is a chain in $M$ and all chains in $M$ are
of this form. See Figure 2.

$q$
---
$\ell$
---
$q_{*}$
---
$\ell_{*}$
---
Figure 2: Chains of the flat path geometry (straight lines).

To prove it, note that $\mathrm{GL}_{3}(\mathbb{R})$ acts naturally on
$(M,L_{1},L_{2})$. We look for a 3-dimensional subgroup
$G\subset\mathrm{GL}_{3}(\mathbb{R})$ acting on $M$ with an open orbit. Fixing
a point $m_{0}=(q_{0},\ell_{0})\in M$ yields we get two left-invariant line
fields $L_{1},L_{2}\subset TG,$ given by their value
$(L_{1})_{\mathrm{id}},(L_{2})_{\mathrm{id}}\subset\mathfrak{g}$, the Lie
algebras of the stabilizers of $q_{0},\ell_{0}$ (resp.). It is then easy to
find left-invariant adapted coframes on $G$ describing $L_{1},L_{2}$ and the
associated Fefferman metric. We consider two such $G$: the Heisenberg group
and the Euclidean group.

####  4.1.1 First proof of Proposition 4.1: via the Heisenberg group

Let $\mathrm{H}$ be the set of matrices of the form

$\left(\begin{array}[]{ccc}1&z&y\\\ 0&1&x\\\ 0&0&1\end{array}\right),\quad
x,y,z\in\mathbb{R}.$ (44)

Its Lie algebra ${\mathfrak{h}}$ consists of matrices of the form

$\left(\begin{array}[]{ccc}0&x^{1}&x^{3}\\\ 0&0&x^{2}\\\
0&0&0\end{array}\right),\quad x^{i}\in\mathbb{R}.$ (45)

Let $\theta^{i}$ be the left-invariant 1-form on $\mathrm{H}$ whose value at
$\mathrm{id}\in\mathrm{H}$ is $x^{i}$, $i=1,2,3$. Then

$\Theta:=\left(\begin{array}[]{ccc}0&\theta^{1}&\theta^{3}\\\
0&0&\theta^{2}\\\ 0&0&0\end{array}\right)$

is the left-invariant Maurer–Cartan form on $\mathrm{H}$, satisfying
$\mathrm{d}\Theta=-\Theta\wedge\Theta$, from which we get

$\mathrm{d}\theta^{1}=\mathrm{d}\theta^{2}=0,\
\mathrm{d}\theta^{3}=-\theta^{1}\wedge\theta^{2}.$ (46)

Identify $\mathbb{R}^{2}$ with an affine plane in $\mathbb{R}^{3}$,
$(x,y)\mapsto(y,x,1)$. It is $\mathrm{H}$-invariant, and the resulting affine
action on $\mathbb{R}^{2}$ is $(x_{0},y_{0})\mapsto(x_{0}+x,y_{0}+y+zx_{0}).$
This action is transitive on $\mathbb{R}^{2}$ and transitive and free on the
set $M$ of incident pairs $(q,\ell)$, where $q\in\mathbb{R}^{2}$ and
$\ell\subset\mathbb{R}^{2}$ is a non-vertical line through $q$. There are
$\mathrm{H}$-invariant line fields $L_{1},L_{2}\subset TM$, where $L_{1}$
(resp. $L_{2}$) is tangent to the fibers of the projection $(q,\ell)\mapsto q$
(resp. $(q,\ell)\mapsto\ell$).

Let $q_{0}=(0,0),\ell_{0}=\\{y=0\\}$ (the real axis). Let
$(X_{1},X_{2},X_{3})$ the (left-invariant) frame on $\mathrm{H}$ dual to
$(\theta^{1},\theta^{2},\theta^{3})$. Then the Lie algebras
$(L_{1})_{\mathrm{id}},(L_{2})_{\mathrm{id}}$ of the stabilizers of
$q_{0},\ell_{0}$ are spanned by $X_{1},X_{2}$ (resp.). Thus,

$D=L_{1}\oplus L_{2}=(\theta^{3})^{\perp},\quad
L_{1}=\\{\theta^{2},\theta^{3}\\}^{\perp},\quad
L_{2}=\\{\theta^{1},\theta^{3}\\}^{\perp},$

with an adapted coframe

$\eta^{1}:=\theta^{1},\quad\eta^{2}:=\theta^{2},\quad\eta^{3}:=-\theta^{3}.$

Solving the structure equations (33)–(34), we get $\alpha=\theta^{4},\ K=0,$
where $\theta^{4}=(\mathrm{d}s)/s$ (the Maurer–Cartan form on
$\mathbb{R}^{*}$), which gives, using equations (35)–(36),
$\sigma=-(2/3)\theta^{4}$ and

$\mathrm{g}=\theta^{1}\cdot\theta^{2}+\frac{2}{3}\theta^{3}\cdot\theta^{4}.$
(47)

######  Lemma 4.2.

Null geodesics of (47), projected to $\mathrm{H}$ and passing through
$\mathrm{id}\in\mathrm{H}$ at $t=0$, are of the form

$x=b(1-e^{-ct}),\ y=-ab(1-e^{-ct}),\ z=a(e^{ct}-1),\qquad a,b,c\in\mathbb{R}.$

They correspond to chains $(q_{t},\ell_{t})\in M$, passing through
$(q_{0},\ell_{0})$ at $t=0,$ where $q_{t}$ moves along the line $\ell_{*}$
through $q_{0}$ of slope $-a$, and $\ell_{t}$ is a line through $q_{t}$ and
$q_{*}=(b,0)$.

###### Proof.

The metric (47) is a left-invariant metric on
$G=\mathrm{H}\times\mathbb{R}^{*},$ with an inertia operator
$A:\mathfrak{g}\to\mathfrak{g}^{*}$

$A={1\over 6}\left(\begin{array}[]{cccc}0&3&0&0\\\ 3&0&0&0\\\ 0&0&0&2\\\
0&0&2&0\\\ \end{array}\right).$

The geodesic flow on $T^{*}G$ projects via left translation to the Euler
equations on $\mathfrak{g}^{*}$, $\dot{P}=\mathrm{ad}_{A^{-1}P}^{*}P$, where
$\mathrm{ad}_{X}^{*}=(\mathrm{ad}_{X})^{t}\in{\rm End}(\mathfrak{g}^{*}),$
$X\in\mathfrak{g}$, $P\in\mathfrak{g}^{*}$ and $\mathrm{ad}_{X}Y=[X,Y].$ These
are the Hamiltonian equations $\dot{P}=\\{H,P\\}$ with respect to the standard
Lie-Poisson structure on $\mathfrak{g}^{*}$, where $H={1\over 2}(P,A^{-1}P).$
See [2, page 66]. Equivalently, $\dot{X}=A^{-1}\mathrm{ad}_{X}^{*}AX$. To
write these down explicitly with respect to our bases, we first represent
$X\in\mathfrak{g}$ and $\mathrm{ad}_{X}^{*}\in{\rm End}(\mathfrak{g}^{*})$ by
$4\times 4$ matrices

$X=\left(\begin{array}[]{cccc}0&x^{1}&x^{3}&0\\\ 0&0&x^{2}&0\\\ 0&0&0&0\\\
0&0&0&x^{4}\\\
\end{array}\right),\qquad\mathrm{ad}_{X}^{*}=(\mathrm{ad}_{X})^{t}=\left(\begin{array}[]{cccc}0&0&-x^{2}&0\\\
0&0&x^{1}&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{array}\right),$

so $\dot{X}=A^{-1}\mathrm{ad}_{X}^{*}AX$ becomes

$\dot{x}^{1}={2\over 3}x^{1}x^{4},\quad\dot{x}^{2}=-{2\over
3}x^{2}x^{4},\quad\dot{x}^{3}=\dot{x}^{4}=0.$ (48)

The general solution, with $H=(AX,X)/2=x^{1}x^{2}/2+x^{3}x^{4}/3=0$ (we are
interested in the zero level set because we are computing the null geodesics),
is

$x^{1}=ae^{ct},\quad x^{2}=be^{-ct},\quad x^{3}=-{ab\over c},\quad
x^{4}={3c\over 2},$ (49)

where $a,b,c\in\mathbb{R},$ $c\neq 0$. (In addition to these solutions there
are some fixed points, which we now ignore).

Now let $g(t)\in H\times\mathbb{R}^{*}$ be a null geodesic, with

$g(t)=\left(\begin{array}[]{cccc}1&z&y&0\\\ 0&1&x&0\\\ 0&0&1&0\\\
0&0&0&s\end{array}\right),\quad x,y,z,s\in\mathbb{R},\ s\neq 0.$

Then $X=g^{-1}\dot{g}\in\mathfrak{g}$ is given by (49). Explicitly,

$\dot{x}=x^{2}=b\,e^{-ct},\quad\dot{y}-z\dot{x}=x^{3}=-{ab\over
c},\quad\dot{z}=x^{1}=a\,e^{ct}$ (50)

(we do not need the $s$ equation). Change the time variable to $\tau=ct$,
denoting derivative with respect to $\tau$ by $(\ )^{\prime}$ and renaming the
constants, $a\mapsto a/c,b\mapsto b/c$, we get

$x^{\prime}=be^{-\tau},\quad y^{\prime}-zx^{\prime}=-ab,\quad
z^{\prime}=ae^{\tau}.$ (51)

Consider chains through $\mathrm{id}\in\mathrm{H}$, i.e.
$x_{0}=y_{0}=z_{0}=0.$ Then $z=a(e^{\tau}-1),$ hence
$y^{\prime}=zx^{\prime}-ab=-ab\,e^{-\tau}.$ The solution of (51) is then

$x=b(1-e^{-\tau}),\quad y=-ab(1-e^{-\tau}),\quad z=a(e^{\tau}-1).$

Thus $(x,y)$ traces a line of slope $-a$ through the origin, and each line of
slope $z$ through $(x,y)$ passes through $(b,0).$ $\square$

####  4.1.2 Second proof of Proposition 4.1: via the Euclidean group

Here
$M=\mathbb{P}(T\mathbb{R}^{2})=\mathbb{R}^{2}\times\mathbb{P}(\mathbb{R}^{2})$
is the set of pairs $(q,\ell)$ with $\ell$ a line through $q$, $L_{1}\subset
TM$ is tangent to the fibers of the projection onto the first factor and
similarly for $L_{2}$. The group $\mathrm{SE}_{2}$ of orientation-preserving
isometries of $\mathbb{R}^{2}$ acts transitively on $M$, with stabilizer
$\mathbb{Z}_{2}$ (reflection about a point), preserving $L_{1},L_{2}$. Fixing
a point $(q_{0},\ell_{0})\in M$ identifies $M$ with
$\mathrm{SE}_{2}/\mathbb{Z}_{2}$, and hence equips $\mathrm{SE}_{2}$ with
left-invariant line fields $L_{1},L_{2}$ given by a pair of 1-dimensional
subspaces $(L_{1})_{\mathrm{id}},(L_{2})_{\mathrm{id}}$, the Lie algebras of
the stabilizers of $q_{0},\ell_{0}$ (resp.).

Identify $\mathbb{R}^{2}=\mathbb{C}$ with the affine plane ${\mathbf{z}}=1$ in
$\mathbb{C}^{2}$, ${\mathbf{z}}\mapsto({\mathbf{z}},1)$; then
$\mathrm{SE}_{2}$ is identified with the subgroup of
$\mathrm{GL}_{2}(\mathbb{C})$ consisting of matrices of the form

$\left(\begin{array}[]{ccc}e^{i\theta}&{\mathbf{z}}\\\
0&1\end{array}\right),\quad{\mathbf{z}}\in\mathbb{C},\ \theta\in\mathbb{R}.$
(52)

Its Lie algebra $\mathfrak{se}_{2}$ consists of matrices of the form

$\left(\begin{array}[]{cc}ix^{1}&x^{2}+ix^{3}\\\ 0&0\end{array}\right),\quad
x^{i}\in\mathbb{R}.$ (53)

Let $\theta^{j}$ be the left-invariant 1-form on $\mathrm{SE}_{2}$ whose value
at $\mathrm{id}$ is $x^{j}$, $j=1,2,3$. Then

$\Theta:=\left(\begin{array}[]{cc}i\theta^{1}&\theta^{2}+i\theta^{3}\\\
0&0\end{array}\right)$

is the left-invariant Maurer–Cartan form on $\mathrm{SE}_{2}$, satisfying
$\mathrm{d}\Theta=-\Theta\wedge\Theta$, from which we get

$\mathrm{d}\theta^{1}=0,\ \mathrm{d}\theta^{2}=\theta^{1}\wedge\theta^{3},\
\mathrm{d}\theta^{3}=-\theta^{1}\wedge\theta^{2}.$ (54)

Let $X_{1},X_{2},X_{3}$ the left-invariant vector fields on $\mathrm{SE}_{2}$
dual to $\theta^{1},\theta^{2},\theta^{3}$. Let $q_{0}=0,\ell_{0}=\mathbb{R}$
(the real axis). Then the Lie algebras of the stabilizers of $q_{0},\ell_{0}$
are spanned by $X_{1},X_{2}$ (resp.). Thus

$D=L_{1}\oplus L_{2}=(\theta^{3})^{\perp},\quad
L_{1}=\\{\theta^{2},\theta^{3}\\}^{\perp},\quad
L_{2}=\\{\theta^{1},\theta^{3}\\}^{\perp},$

with an adapted coframe

$\eta^{1}:=\theta^{1},\quad\eta^{2}:=\theta^{2},\quad\eta^{3}:=-\theta^{3}.$

Solving the structure equations (33)–(34), we get $\alpha=\theta^{4},\ K=0,$
where $\theta^{4}=(\mathrm{d}s)/s$ (the MC form on $\mathbb{R}^{*}$), which
gives, using equations (35)–(36), $\sigma=-(2/3)\theta^{4}$ and

$\mathrm{g}=\theta^{1}\cdot\theta^{2}+\frac{2}{3}\theta^{3}\cdot\theta^{4}.$
(55)

This is a left-invariant metric on $G=\mathrm{SE}_{2}\times\mathbb{R}^{*},$
with an inertia operator $A:\mathfrak{g}\to\mathfrak{g}^{*}$

$A={1\over 6}\left(\begin{array}[]{cccc}0&3&0&0\\\ 3&0&0&0\\\ 0&0&0&2\\\
0&0&2&0\\\ \end{array}\right).$

The geodesic flow on $T^{*}G$ projects via left translation to the Euler
equations on $\mathfrak{g}^{*}$, $\dot{P}=\mathrm{ad}_{A^{-1}P}^{*}P$, where
$\mathrm{ad}_{X}^{*}=-(\mathrm{ad}_{X})^{t}\in{\rm End}(\mathfrak{g}^{*}).$
These are the Hamiltonian equations $\dot{P}=\\{H,P\\}$ with respect to the
standard Lie-Poisson structure on $\mathfrak{g}^{*}$, where $H={1\over
2}(P,A^{-1}P).$ See [2, p. 66]. To write these down explicitly with respect to
our bases, we first represent $X\in\mathfrak{g}$ and
$\mathrm{ad}_{X}^{*}\in{\rm End}(\mathfrak{g}^{*})$ by $4\times 4$ matrices

$X=\left(\begin{array}[]{cccc}0&-x^{1}&x^{2}&0\\\ x^{1}&0&x^{3}&0\\\
0&0&0&0\\\ 0&0&0&x^{4}\\\
\end{array}\right),\qquad\mathrm{ad}_{X}^{*}=(\mathrm{ad}_{X})^{t}=\left(\begin{array}[]{cccc}0&x^{3}&-x^{2}&0\\\
0&0&x^{1}&0\\\ 0&-x^{1}&0&0\\\ 0&0&0&0\\\ \end{array}\right),$

so $\dot{P}=\mathrm{ad}_{A^{-1}P}^{*}P$ becomes

$\dot{P}_{1}=-2P_{1}P_{3}+3P_{2}P_{4},\quad\dot{P}_{2}=2P_{2}P_{3},\quad\dot{P}_{3}=-2P_{2}^{2},\quad\dot{P}_{4}=0.$
(56)

with constants of motion (in addition to $P_{4}$),

$\displaystyle H$ $\displaystyle={1\over
2}(P,A^{-1}P)=2P_{1}P_{2}+3P_{3}P_{4}=0,\qquad k=(P_{2})^{2}+(P_{3})^{2}.$

Let us use polar coordinates in the $P_{2}P_{3}$-plane:

$P_{2}=r\cos\phi,\quad P_{3}=r\sin\phi.$

Then

$\dot{\phi}=-2r\cos\phi,\quad P_{1}=c\tan\phi,\quad P_{4}=-2c/3,\quad
c=const.,\quad r=const.$

Now let $g(t)\in\mathrm{SE}_{2}\times\mathbb{R}^{*}$ be a null geodesic, with

$g(t)=\left(\begin{array}[]{ccc}e^{i\theta}&{\mathbf{z}}&0\\\ 0&1&0\\\
0&0&s\end{array}\right),\quad{\mathbf{z}}\in\mathbb{C},\
\theta,s\in\mathbb{R}^{*}.$

Let $X=g^{-1}\dot{g}\in\mathfrak{g}$. Then $P=AX$ satisfies Euler equations
(56). Explicitly,

$\dot{\theta}=x^{1}=2P_{2}=-\dot{\phi},\quad\dot{\mathbf{z}}=e^{i\theta}(x^{2}+ix^{3})=e^{i\theta}(2P_{1}+i3P_{4})=2ce^{i\theta}(\tan\phi-i).$

(The $s$ equation is omitted; it will not be used). Assume, without loss of
generality, that $g(0)=\mathrm{id},$ i.e. $\theta(0)=0$ and
${\mathbf{z}}(0)=0$, so $\theta=\phi_{0}-\phi$. We reparametrize $g(t)$ by
$\phi$, denote derivative with respect to $\phi$ by $(\ )^{\prime}$, and get
${\mathbf{z}}^{\prime}=i(c/r)e^{i\phi_{0}}\dot{\mathbf{z}}/\dot{\phi}=i(c/r)e^{i\phi_{0}}\sec^{2}\phi.$
Integrating yields ${\mathbf{z}}=i(c/r)e^{i\phi_{0}}\tan\phi.$ Now we rotate
the chain by $-\phi_{0}$, reflect about the $x$-axis and rename $c$, so that

${\mathbf{z}}=ic\tan\phi,\quad\theta=\phi,\quad c\in\mathbb{R}.$ (57)

This corresponds to a chain $(q_{\phi},\ell_{\phi})$, where $q_{\phi}$ moves
along $\ell^{*}=$the $y$ axis and $\ell_{\phi}$ is the line connecting
$q_{*}=(-c,0)$ with $q_{\phi}$.

###  4.2 Circles of fixed radius

Here $M\subset\mathbb{C}\times\mathbb{C}$ is the set of pairs of points
$(p,q)$ with $|p-q|=1$, $L_{1},L_{2}\subset TM$ are tangent to the fibers of
the projection onto the first (resp. second) factor. The first projection maps
the fibers of the second projection to the set of plane circles of radius 1.
The group $\mathrm{SE}_{2}$ of orientation preserving isometries of
$\mathbb{C}=\mathbb{R}^{2}$ acts transitively and freely on $M$, preserving
$L_{1},L_{2}$. We use the same notation for this group as in Section 4.1.2.
Let $p_{0}=0,q_{0}=1.$ Then the Lie algebras
$(L_{1})_{\mathrm{id}},(L_{2})_{\mathrm{id}}$ of the stabilizers of these
points are spanned by $X_{1},X_{1}-X_{3}$ (resp.). Thus

$D=L_{1}\oplus L_{2}=(\theta^{2})^{\perp},\quad
L_{1}=\\{\theta^{2},\theta^{3}\\}^{\perp},\quad
L_{2}=\\{\theta^{2},\theta^{1}+\theta^{3}\\}^{\perp},$

with an adapted coframe

$\eta^{1}:=\theta^{3},\quad\eta^{2}:=\theta^{1}+\theta^{3},\quad\eta^{3}:=-\theta^{2}.$

Solving the structure equations (33)–(34), we get
$\alpha=\theta^{2}+\theta^{4},\ K=-1,$ where $\theta^{4}=(\mathrm{d}s)/s$ (the
Maurer–Cartan form on $\mathbb{R}^{*}$), which gives, using equations
(35)–(36), $\sigma=-\theta^{2}/2-2\theta^{4}/3$ and

$\mathrm{g}={1\over
2}(\theta^{2})^{2}+(\theta^{3})^{2}+\theta^{1}\cdot\theta^{3}+\frac{2}{3}\theta^{2}\cdot\theta^{4}.$
(58)

This is a left-invariant metric on $G=\mathrm{SE}_{2}\times\mathbb{R}^{*},$
with an inertia operator $A:\mathfrak{g}\to\mathfrak{g}^{*}$

$A={1\over 6}\left(\begin{array}[]{cccc}0&0&3&0\\\ 0&3&0&2\\\ 3&0&6&0\\\
0&2&0&0\\\ \end{array}\right).$

The geodesic flow on $T^{*}G$ projects via left translation to the Euler
equations on $\mathfrak{g}^{*}$, $\dot{P}=\mathrm{ad}_{A^{-1}P}^{*}P$. To
write these down explicitly with respect to our bases, we first represent
$X\in\mathfrak{g}$ and $\mathrm{ad}_{X}^{*}\in{\rm End}(\mathfrak{g}^{*})$ by
$4\times 4$ matrices

$X=\left(\begin{array}[]{cccc}0&-x^{1}&x^{2}&0\\\ x^{1}&0&x^{3}&0\\\
0&0&0&0\\\ 0&0&0&x^{4}\\\
\end{array}\right),\qquad\mathrm{ad}_{X}^{*}=(\mathrm{ad}_{X})^{t}=\left(\begin{array}[]{cccc}0&x^{3}&-x^{2}&0\\\
0&0&x^{1}&0\\\ 0&-x^{1}&0&0\\\ 0&0&0&0\\\ \end{array}\right),$

so $\dot{P}=\mathrm{ad}_{A^{-1}P}^{*}P$ become

$\displaystyle\begin{split}\dot{P}_{1}&=2P_{1}P_{2}-3P_{3}P_{4},\quad\dot{P}_{2}=2P_{3}(P_{3}-2P_{1}),\\\
\dot{P}_{3}&=-2P_{2}(P_{3}-2P_{1}),\quad\dot{P}_{4}=0,\end{split}$ (59)

with constants of motion (in addition to $P_{4}$),

$\displaystyle H$ $\displaystyle={1\over
2}(P,A^{-1}P)=-2P_{1}^{2}+2P_{1}P_{3}+3P_{2}P_{4}-\frac{9}{4}M_{4}^{2}$
$\displaystyle r^{2}$ $\displaystyle=(P_{2})^{2}+(P_{3})^{2}.$

We make the following change of variables:

$y=4P_{1}-2P_{3},\quad P_{2}=r\cos\phi,\quad P_{3}=-r\sin\phi,\quad
P_{4}=c/3.$ (60)

Then (59) reduces to

$\dot{\phi}=-y,\quad\dot{y}=4r(c-r\cos\phi)\sin\phi,\qquad c,r\in\mathbb{R},\
r\geq 0,$ (61)

and the nullity condition $H=0$ becomes

$y^{2}=8cr\cos\phi+4r^{2}\sin^{2}\phi-2c^{2}.$ (62)

######  Remark 4.3.

Equations (61) can be written as a single Newton type second order ODE,
$\ddot{\phi}=f(\phi),$ where $f(\phi)=4r(r\cos\phi-c)\sin\phi.$ As usual, one
writes $f(\phi)=-U^{\prime}(\phi)$, say
$U(\phi)=-4cr\cos\phi-2r^{2}\sin^{2}\phi.$ Then $\dot{\phi}^{2}/2+U(\phi)$ is
constant along solutions of $\ddot{\phi}=f(\phi)$ (‘conservation of energy’).
Equation (62) fixes the value of this energy.

Now let $g(t)\in\mathrm{SE}_{2}\times\mathbb{R}^{*}$ be a null geodesic, with

$g(t)=\left(\begin{array}[]{ccc}e^{i\theta}&{\mathbf{z}}&0\\\ 0&1&0\\\
0&0&s\end{array}\right),\quad{\mathbf{z}}\in\mathbb{C},\
\theta,s\in\mathbb{R}^{*}.$

Let $X=g^{-1}\dot{g}\in\mathfrak{g}$. Then $P=AX$ satisfies (59). Explicitly,

$\dot{\mathbf{z}}=e^{i\theta}(x^{2}+ix^{3})=e^{i\theta}(3P_{4}+i2P_{1}),\quad\dot{\theta}=x^{1}=-4P_{1}+2P_{3}.$

Using the change of variables (60), we get

$\dot{\mathbf{z}}=e^{i\theta}[c-i(\dot{\phi}/2+r\sin\phi)],\
\dot{\theta}=\dot{\phi},$

where $\phi(t)$ satisfies equations (61)–(62). For a fixed $\phi(t)$ these
equations are invariant under rigid motions (adding a constant angle to
$\theta$, rotating ${\mathbf{z}}$ by this angle and translating ${\mathbf{z}}$
by some constant vector). So we can assume, without loss of generality, that
$\theta=\phi$. Hence

$\dot{\mathbf{z}}=e^{i\theta}[c-i(\dot{\theta}/2+r\sin\theta)],\quad\dot{\theta}^{2}=8cr\cos\theta+4r^{2}\sin^{2}\theta-2c^{2}.$

Next we use the scaling invariance, $t\mapsto\lambda t,\ c\mapsto\lambda c,\
r\mapsto\lambda r,$ to assume $r=1$. We can also use the reflection symmetry
$t\mapsto-t,\theta\mapsto\theta+\pi,{\mathbf{z}}\mapsto-{\mathbf{z}},c\mapsto-c$
to assume that $c\geq 0$. Thus every chain, up to a rigid motion and
reparametrization, is a solution to

$\displaystyle\dot{\mathbf{z}}$
$\displaystyle=e^{i\theta}[c-i(\dot{\theta}/2+\sin\theta)],$ (63)
$\displaystyle(\dot{\theta})^{2}$
$\displaystyle=8c\cos\theta+4\sin^{2}\theta-2c^{2},$ (64)

with $c\in\mathbb{R},\ c\geq 0.$

######  Lemma 4.4.

Let $F(\theta,c)=8c\cos\theta+4\sin^{2}\theta-2c^{2}$ (the right-hand side of
equation (64)). Then $F\geq 0$ has a solution if and only if $|c|\leq 4.$ For
every $c\in[0,4]$ the set of $\theta\in[-\pi,\pi]$ such that $F(\theta,c)\geq
0$ is an interval $[-\theta_{max},\theta_{max}]$, where
$\theta_{max}=\cos^{-1}\left(c-\sqrt{c^{2}/2+1}\right)\in[0,\pi]$. For
$c\in(0,4)$ every solution $\theta(t)$ of (64) oscillates between
$-\theta_{max}$ and $\theta_{max}$. If $c=0$ then $\lim\theta$ is $0$ or $\pi$
as $t\to\pm\infty$. If $c=4$ then $\theta\equiv 0$.

###### Proof.

We write $F=-4x^{2}+8cx+4-2c^{2},$ where $x=\cos\theta$. The roots of this
polynomial are $x_{\pm}=c\pm\sqrt{1+c^{2}/2}$ and $F>0$ in the interval
$(x_{-},x_{+})$. To be able to solve for $\theta$ we need $[x_{-},x_{+}]$ to
intersect the interval $[-1,1]$. It is elementary to show that this occurs if
and only if $|c|\leq 4.$ $\square$

$c=0$$c=.5$$c=1$$c=1.5$$c=2.5$$c=3.5$$c=4$$\theta$$\dot{\theta}$${\pi\over
2}$$\theta_{max}$$\pi$03124 Figure 3: Chains of circles path geometry
(solutions of equations (63)–(64)). Top left: plot of the maximum amplitude of
oscillation of $\theta$ as a function of the chain parameter $c\in[0,4].$
Bottom left: phase curves of equation (64) for various $c$ values. Right: each
red curve is the projection of the chain on the Euclidean plane. The blue
curve represents the projection of the chain on the dual plane; it is formed
by joining the tips of the unit vectors in the direction $\theta$ at each
point ${\mathbf{z}}$ of the red curve (the thin blue lines).

######  Proposition 4.5.

Every chain of the path geometry of circles of radius 1 in the Euclidean
plane, up to an affine reparametrization and rigid motion, is given by a
unique solution of equations (63)–(64) with $c\in[0,4)$,
${\mathbf{z}}(0)=\theta(0)=0$ (for $c=0$ one should take $\theta(0)\neq
0,\pi.$

See Figure 3. The projection of the chains on the Euclidean plane (the curves
${\mathbf{z}}(t)$) look like inflectional elastica, but they are not (checked
numerically).

### Further properties/questions about these chains:

  1. 1.

From the pictures, ${\mathbf{z}}(t)+e^{i\theta(t)}$ (the red curve) is
obtained from ${\mathbf{z}}(t)$ by translation and parameter shift.
Presumably, this comes out of equations (63)–(64). Is this a manifestation of
the self-duality of this path geometry? How exactly?

  2. 2.

One should be able to write explicit solutions of equations (63)–(64) using
elliptic functions. See [23].

Note. One can write down an explicit general solution for the case $c=0$
without any special functions, and one can verify analytically that the arcs
are semicircles. So, as embedded submanifolds they are $C^{1}$ but not $C^{2}$
at inflection points. In particular, the chain ODE is not satisfied at these
inflection points.

  3. 3.

Equation (64) is the equation of a pendulum under a strange force law:
$f(\theta)=4(\cos\theta-c)\sin\theta,$ with special initial conditions:
$\theta(0)=0,\dot{\theta}(0)=8c-c^{2}$ (for $c=0$ it is the homoclinic
solution of the pendulum equation $\ddot{\theta}=2\sin 2\theta$. Is there a
good mechanical/geometrical interpretation of this motion?

  4. 4.

The chains of this geometry project to a 1-parameter family of curves in
$\mathbb{R}^{2}$ (up to rigid motion). Is there a simple geometric description
of this family? Our first guess was elastica but it is not the case.

  5. 5.

In the pictures, there are points along ${\mathbf{z}}(t)$ at which $\theta(t)$
is the direction of the tangent $\dot{\mathbf{z}}(t)$ (the inflection points
of the red curves on the right of Figure 3). Is this phenomenon unavoidable?

###  4.3 Hooke ellipses of fixed area

The manifold

$M=\\{(x,y,E,F,G)\in\mathbb{R}^{5}\,|\,Ex^{2}+2Fxy+Gz^{2}=1,\ EG-F^{2}=1,\
E>0\\}$

parametrizes the set of incident pairs $({\mathbf{r}},\mathcal{E})$, where
${\mathbf{r}}=(x,y)^{t}\in\mathbb{R}^{2}\setminus\\{0\\}$ and $\mathcal{E}$ is
an ellipse centered at the origin (a ‘Hooke ellipse’) of area $\pi$.

######  Proposition 4.6.

The path geometry in $\mathbb{R}^{2}\setminus\\{0\\}$ of Hooke ellipses of
fixed area is projective (the paths are the unparametrized geodesics of a
torsion-free affine connection).

###### Proof.

As mentioned before, this is equivalent to showing that the associated ODE
$y^{\prime\prime}=f(x,y,y^{\prime})$ is cubic in $y^{\prime}$. Let
$\mathbb{H}=\\{(E,F,G)\,|\,EG-F^{2}=1,E>0\\}$ be the path space. We
parametrize $\mathbb{H}$ by the upper half-plane
$\mathbb{R}^{2}_{+}=\\{(a,b)\,|\,b>0\\}$,

$(a,b)\mapsto{1\over b}(1,-a,a^{2}+b^{2}).$ (65)

Hooke ellipses of area $\pi$ are then given by equations of the form

$x^{2}-2axy+(a^{2}+b^{2})y^{2}=b,\quad a,b\in\mathbb{R},b>0.$ (66)

Assuming $y=y(x)$ in this equation and taking two derivatives with respect to
$x$, we get

$\displaystyle x-a(y+xy^{\prime})+(a^{2}+b^{2})yy^{\prime}=0,$ $\displaystyle
1-a(2y^{\prime}+xy^{\prime\prime})+(a^{2}+b^{2})\left[(y^{\prime})^{2}+yy^{\prime\prime}\right]=0.$

Eliminating $a,b$ from the last 3 equations and solving for
$y^{\prime\prime}$, we obtain

$y^{\prime\prime}=(xy^{\prime}-y)^{3}.$

Another proof, more direct, consists of showing that Hooke ellipses of area
$\pi$ are the (unparametrized) geodesics of a Riemannian metric in
$\mathbb{R}^{2}\setminus\\{0\\}$, given in polar coordinates by
$ds^{2}={dr^{2}}/\Delta^{2}+{r^{2}d\theta^{2}}/\Delta,$
$\Delta=1+cr^{2}+r^{4},$ $c\in\mathbb{R}$.

See [3] for yet another proof, via equivalence with the path geometry of
Kepler ellipses of fixed major axis, which is projective since these are
geodesics of the Jacobi-Maupertuis metric of the Kepler problem. $\square$

##### Fefferman metric.

Let $L_{1}\subset TM$ be the tangents to the fibers of the projection on the
first component, $(q,\mathcal{E})\mapsto q$, and similarly for $L_{2}$. The
group ${\mathrm{SL}_{2}(\mathbb{R})}$ acts transitively and freely on $M$ via
its standard linear action on $\mathbb{R}^{2}$, preserving $L_{1},L_{2}$.
Fixing a point $(q_{0},\mathcal{E}_{0})\in M$ identifies $M$ with
${\mathrm{SL}_{2}(\mathbb{R})}$, and $L_{1},L_{2}$ with two left-invariant
line fields on ${\mathrm{SL}_{2}(\mathbb{R})}$, given at
$\mathrm{id}\in{\mathrm{SL}_{2}(\mathbb{R})}$ by the Lie algebras of the
stabilizers of $q_{0},\mathcal{E}_{0}$, respectively.

The Lie algebra $\mathfrak{sl}_{2}(\mathbb{R})$ of
${\mathrm{SL}_{2}(\mathbb{R})}$ consists of matrices of the form

$\left(\begin{array}[]{rr}x^{1}&x^{2}\\\ x^{3}&-x^{1}\end{array}\right),\quad
x^{i}\in\mathbb{R}.$

The left-invariant $\mathfrak{sl}_{2}(\mathbb{R})$-valued Maurer–Cartan form
on ${\mathrm{SL}_{2}(\mathbb{R})}$ is

$\Theta=g^{-1}\mathrm{d}g=\left(\begin{array}[]{rr}\theta^{1}&\theta^{2}\\\
\theta^{3}&-\theta^{1}\end{array}\right).$ (67)

The Maurer–Cartan equation $\mathrm{d}\Theta=-\Theta\wedge\Theta$ gives

$\mathrm{d}\theta^{1}=-\theta^{2}\wedge\theta^{3},\
\mathrm{d}\theta^{2}=-2\theta^{1}\wedge\theta^{2},\
\mathrm{d}\theta^{3}=2\theta^{1}\wedge\theta^{3}.$ (68)

Fix $q_{0}:=(1,0)^{t}$, $\mathcal{E}_{0}:=\\{x^{2}+y^{2}=1\\}$. Then

$(L_{1})_{\mathrm{id}}=\mathrm{Span}\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right),\
(L_{2})_{\mathrm{id}}=\mathrm{Span}\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right),\ D=L_{1}\oplus L_{2}={\rm Ker}(\theta^{1}).$

An adapted coframe is thus

$\eta^{1}=\theta^{2}+\theta^{3},\quad\eta^{2}=\theta^{3},\quad\eta^{3}:=-\theta^{1}.$

We use this coframe to trivialize the associated $\mathbb{R}^{*}$-structure
$B\simeq{\mathrm{SL}_{2}(\mathbb{R})}\times\mathbb{R}^{*}$ and put the
standard coordinate $s$ on the $\mathbb{R}^{*}$ factor. The associated 1-forms
on $B$ are

$\omega^{1}={1\over
s}(\theta^{2}+\theta^{3}),\quad\omega^{2}=s\theta^{3},\quad\omega^{3}=-\theta^{1}.$

Solving the structure equations (33)-(34), we get
$\alpha=2\theta^{1}+\theta^{4},\ a_{1}=4/s^{2},\ a_{2}=0,\ K=-2,$ where
$\theta^{4}=(\mathrm{d}s)/s$ (the MC form on $\mathbb{R}^{*}$), which gives,
using equations (35)–(36), $\sigma=-\theta^{1}-(2/3)\theta^{4}$ and

$\mathrm{g}=(\theta^{1})^{2}+(\theta^{3})^{2}+\theta^{2}\theta^{3}+\frac{2}{3}\theta^{1}\theta^{4}.$
(69)

Hooke chains (null geodesics of the Fefferman metric). The pseudo-Riemannian
metric (69) is a left-invariant metric on the Lie group
$G:={\mathrm{SL}_{2}(\mathbb{R})}\times\mathbb{R}^{*}.$ Let
$\mathfrak{g}=\mathfrak{sl}_{2}(\mathbb{R})\times\mathbb{R}$ be its Lie
algebra and $A:\mathfrak{g}\to\mathfrak{g}^{*}$ the ‘inertia’ operator
corresponding to the quadratic form (69); that is, $\mathrm{g}(X,Y)=(AX)Y,$
$X,Y\in\mathfrak{g}$. Then

$A={1\over 6}\left(\begin{array}[]{cccc}6&0&0&2\\\ 0&0&3&0\\\ 0&3&6&0\\\
2&0&0&0\\\ \end{array}\right)$

(with respect to the basis $\\{\theta^{i}\\}$ and its dual). As in previous
examples, the geodesic flow on $T^{*}G$ projects to
$\dot{P}=\mathrm{ad}_{A^{-1}P}^{*}P$ on $\mathfrak{g}^{*}$, the Hamiltonian
equations with respect to the standard Lie-Poisson structure on
$\mathfrak{g}^{*}$ with Hamiltonian $H={1\over 2}(P,A^{-1}P).$ To write these
down explicitly, we first represent $X\in\mathfrak{g}$ and
$\mathrm{ad}_{X}^{*}\in{\rm End}(\mathfrak{g}^{*})$ by the matrices

$X=\left(\begin{array}[]{ccc}x^{1}&x^{2}&0\\\ x^{3}&-x^{1}&0\\\
0&0&x^{4}\end{array}\right),\qquad\mathrm{ad}_{X}^{*}=\left(\begin{array}[]{cccc}0&-2x^{2}&2x^{3}&0\\\
-x^{3}&2x^{1}&0&0\\\ x^{2}&0&-2x^{1}&0\\\ 0&0&0&0\\\ \end{array}\right),$

so $\dot{P}=\mathrm{ad}_{A^{-1}P}^{*}P$ becomes

$\displaystyle\begin{split}\dot{P}_{1}&=8(P_{2})^{2},\quad\dot{P}_{2}=2P_{2}(3P_{4}-P_{1}),\\\
\dot{P}_{3}&=2P_{1}(P_{3}-2P_{2})-6P_{3}P_{4},\quad\dot{P}_{4}=0,\end{split}$
(70)

with constants of motion $P_{4},k,H$, where

$k=(P_{1})^{2}+4P_{2}P_{3},\quad
H=2P_{2}P_{3}-2(P_{2})^{2}+3P_{1}P_{4}-9P_{4}^{2}/2=0.$ (71)

Note that $k$ is a Casimir of $\mathfrak{g}^{*}$ coming from the Killing form
of $\mathfrak{sl}_{2}(\mathbb{R})$. We set $H=0$ since we are looking for null
geodesics. Next we make the following change of variables

$P_{1}=b(c+\sin\phi),\quad P_{2}={b\over 2}\cos\phi,\quad
P_{3}=b\left(\cos\phi-{p\over 2}\right),\quad P_{4}={bc\over 3}.$

We have $k-2H=b^{2}$, hence $b$ is constant. Since $P_{4}=bc/2$ is constant
$c$ is constant as well. Equations (70)-(71) then reduce to

$\dot{\phi}=2b\cos\phi,\quad p=\cos\phi+c(c+2\sin\phi)\sec\phi.$ (72)

Next let $g(t)\in{\mathrm{SL}_{2}(\mathbb{R})}\times\mathbb{R}^{*}$ be a null
geodesic, with

$g(t)=\left(\begin{array}[]{ccc}x&z&0\\\ y&w&0\\\
0&0&s\end{array}\right),\quad x,y,z,w,s\in\mathbb{R},\ s\neq 0,\ xw-yz=1.$

Let $X=g^{-1}\dot{g}\in\mathfrak{g}$. Then $P=AX$ satisfies equations (70).
Explicitly,

$\displaystyle\dot{x}=x^{1}x+x^{3}z=b[cx+(\cos\phi)z],\
\quad\dot{z}=x^{2}x-x^{1}z=-b[p\,x+cz],$
$\displaystyle\dot{y}=x^{1}y+x^{3}w=b[cy+(\cos\phi)w],\quad\dot{w}=x^{2}y-x^{1}w=-b[p\,y+cw],$

where $p,\phi$ are given by equation (72). Denote ${\mathbf{r}}:=(x,y),\
\mathbf{h}:=(z,w)\in\mathbb{R}^{2}$, then the last system is

$\dot{\mathbf{r}}=b\left[c{\mathbf{r}}+(\cos\phi)\mathbf{h}\right],\quad\dot{\mathbf{h}}=-b\left[p{\mathbf{r}}+c\mathbf{h}\right].$
(73)

######  Lemma 4.7.

$\phi$ is twice the centro-affine arclength of the projection of the chain to
the Hooke plane (the ${\mathbf{r}}$ plane).

###### Proof.

${\mathbf{r}},\mathbf{h}$ are the columns of a matrix in
${\mathrm{SL}_{2}(\mathbb{R})}$, hence $[{\mathbf{r}},\mathbf{h}]=1$. It then
follows from equations (73) that
$[{\mathbf{r}},\mathrm{d}{\mathbf{r}}/\mathrm{d}\phi]=[{\mathbf{r}},\dot{\mathbf{r}}/\dot{\phi}]=[{\mathbf{r}},\mathbf{h}/2]=1/2.$
$\square$

Let us reparametrize the chains by $\tau:=\phi/2$ (the centro-affine
arclength) and denote derivative with respect to $\tau$ by $(\ )^{\prime}$.
Equations (73) now become

$\displaystyle\begin{split}&{\mathbf{r}}^{\prime}=c(\sec
2\tau){\mathbf{r}}+\mathbf{h},\\\ &\mathbf{h}^{\prime}=-\left[1+c(c+2(\sin
2\tau))\sec^{2}2\tau\right]{\mathbf{r}}-c(\sec 2\tau)\mathbf{h}.\end{split}$
(74)

######  Lemma 4.8.

${\mathbf{r}}^{\prime\prime}=-{\mathbf{r}}.$

###### Proof.

Straightforward calculation from equations (74). $\square$

Thus, combined with $[{\mathbf{r}},{\mathbf{r}}^{\prime}]=1$ (Lemma 4.7), each
Hooke chain projects to a Hooke ellipse of area $\pi$ in the ${\mathbf{r}}$
plane, as expected from Proposition 4.6 and Theorem 1.

######  Proposition 4.9.

Every chain in ${\mathrm{SL}_{2}(\mathbb{R})}$ of the path geometry of Hooke
ellipses of area $\pi$, up to left translation, is of the form

${\mathbf{r}}=e^{i\tau},\quad\mathbf{h}=e^{i\tau}(-c\sec(2\tau)+i)$

(using complex notation), for some $c\in\mathbb{R}$, $c\neq 0$. See Figure 4.

###### Proof.

${\mathrm{SL}_{2}(\mathbb{R})}$ acts transitively on Hooke ellipses of area
$\pi$, hence the projection of the chain to the ${\mathbf{r}}$ plane can be
brought to the unit circle. Parametrized by centro affine arc length, it is
${\mathbf{r}}=e^{i\tau}$. Then the 1st equation of (74) implies the formula
for $\mathbf{h}(\tau).$ For $c=0$ this formula produces a curve tangent to the
contact distribution, which is excluded. $\square$

Figure 4: Hooke’s chains, given by Proposition 4.9, for $c=-1,1,2$.

###  4.4 Horocycles in the hyperbolic plane

The space of Hooke ellipses is $\mathbb{H}=\\{(E,F,G)\,|\,EG-F^{2}=1,E>0\\}$,
the hyperboloid model of the hyperbolic plane. The curves in $\mathbb{H}$ of
constant (hyperbolic) curvature 1 are called horocycles and are the sections
of $\mathbb{H}$ by planes parallel to a generator of the cone $EG-F^{2}=0$. In
the upper half-plane model these are (Euclidean) circles tangent to the real
axis.

######  Lemma 4.10.

For each fixed $(x,y)\in\mathbb{R}^{2}\setminus\\{0\\}$, the set of Hooke
ellipses passing through $(x,y)$ is a horocycle in $\mathbb{H}$. This defines
a bijection between the punctured plane $\mathbb{R}^{2}\setminus\\{0\\}$ and
the space of horocycles in $\mathbb{H}$.

###### Proof.

For each $(x,y)\in\mathbb{R}^{2}\setminus 0$, equation (66),

$x^{2}-2axy+(a^{2}+b^{2})y^{2}=b,$

defines in the upper half-plane $\\{(a,b)\,|\,b>0\\}$ either the circle of
radius ${1\over 2y^{2}}$ centered at $\left({x\over y},{1\over 2y^{2}}\right)$
if $y\neq 0$, or the horizonal line $b=x^{2}$ if $y=0.$ These are precisely
all the horocycles of the upper half plane model of the hyperbolic plane.
$\square$

It follows that the horocycle path geometry in $\mathbb{H}$ is dual to the
path geometry in $\mathbb{R}^{2}\setminus\\{0\\}$ of Hooke ellipses of fixed
area. Thus we can use the analysis of the previous section to determine the
projection of the chains to $\mathbb{H}$.

######  Proposition 4.11.

Each chain of the horocycle path geometry, up to the action of
${\mathrm{SL}_{2}(\mathbb{R})}$, projects to a curve in the hyperbolic plane,
given in the upper half-plane model $\\{(x,y)\,|\,y>0\\}$ by

$(x^{2}+y^{2})^{2}-[4cx+(c^{2}+4)y](x^{2}+y^{2})+(6c^{2}-2)x^{2}+2c^{3}xy+6y^{2}\\\
-4c(c^{2}-1)x-(c^{4}-3c^{2}+4)y+(c^{2}-1)^{2}=0$ (75)

where $c\neq 0.$ See Figure 5.This curve is the projection of a chain in
${\mathrm{SL}_{2}(\mathbb{R})}$, the solution to equations (74) that passes
through $\mathrm{id}\in{\mathrm{SL}_{2}(\mathbb{R})}$. The projection of this
chain to the Hooke plane is the Hooke ellipse $(x-cy)^{2}+y^{2}=1$. The
horocycles along this chain, in the upper half plane model, all pass through
$(c,1)$, the point corresponding to this Hooke ellipse. The chains
corresponding to $c$ and $-c$ are congruent via an outer automorphism of
${\mathrm{SL}_{2}(\mathbb{R})}$ (conjugation by
$\mathrm{diag}(-1,1)\in\mathrm{GL}_{2}(\mathbb{R})$), acting by reflection
about the $y$-axis.

Figure 5: Horocycle chains, given by Proposition 4.11, projected to the
hyperbolic plane (the upper half plane model), are rational bicircular
quartics. Left: crunodal (one node), $|c|<2.$ Middle: cuspidal (one cusp),
$c=2$. Right: acnodal (smooth), $|c|>2$.

###### Proof.

Using ${\mathbf{r}}^{\prime\prime}=-{\mathbf{r}}$ and
${\mathbf{r}}^{\prime}=c(\sec 2\tau){\mathbf{r}}+\mathbf{h}$ (Lemma 4.8 and
equation (74)), the chain $g(\tau)$ in ${\mathrm{SL}_{2}(\mathbb{R})}$ with
$g(0)=\mathrm{id}$ is

$g(\tau)=\left(\begin{array}[]{cc}\cos\tau+c\sin\tau&-\sin\tau\left(\sec(2\tau)c^{2}+\tan(2\tau)c+1\right)\\\
\sin\tau&\cos\tau-c\sec(2\tau)\sin\tau\\\ \end{array}\right).$ (76)

The projection of this chain to $\mathbb{H}$ is obtained by acting by
$g(\tau)$ on the point in $\mathbb{H}$ corresponding to the Hooke ellipse
$\mathcal{E}_{0}=\\{x^{2}+y^{2}=1\\}.$ One can check that the parametrization
of $\mathbb{H}$ by the upper half-plane in equation (65) is
${\mathrm{SL}_{2}(\mathbb{R})}$-equivariant, so one can act instead by
$g(\tau)$ via fractional linear transformations on $(0,1)$, the point in the
upper half-plane corresponding to $\mathcal{E}_{0}$. Reverting to
$\phi=2\tau$, the outcome is

$(x,y)={\left(c^{2}\left[(c+2\sin\phi)\cos\phi-c-\sin\phi\right],\
-2\cos^{2}\phi\right)\over-2\cos^{2}\phi+c(c+2\sin\phi)\cos\phi-c^{2}.}$

Eliminating $\phi$ in the above equation (we used Maple for this), one obtains
equation (75). $\square$

######  Remark 4.12.

The curves of Proposition 4.11 are examples of bicircular quartics, a
remarkable class of plane curves introduced by J. Casey in 1871 [8]. They have
many equivalent geometric and algebraic definitions, the simplest being the
inversion of a conic (with respect to a circle).

## References

  * [1] V. I. Arnol’d. Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd edition, Springer-Verlag, New York, 1988.
  * [2] V. I. Arnol’d, A. B. Givental. Symplectic geometry, Encyclopaedia of Mathematical Sciences (Dynamical systems IV), vol. 4. (1990), 1–136.
  * [3] G. Bor, H. Jacobowitz. Left-invariant CR structures on 3-dimensional Lie groups. Complex Anal. Synerg. 7, 23 (2021).
  * [4] G. Bor, C. Jackman. Revisiting Kepler: new symmetries of an old problem. Preprint (2021). https://arxiv.org/abs/2106.02823
  * [5] D. Burns Jr., K. Diederich, S. Shnider. Distinguished curves in pseudoconvex boundaries, Duke Math. J. 44.2 (1977), 407–431.
  * [6] A. Čap, J. Slovák. Parabolic Geometries I: Background and General Theory, Math. Surv. and Monographs 154, Amer. Math. Soc. (2009).
  * [7] A. Čap, V. Žádník. On the geometry of chains, J. Differential Geom. 82 (2009), 1–33.
  * [8] J. Casey. On bicircular quartics, The Transactions of the Royal Irish Academy 24 (1871), 457–569.
  * [9] E. Cartan. Sur les variétés à connexion projective, Bull. Soc. Math. France 52 (1924), 205–241.
  * [10] E. Cartan. Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes. Part I: Ann. Math. Pura Appl. 11.4 (1932), 17–90. Part II: Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 2e série 1.4 (1932), 333–354.
  * [11] A. Castro, R. Montgomery. The chains of left-invariant Cauchy-Riemann structures on $SU(2)$, Pacific Journal of Mathematics, 238 (2008), 41–71.
  * [12] J. H. Cheng, T. Marugame, V. S. Matveev, R. Montgomery. Chains in CR geometry as geodesics of a Kropina metric. Advances in Mathematics 350 (2019),973–999.
  * [13] B. Doubrov, B. Komrakov. The geometry of second-order ordinary differential equations. Preprint (2016). https://arxiv.org/abs/1602.00913
  * [14] J. Douglas. The general geometry of paths, Ann. of Math. 29 (1928), 143–168.
  * [15] F. A. Farris. An intrinsic construction of Fefferman’s CR metric, Pacific J. Math. 123.1 (1986), 33–45.
  * [16] C. L. Fefferman. Monge-Ampére equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. 2103:2 (1976), 395–416.
  * [17] K. Hughen. The geometry of subriemannian 3-manifolds, PhD Thesis (1995). https://pdfs.semanticscholar.org/4069/84ef45565eae1bfe4241e70eb3ed8b60f88b.pdf
  * [18] T. A. Ivey, J. M. Landsberg. Cartan for beginners: differential geometry via moving frames and exterior differential systems. Vol. 61. Providence, RI: American Mathematical Society, 2003.
  * [19] H. Jacobowitz. An introduction to CR structures. No. 32. American Mathematical Soc., 1990.
  * [20] B. Kruglikov. Point classification of second order ODEs: Tresse classification revisited and beyond, in: Differential Equations-Geometry, Symmetries and Integrability. Springer, Berlin, Heidelberg, 2009. 199-221.
  * [21] J.M. Lee, The Fefferman metric and pseudo-Hermitian invariants, Trans. Amer. Math. Soc. 296.1 (1986), 411–429.
  * [22] P. Nurowski, G. Sparling. Three-dimensional Cauchy-Riemann structures and second-order ordinary differential equations, Class. Quantum Grav. 20 (2003), 4995–5016.
  * [23] G. Pastras. Four Lectures on Weierstrass Elliptic Function and Applications in Classical and Quantum Mechanics. Preprint 2017. https://arxiv.org/abs/1706.07371
  * [24] A. M. L. Tresse. Détermination des invariants ponctuels de l’équation difféentielle ordinaire du second ordre $y^{\prime\prime}\omega(x,y,y^{\prime})$, Hirzel, Leipzig, 1896.
  * [25] S. Webster. Pseudo-hermitian Structures on a Real Hypersurface, J. Differential Geom. 13 (1978), 25–41.