# Piecewise circular curves and Positivity

Jean-Philippe Burelle  and  Ryan Kirk

###### Abstract.

We introduce the moduli space of generic piecewise circular $n$-gons in the
Riemann sphere and relate it to a moduli space of Legendrian polygons. We
prove that when $n=2k$, this moduli space contains a connected component
homeomorphic to the Fock-Goncharov space of $k$-tuples of positive flags for
$\mathsf{PSp}(4,\mathbb{R})$ and hence is a topological ball. We characterize
this component geometrically as the space of simple piecewise circular curves
with decreasing curvature.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme (ERC
starting grant DiGGeS, grant agreement No 715982). The first author
acknowledges the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC), [funding reference number RGPIN-2020-05557]

##  1\. Introduction

A _piecewise circular curve_ is a curve in the plane made of finitely many
arcs of circles, such that tangents agree at the intersection of pieces. We
will call a closed piecewise circular curve made of $n$ circular segments a
circular $n$-gon. Since Möbius transformations map circular arcs to circular
arcs or line segments, it makes sense to consider piecewise circular curves in
the Riemann sphere up to Möbius transformations, allowing circular and linear
pieces.

A co-orientation on a piecewise circular curve is a continuous choice of
perpendicular orientation on the curve. There is a natural map on the set of
co-oriented piecewise circular curves which moves each point on the curve a
fixed distance $d$ in the direction of the co-orientation. We will call this
map the radial translation with parameter $d$. If the piecewise circular curve
is interpreted as a wavefront, this map models wave propagation with $d$ being
the time parameter.

We consider the natural problem of classifying co-oriented piecewise circular
curves up to Möbius transformations and radial translations.

Piecewise circular curves have been previously been investigated, especially
in the setting of curve approximation ([BG93], [BG94], [Ros99]) but as far as
we know this classification problem has not been considered in the literature.

Since the radial translation may introduce singularities by collapsing a
circular arc to a point, we allow such singularities in our piecewise circular
curves. Keeping track of the co-orientation, collapsed circular arcs are not
degenerate as curves in the space of oriented contact elements of the sphere,
and the action of radial translations and Möbius transformation is well-
defined.

Figure 1. A simple co-oriented piecewise circular quadrilateral and its images
under radial translation. The central figure shows a collapsed circular arc.

Even though we cannot compare the radii of any two co-oriented circles in the
Riemann sphere, there is a partial cyclic order coming from the Maslov index
which detects when three co-oriented circles are nested, and is invariant
under Möbius transformations and radial translations. When the collection of
circles extending the edges of a co-oriented piecewise circular curve is
cyclically ordered, we will say that this curve has decreasing curvature. This
property means that piecewise circular curves with decreasing curvature are
locally spirals.

We say that a co-oriented piecewise circular curve is _generic_ if non-
adjacent vertices are never part of a common co-oriented circle, and non-
adjacent edges are never arcs of tangent co-oriented circles. We say it is
_simple_ if _all_ of its radial translates have no self-intersections. This is
equivalent to there being no co-oriented circle which is tangent to two non-
adjacent edges (see Prop. 4.10).

We define the moduli spaces rigorously and make the following simple
observations.

######  Proposition 1.1.

The moduli space of generic circular $n$-gons with $n\geq 5$ is a smooth
$2(n-5)$-dimensional manifold.

######  Proposition 1.2.

The moduli space of simple circular $4$-gons consists of two points.

######  Proposition 1.3.

As a curve in the space of oriented contact elements to the $2$-sphere, a
simple piecewise circular polygon is non-contractible.

A related moduli space, the space of _$(2,n)$ -Lagrangians configurations_ in
$\mathbb{R}^{4}$ was investigated in [CO19] and [MG19]. The precise relation
is:

######  Proposition 1.4.

The moduli space of simple circular $n$-gons is a two-to-one trivial cover of
the space $\mathcal{L}(2,n)$ of generic configurations of $n$ Lagrangians in
$\mathbb{R}^{4}$.

Among generic circular $2k$-gons, there is a particular class consisting of
curves which are simple and have decreasing curvature. Our main result is:

######  Theorem 1.5.

Let $k\geq 1$. In the moduli space of generic circular $2k$-gons, the subspace
of simple curves with decreasing curvature is a connected component
homeomorphic to an open ball.

We also show that for hexagons, the curvature condition is not necessary, a
result which was proved in the second author’s Ph.D thesis [Kir19]:

######  Proposition 1.6.

A simple circular hexagon either has increasing or decreasing curvature.

The strategy of the proof is to construct a homeomorphism between the moduli
space of pairwise transverse $k$-tuples of $\mathsf{PSp}(4,\mathbb{R})$ flags
and the moduli space of generic circular $2k$-gons. Under this homeomorphism,
positive $k$-tuples of flags are in bijection with positive circular
$2k$-gons. Fock and Goncharov showed in [FG06] that the subspace of positive
$k$-tuples of flags is a connected component homeomorphic to a ball, and so
the theorem follows.

Positive $k$-tuples of flags for the group $\mathsf{PSL}(3,\mathbb{R})$ also
parameterize a natural moduli space of geometric objects: pairs of nested
convex polygons in the projective plane [FG07]. Our result provides a similar
interpretation for positive $k$-tuples of flags in
$\mathsf{PSp}(4,\mathbb{R})$. Both can be considered toy models of higher
Teichmüller spaces.

As a consequence of the techniques used to prove the main theorem, we also
obtain the following result about positive curves and Hitchin representations
of surface groups into $\mathsf{PSp}(4,\mathbb{R})$:

######  Theorem 1.7.

Positive Frenet $\mathsf{PSp}(4,\mathbb{R})$ flag curves are lifts of simple
closed curves with decreasing curvature on the sphere.

######  Corollary 1.8.

Let $\Sigma$ be a closed surface and
$\rho:\pi_{1}(\Sigma)\rightarrow\mathsf{PSp}(4,\mathbb{R})$ a Hitchin
representation. Then, the limit curve of $\rho$ is the contact lift of a
simple closed curve with decreasing curvature on the sphere.

This corollary gives an alternative geometric interpretation of the
$\mathsf{PSp}(4,\mathbb{R})$ Hitchin component to the one presented in [GW08].
Guichard and Wienhard interpret the Hitchin component as the moduli space of
convex-foliated contact projective structures on the tangent bundle of
$\Sigma$. Corollary 1.8 gives an interpretation of this component as the
moduli space of $\pi_{1}(\Sigma)$-equivariant simple closed curves with
decreasing curvature on the $2$-sphere.

We now describe the structure of the paper. After establishing some notation,
in Section 3 we introduce Legendrian polygons and their moduli spaces. In
Section 4 we develop the dictionary between the contact geometry of projective
$3$-space and the geometry of circles on the $2$-sphere. Most of these results
are known and have been described using the perspective of the Lie group
$\mathsf{SO}^{0}(3,2)$ (for instance in [Cec08]). For our purposes, the group
$\mathsf{PSp}(4,\mathbb{R})$ which is isomorphic to $\mathsf{SO}^{0}(3,2)$ is
more convenient and so we develop _Lie circle geometry_ from that point of
view in a coordinate-free way. Finally, in Section 5 we recall the notion of
positivity of a configuration of flags and prove the main theorem and describe
the applications to higher Teichmüller theory.

##  2\. Notation

When working with indexed objects $A_{i}$ with a natural cyclic structure (for
instance vertices of a polygon) we will consider indices from the set
$\\{1,2,\dots,n\\}$ _modulo $n$_, meaning for instance that $A_{n+1}=A_{1}$
and $A_{n+2}=2$. When writing inequalities of the form $i<j$ for those
indices, we use the unique representatives in $\\{1,2,\dots,n\\}$.

If $\mathsf{V}$ is a vector space over a field $k$, we denote by
$\mathbb{P}(\mathsf{V})$ the associated projective space. If
$\mathsf{U}\subset\mathsf{V}$ is a vector subspace, we also denote by
$\mathbb{P}(\mathsf{U})\subset\mathbb{P}(\mathsf{V})$ the corresponding
projective subspace. Over $\mathbb{R}$, we also define the _sphere of
directions_ $\mathbb{S}(\mathsf{V}):=(\mathsf{V}-\\{0\\})/\mathbb{R}^{+}$ and
we use the same notation $\mathbb{S}(E)$ for the sphere bundle associated to a
real vector bundle $E\rightarrow M$ over a manifold $M$. If $\mathsf{V}$ is a
vector space over two different fields, we will use
$\mathbb{P}_{k}(\mathsf{V})$ to specify which field we are considering. We use
bold variables $\mathbf{v}\in\mathsf{V}$ to denote elements of $\mathsf{V}$
and brackets $[\mathbf{v}]_{F}\in\mathbb{P}_{F}(\mathsf{V})$ to denote
projective equivalence classes.

A _symplectic form_ $\omega$ on $\mathsf{V}$ is a skew-symmetric bilinear
form. When $\mathsf{V}$ is equipped with a symplectic form, we will call it a
symplectic vector space. Let $S\subset\mathsf{V}$ be a subset. Then,

$S^{\perp}=\\{\mathbf{v}\in\mathsf{V}~{}|~{}\omega(\mathbf{v},\mathbf{u})=0,~{}\forall\mathbf{u}\in
S\\}$

denotes the orthogonal subspace to $S$.

A Lagrangian subspace $L\subset V$ is a subspace such that $L^{\perp}=L$. The
Lagrangian Grassmannian $\mathsf{Lag}(\mathsf{V})$ is the space of Lagrangian
subspaces in $\mathsf{V}$. When a basis of $\mathsf{V}$ is fixed and
$\dim(\mathsf{V})=2n$, we will use $2n\times n$ matrices to denote elements of
$\mathsf{Lag}(\mathsf{V})$, understood as maps
$\mathbb{R}^{n}\rightarrow\mathsf{V}$. This notation is unique up to right-
multiplication by an invertible $n\times n$ matrix.

##  3\. Legendrian Polygons

Let $(\mathsf{V},\omega)$ be a $4$-dimensional symplectic vector space over
$\mathbb{R}$. A _Lagrangian_ in $\mathsf{V}$ is a $2$-dimensional subspace on
which the symplectic form $\omega$ vanishes.

Orthogonality with respect to $\omega$ defines a hyperplane distribution
$\mathbf{v}^{\perp}\subset T_{\mathbf{v}}\mathsf{V}$. This distribution
projects to a contact structure on the projective space
$\mathbb{P}(\mathsf{V})$, making it a contact manifold.

######  Definition 3.1.

Two points $p,q\in\mathbb{P}(\mathsf{V})$ will be called _incident_ whenever
$p\in\mathbb{P}(q^{\perp})$, or equivalently $q\in\mathbb{P}(p^{\perp})$.

A path in $\mathbb{P}(\mathsf{V})$ which is always tangent to the contact
distribution is called _Legendrian_.

######  Definition 3.2.

A _Legendrian polygon_ in $\mathbb{P}(\mathsf{V})$ is the image of a closed,
piecewise linear path such that each segment is Legendrian, with labeled
vertices.

We will assume that all Legendrian polygons are nondegenerate in the following
sense: no two adjacent segments are part of the same projective line and no
segment is degenerated to a point.

Let $(\mathbf{v}_{1},\dots,\mathbf{v}_{n})$ be nonzero vectors in $\mathsf{V}$
such that $\mathbf{v}_{k},\mathbf{v}_{k+1},\mathbf{v}_{k+2}$ are linearly
independent, and $\omega(\mathbf{v}_{k},\mathbf{v}_{k+1})=0$ for $1\leq k\leq
n$. We define two Legendrian polygons
$P_{\pm}(\mathbf{v}_{1},\dots,\mathbf{v}_{n})$ with $n$ segments
$\gamma_{1},\dots,\gamma_{n}$ given by:

$\gamma_{k}(t)=[(1-t)\mathbf{v}_{k}+t\mathbf{v}_{k+1}]$

for $0\leq t\leq 1$ and $1\leq k\leq n-1$ and

$\gamma_{n}(t)=[(1-t)\mathbf{v}_{n}\pm t\mathbf{v}_{1}].$

Any Legendrian polygon can be parametrized this way, and the ambiguity in the
choice of representatives $\mathbf{v}_{1},\dots,\mathbf{v}_{n}$ is
multiplication of each $\mathbf{v}_{j}$ by a positive scalar $\lambda_{j}>0$
and simultaneously multiplying all $\mathbf{v}_{j}$ by $-1$.

######  Proposition 3.3.

For any nonzero vectors $\mathbf{v}_{1},\dots,\mathbf{v}_{n}\in\mathsf{V}$,
the polygon $P_{+}(\mathbf{v}_{1},\dots,\mathbf{v}_{n})$ is contractible, and
the polygon $P_{-}(\mathbf{v}_{1},\dots,\mathbf{v}_{n})$ generates
$\pi_{1}(\mathbb{P}(\mathsf{V}))$.

###### Proof.

Let $[\mathbf{u}_{1}],\dots,[\mathbf{u}_{n}]$ be $n$ distinct points on a
projective line $\ell\subset\mathbb{P}(\mathsf{V})$ disjoint from the polygon,
placed in that order. Multiplying certain $\mathbf{u}_{k}$ by $-1$ if needed,
we may assume that the full line is parametrized by
$P_{-}(\mathbf{u}_{1},\dots,\mathbf{u}_{n})$.

For each vector $\mathbf{v}_{i}$, choose a path
$\mathbf{v}_{i}(s)\in\mathsf{V}-\\{0\\}$ such that
$\mathbf{v}_{i}(0)=\mathbf{v}_{i}$ and $\mathbf{v}_{i}(1)=\mathbf{u}_{i}$.
Perturbing the path if needed, $\mathbf{v}_{k}(s)$ and $\mathbf{v}_{k+1}(s)$
are linearly independent for all $s$. Then, the homotopy between
$P_{\pm}(\mathbf{v}_{1},\dots,\mathbf{v}_{n})$ and
$P_{\pm}(\mathbf{u}_{1},\dots,\mathbf{u}_{n})$

$\gamma_{k}(t,s):=[(1-t)\mathbf{v}_{k}(s)+t\mathbf{v}_{k+1}(s)]$
$\gamma_{n}(t,s):=[(1-t)\mathbf{v}_{n}(s)\pm t\mathbf{v}_{k+1}(s)]$

is well-defined. The path $P_{+}(\mathbf{u}_{1},\dots,\mathbf{u}_{n})$ is
contained in a line segment, hence contractible, and the path
$P_{-}(\mathbf{u}_{1},\dots,\mathbf{u}_{n})$ is a full projective line, hence
generates $\pi_{1}(\mathbb{P}(\mathsf{V}))$.

∎

######  Definition 3.4.

A Legendrian polygon $P$ is _generic_ if its non-adjacent vertices are non-
incident and its non-adjacent edges are parts of transverse projective lines.

The moduli space of generic Legendrian $n$-gons up to the action of
$\mathsf{PSp}(\mathsf{V})$ will be denoted by $\mathscr{P}_{n}$.

As a consequence of Proposition 3.3, $\mathscr{P}_{n}$ naturally separates
into a disjoint union of the space of generic contractible $n$-gons
$\mathscr{P}_{n}^{+}$ and the space of generic non-contractible $n$-gons
$\mathscr{P}_{n}^{-}$, and each subspace
$\mathscr{P}_{n}^{+},\mathscr{P}_{n}^{-}$ is a union of connected components.

######  Definition 3.5.

A Legendrian polygon $P$ is _transverse_ if no pair of points $p,q\in P$
belonging to distinct, non-adjacent closed segments is incident.

We first observe that there are no Legendrian triangles, so the first
interesting case of Legendrian polygon is $n=4$. This is because the vertices
of a Legendrian triangle would span a $3$-dimensional totally isotropic
subspace, which is impossible by nondegeneracy of $\omega$.

As a first example, in a basis
$\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}$ in which the
symplectic form $\omega$ is represented by the matrix

(1) $\Omega=\begin{pmatrix}0&0&0&1\\\ 0&0&-1&0\\\ 0&1&0&0\\\
-1&0&0&0\end{pmatrix},$

the Legendrian quadrilateral
$P_{-}(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{4},-\mathbf{e}_{3})$ is
generic and transverse. This is a consequence of the following lemma:

######  Lemma 3.6.

Let $\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{v}_{1},\mathbf{v}_{2}\in\mathsf{V}$
such that
$\omega(\mathbf{u}_{1},\mathbf{u}_{2})=\omega(\mathbf{v}_{1},\mathbf{v}_{2})=0$
and $\omega(\mathbf{u}_{1},\mathbf{v}_{2})\neq 0$. Then, the points
$[(1-t)\mathbf{u}_{1}+t\mathbf{u}_{2}]$ and
$[(1-s)\mathbf{v}_{1}+s\mathbf{v}_{2}]$ are non-incident for all $t,s\in(0,1)$
if and only if the four symplectic products
$\omega(\mathbf{u}_{1},\mathbf{v}_{1})$,
$\omega(\mathbf{u}_{1},\mathbf{v}_{2})$,
$\omega(\mathbf{u}_{2},\mathbf{v}_{1})$, and
$\omega(\mathbf{u}_{2},\mathbf{v}_{2})$ are all nonpositive or all
nonnegative.

###### Proof.

The points $[(1-t)\mathbf{u}_{1}+t\mathbf{u}_{2}]$ and
$[(1-s)\mathbf{v}_{1}+s\mathbf{v}_{2}]$ are incident if and only if
$\omega((1-t)\mathbf{u}_{1}+t\mathbf{u}_{2},(1-s)\mathbf{v}_{1}+s\mathbf{v}_{2})=0$,
which expands to

$(1-t)(1-s)\,\omega(\mathbf{u}_{1},\mathbf{v}_{1})+(1-t)s~{}\,\omega(\mathbf{u}_{1},\mathbf{v}_{2})+t(1-s)\,\omega(\mathbf{u}_{2},\mathbf{v}_{1})+ts\,\omega(\mathbf{u}_{2},\mathbf{v}_{2})=0.$

Suppose that the four symplectic products
$\omega(\mathbf{u}_{i},\mathbf{v}_{j})$ are all nonpositive or all
nonnegative. Then, since one of them is nonzero the above equation has no
solutions for $0<t,s<1$, and so $[(1-t)\mathbf{u}_{1}+t\mathbf{u}_{2}]$ and
$[(1-s)\mathbf{v}_{1}+s\mathbf{v}_{2}]$ are non-incident.

Conversely, note that for fixed values of $a,b,c,d$ the function

$\phi(t,s)=(1-t)(1-s)a+(1-t)s\,b+t(1-s)c+ts\,d$

satisfies $\phi(0,0)=a$, $\phi(0,1)=b$, $\phi(1,0)=c$, and $\phi(1,1)=d$.

If any two of $a,b,c,d$ have different signs we can, by applying the
intermediate value theorem to a path joining the vertices of the square
$[0,1]\times[0,1]$, find $(t,s)\in(0,1)\times(0,1)$ such that $\phi(t,s)=0$. ∎

######  Proposition 3.7.

The Legendrian polygon $P_{+}(\mathbf{v}_{1},\dots,\mathbf{v}_{n})$ is not
transverse for $n\geq 4$.

###### Proof.

Assume $P_{+}(\mathbf{v}_{1},\dots,\mathbf{v}_{n})$ is transverse. Then, by
Lemma 3.6 the symplectic products $\omega(\mathbf{v}_{i},\mathbf{v}_{j})$ have
the same sign for all $i<j-1$. Moreover, if $n>4$ the segment
$(1-t)\mathbf{v}_{3}+t\mathbf{v}_{4}$ is transverse to the segment
$(1-t)\mathbf{v}_{n}+t\mathbf{v}_{1}$, so
$\omega(\mathbf{v}_{3},\mathbf{v}_{n})$ has the same sign as
$\omega(\mathbf{v}_{3},\mathbf{v}_{1})=-\omega(\mathbf{v}_{1},\mathbf{v}_{3})$,
a contradiction. The case $n=4$ can be normalized to have the matrix of
symplectic products $\Omega$ (equation (1)) and the finitely many cases which
can be checked by hand are all non-transverse (see Prop. 3.11). ∎

Combining Propositions 3.7 and 3.3, we obtain

######  Corollary 3.8.

A transverse Legendrian polygon is non-contractible.

For non-contractible Legendrian polygons, we now give a simple criterion for
transversality.

######  Proposition 3.9.

The Legendrian polygon $P_{-}(\mathbf{v}_{1},\dots,\mathbf{v}_{n})$ is
transverse if and only if $\omega(\mathbf{v}_{i},\mathbf{v}_{j})>0$ for all
$i<j-1$, or $\omega(\mathbf{v}_{i},\mathbf{v}_{j})<0$ for all $i<j-1$.

###### Proof.

We start with the reverse direction. The transversality for segments
$(1-t)\mathbf{v}_{i}+t\mathbf{v}_{i+1}$ and
$(1-t)\mathbf{v}_{j}+t\mathbf{v}_{j+1}$ with $i<j-1$ follows at once from
Lemma 3.6. It remains to show that segments
$(1-t)\mathbf{v}_{i}+t\mathbf{v}_{i+1}$ are transverse to the final segment
$(1-t)\mathbf{v}_{n}-t\mathbf{v}_{1}$. This follows from another application
of Lemma 3.6 since
$\omega(\mathbf{v}_{i},-\mathbf{v}_{1})=\omega(\mathbf{v}_{1},\mathbf{v}_{i})$
and
$\omega(\mathbf{v}_{i+1},-\mathbf{v}_{1})=\omega(\mathbf{v}_{1},\mathbf{v}_{i+1})$.

For the forwards direction, if the polygon is transverse then the non-adjacent
vertices are non-incident and so $\omega(\mathbf{v}_{i},\mathbf{v}_{j})\neq 0$
for $i<j-1$. Using Lemma 3.6 we find that
$\omega(\mathbf{v}_{i},\mathbf{v}_{j})$ all have the same sign. ∎

The common sign of $\omega(\mathbf{v}_{i},\mathbf{v}_{j})$ for a transverse
Legendrian polygon $P_{-}(\mathbf{v}_{1},\dots,\mathbf{v}_{n})$ does not
depend on the choice of representatives for the vertices
$\mathbf{v}_{1},\dots,\mathbf{v}_{n}$. Indeed, multiplication by a positive
scalar does not change this sign, and simultaneously changing the sign of all
$\mathbf{v}_{j}$ also does not change this sign. We will call the
corresponding classes of Legendrian polygons _positive-transverse_ and
_negative-transverse_ according to this sign. Reversing the cyclic order
exchanges positive-transverse and negative-transverse polygons.

We now recall the definition of the space of generic Lagrangian configurations
from [CO19] and relate it to the space of transverse Legendrian polygons.

######  Definition 3.10.

The moduli space of generic Lagrangian configurations $\mathcal{L}_{2,n}$ is
the space of $n$-tuples $(p_{1},\dots,p_{n})$ in $\mathbb{P}(\mathsf{V})$ such
that a pair $p_{i},p_{j}$ is incident if and only if it is cyclically
adjacent, up to the action of $\mathsf{PSp}(\mathsf{V})$.

The projection sending a transverse Legendrian $n$-gon to its tuple of
vertices is two-to-one, each fiber consisting of a polygon with positive
symplectic products of vertices and one with negative products. It is
surjective onto $\mathcal{L}_{2,n}$ since is it always possible to choose
representative vectors $(\mathbf{v}_{1},\dots,\mathbf{v}_{n})$ for the points
$(p_{1},\dots,p_{n})\in\mathcal{L}_{2,n}$ such that
$\omega(\mathbf{v}_{i},\mathbf{v}_{j})>0$ for all $1<i+1<j\leq n$. Then, the
resulting polygon $P_{-}(\mathbf{v}_{1},\dots,\mathbf{v}_{n})$ is transverse
and projects to $(p_{1},\dots,p_{n})$.

As a consequence of Proposition 2.7 of [CO19], these two moduli spaces are
smooth manifolds of dimension $2(n-5)$ when $n\geq 5$.

If we consider instead generic Legendrian $n$-gons, then the map
$\mathscr{P}_{n}\rightarrow\mathcal{L}_{2,n}$ is no longer surjective, but it
is a $2^{n}$-sheeted covering map of its image which is an open submanifold.
We conclude that $\mathscr{P}_{n}$ is a manifold of dimension $2(n-5)$.

######  Proposition 3.11.

The moduli space of transverse Legendrian $4$-gons is a pair of points.

###### Proof.

Let $P$ be a Legendrian $4$-gon. Then, up to rescaling, its vertices form a
basis in which the symplectic form is the standard

$\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\ -1&0&0&0\\\ 0&-1&0&0\end{pmatrix}.$

In this basis, the pointwise stabilizer of the four projective vertices in
$\mathsf{PSp}(V)$ is the collection of diagonal matrices. Naming the four
basis vectors $\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}$,
the eight possible Legendrian quadrilaterals are
$P_{\pm}(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4})$,
$P_{\pm}(\mathbf{e}_{1},\mathbf{e}_{2},-\mathbf{e}_{3},\mathbf{e}_{4})$,
$P_{\pm}(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},-\mathbf{e}_{4})$,
$P_{\pm}(\mathbf{e}_{1},\mathbf{e}_{2},-\mathbf{e}_{3},-\mathbf{e}_{4})$,
where we normalized the first two basis vectors by applying diagonal matrices
with diagonal entries $(-1,1,1,-1)$ and $(1,-1,-1,1)$. By Proposition 3.7 and
Lemma 3.6, we conclude that out of these only
$P_{-}(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4})$ and
$P_{-}(\mathbf{e}_{1},\mathbf{e}_{2},-\mathbf{e}_{3},-\mathbf{e}_{4})$ are
transverse. ∎

We now give a definition of the _Maslov index_ , a classical invariant of
triples of real Lagrangians taking values in the integers. Let $L_{1},L_{2}$
be transverse Lagrangians. Define an anti-symplectic involution
$\sigma_{L_{1},L_{2}}:=I\oplus-I$ according to the splitting
$\mathsf{V}=L_{1}\oplus L_{2}$.

######  Definition 3.12.

Let $L_{1},L_{2},L_{3}$ be pairwise transverse Lagrangians in $\mathsf{V}$.
The _Maslov form_ is the symmetric nondegenerate bilinear form on $L_{2}$
defined by

$b_{L_{1},L_{2},L_{3}}(\mathbf{u},\mathbf{v})=\omega(\mathbf{u},\sigma_{L_{1},L_{3}}\mathbf{v})|_{L_{2}}.$

The _Maslov index_ $\mathcal{M}(L_{1},L_{2},L_{3})$ is the signature of
$b_{L_{1},L_{2},L_{3}}$, that is,
$\mathcal{M}(L_{1},L_{2},L_{3})=\frac{k_{+}-k_{-}}{2}$ where $k_{+}$ is the
number of positive eigenvalues and $k_{-}$ is the number of negative
eigenvalues of $b_{L_{1},L_{2},L_{3}}$.

The Maslov index takes its values in the set $\\{-1,0,1\\}$ and is a complete
$\mathsf{PSp}(2n,\mathbb{R})$ invariant of triples of pairwise-transverse
Lagrangians.

##  4\. Legendrian polygons and piecewise circular curves

In this section, we establish a dictionary between the contact projective
geometry of $\mathbb{P}(\mathsf{V})$ and the geometry of co-oriented circles
in the $2$-sphere.

Let $(\mathsf{V},\omega)$ be a $4$-dimensional real symplectic vector space as
before. Let $J$ be a complex structure on $\mathsf{V}$ which is anti-
symplectic, meaning $J$ is a linear automorphism of $\mathsf{V}$ such that
$J^{2}=-I$ and $J^{*}\omega=-\omega$.

As the name suggests, $J$ endows $\mathsf{V}$ with the structure of a complex
vector space where multiplication by $i$ is given by the automorphism $J$.
Moreover, since $J$ is anti-symplectic, the bilinear form

$\omega_{J}(\mathbf{u},\mathbf{v}):=\omega(\mathbf{u},J\mathbf{v})$

is also symplectic. Together, $\omega$ and $\omega_{J}$ define a
$\mathbb{C}$-linear symplectic form $\omega_{\mathbb{C}}$ on $\mathsf{V}$:

$\omega_{\mathbb{C}}:=\omega-i\omega_{J}.$

Consider the projection

$\pi:\mathbb{P}_{\mathbb{R}}(\mathsf{V})\rightarrow\mathbb{P}_{\mathbb{C}}(\mathsf{V})$

which maps the real projectivisation of a vector $\mathbf{v}\in\mathsf{V}$,
denoted by $[\mathbf{v}]_{\mathbb{R}}$, to its complex projectivisation
$[\mathbf{v}]_{\mathbb{C}}$. This projection $\pi$ is a circle bundle, and it
is related to the Hopf fibration
$p:\mathbb{S}^{3}\rightarrow\mathbb{S}^{2}\cong\mathbb{CP}^{1}$ by
$p=\pi\circ\iota$, where $\iota$ is the covering map
$\mathbb{S}^{3}\rightarrow\mathbb{RP}^{3}$.

######  Definition 4.1.

The _space of oriented contact elements_ of
$\mathbb{P}_{\mathbb{C}}(\mathsf{V})$ is the spherical cotangent bundle
$\mathbb{S}(T^{*}\mathbb{P}_{\mathbb{C}}(\mathsf{V}))$. It is naturally a
contact manifold with the contact distribution given by the tautological
$1$-form (or Liouville $1$-form).

The tangent space at a point
$[\mathbf{v}]_{\mathbb{C}}\in\mathbb{P}_{\mathbb{C}}(\mathsf{V})$ identifies
with the space of $\mathbb{C}$-linear maps
$\mathrm{Hom}_{\mathbb{C}}([\mathbf{v}]_{\mathbb{C}},\mathsf{V}/[\mathbf{v}]_{\mathbb{C}})$.
Similarly, the cotangent space at a point
$[\mathbf{v}]_{\mathbb{C}}\in\mathbb{P}_{\mathbb{C}}(\mathsf{V})$ identifies
with the space of complex linear maps
$\mathrm{Hom}_{\mathbb{C}}(\mathsf{V}/[\mathbf{v}]_{\mathbb{C}},[\mathbf{v}]_{\mathbb{C}})$.

We can evaluate a covector $\alpha\in
T^{*}_{[\mathbf{v}]_{\mathbb{C}}}\mathbb{P}_{\mathbb{C}}(\mathsf{V})$ on a
vector $X\in T_{[\mathbf{v}]_{\mathbb{C}}}\mathbb{P}_{\mathbb{C}}(\mathsf{V})$
by composition:

$\alpha\circ
X\in\mathrm{Hom}([\mathbf{v}]_{\mathbb{C}},[\mathbf{v}]_{\mathbb{C}})\cong\mathbb{C}.$

Since we are mostly concerned with real manifolds, when $U$ is a complex
vector space we use the isomorphism $\alpha\mapsto\Re(\alpha)$ between
$\mathrm{Hom}_{\mathbb{C}}(U,\mathbb{C})$ and
$\mathrm{Hom}_{\mathbb{R}}(U,\mathbb{R})$ to identify complex covectors with
real covectors.

In order to identify
$\mathbb{S}\mathrm{Hom}_{\mathbb{C}}(\mathsf{V}/[\mathbf{v}]_{\mathbb{C}},[\mathbf{v}]_{\mathbb{C}})$
with the space of oriented contact elements at $[\mathbf{v}]_{\mathbb{C}}$, we
note that an element
$\alpha\in\mathrm{Hom}_{\mathbb{C}}(\mathsf{V}/[\mathbf{v}]_{\mathbb{C}},[\mathbf{v}]_{\mathbb{C}})$
defines a halfspace of the tangent space at $[\mathbf{v}]_{\mathbb{C}}$ by
$\Re(\alpha(X))>0$, and this is invariant under multiplication by a positive
real. In figures we depict an oriented contact element as an arrow
perpendicular to the boundary of this halfspace and pointing to its interior
(Figures 1, 4, and 5).

The following proposition will allow us to interpret the points of
$\mathbb{P}_{\mathbb{R}}(\mathsf{V})$ as oriented contact elements of
$\mathbb{P}_{\mathbb{C}}(\mathsf{V})$.

######  Proposition 4.2.

The map

$\displaystyle F:\mathbb{P}_{\mathbb{R}}(\mathsf{V})$
$\displaystyle\rightarrow\mathbb{S}(T^{*}\mathbb{P}_{\mathbb{C}}(\mathsf{V}))$
$\displaystyle[\mathbf{v}]_{\mathbb{R}}$
$\displaystyle\mapsto(\mathbf{u}\mapsto\omega_{\mathbb{C}}(\mathbf{u},\mathbf{v})\mathbf{v}),$

where the right hand side is viewed as an element of
$\mathrm{Hom}_{\mathbb{C}}(\mathsf{V}/[\mathbf{v}]_{\mathbb{C}},[\mathbf{v}]_{\mathbb{C}})$
up to multiplication by a positive real, is an isomorphism of circle bundles
over $\mathbb{P}_{\mathbb{C}}(\mathsf{V})$.

###### Proof.

We first verify that the formula
$(\mathbf{u}\mapsto\omega_{\mathbb{C}}(\mathbf{u},\mathbf{v}))$ defines an
element of
$\mathrm{Hom}_{\mathbb{C}}(\mathsf{V}/[\mathbf{v}]_{\mathbb{C}},[\mathbf{v}]_{\mathbb{C}})$.

Since $\omega_{\mathbb{C}}$ is $\mathbb{C}$-bilinear, the map
$\mathbf{u}\rightarrow\omega_{\mathbb{C}}(\mathbf{u},\mathbf{v})\mathbf{v}$ is
$\mathbb{C}$-linear. Moreover, it is well-defined on
$\mathsf{V}/[\mathbf{v}]_{\mathbb{C}}$ because
$\omega_{\mathbb{C}}(\mathbf{u}+\mathbf{v},\mathbf{v})=\omega_{\mathbb{C}}(\mathbf{u},\mathbf{v})$
and
$\omega_{\mathbb{C}}(\mathbf{u}+J\mathbf{v},\mathbf{v})=\omega_{\mathbb{C}}(\mathbf{u},\mathbf{v})$.

For any $t\in\mathbb{R}^{*}$, we have

$\omega_{\mathbb{C}}(\mathbf{u},t\mathbf{v})t\mathbf{v}=t^{2}\omega_{\mathbb{C}}(\mathbf{u},\mathbf{v})\mathbf{v},$

so the map is well-defined on the quotients.

Identifying $\mathsf{V}\cong\mathbb{C}^{2}$, we can locally trivialize the
bundle $\mathbb{P}_{\mathbb{R}}(\mathsf{V})$ over the affine patch
$\left\\{\begin{bmatrix}z\\\
1\end{bmatrix}\in\mathbb{CP}^{1}\right\\}\subset\mathbb{P}_{\mathbb{C}}(\mathsf{V})$
by

$\displaystyle\varphi:\mathbb{C}\times S^{1}$
$\displaystyle\rightarrow\mathbb{P}_{\mathbb{R}}(\mathbb{C}^{2})$
$\displaystyle(z,e^{i\theta})$
$\displaystyle\mapsto\begin{bmatrix}ze^{\frac{i\theta}{2}}\\\
e^{\frac{i\theta}{2}}\end{bmatrix}.$

Similarly, we can locally trivialize
$\mathrm{Hom}_{\mathbb{C}}(\mathsf{V}/[\mathbf{v}]_{\mathbb{C}},[\mathbf{v}]_{\mathbb{C}})$
by

$\displaystyle\psi:\mathbb{C}\times S^{1}$
$\displaystyle\rightarrow\mathbb{S}\left(\mathrm{Hom}_{\mathbb{C}}(\mathsf{V}/[\mathbf{v}]_{\mathbb{C}},[\mathbf{v}]_{\mathbb{C}})\right)$
$\displaystyle(z,e^{i\theta})$ $\displaystyle\mapsto\left[\mathbf{u}\mapsto
e^{i\theta}\omega_{\mathbb{C}}\left(\mathbf{u},\begin{pmatrix}z\\\
1\end{pmatrix}\right)\begin{pmatrix}z\\\ 1\end{pmatrix}\right].$

In this pair of trivializations, the map $F$ is the identity and hence is
continuous and an isomorphism on each fiber. Similar trivilizations on the
affine patch $\left\\{\begin{bmatrix}1\\\
z\end{bmatrix}\in\mathbb{CP}^{1}\right\\}$ yield the same result, and we
conclude that $F$ is a bundle isomorphism. ∎

The group

$\mathsf{PSp}(2,\mathbb{C}):=\\{M\in\mathsf{PSp}(\mathsf{V})~{}|~{}MJ=JM\\}$

acts transitively on the base $\mathbb{P}_{\mathbb{C}}(\mathsf{V})$ and also
acts on the total spaces $\mathbb{P}_{\mathbb{R}}(\mathsf{V})$ and
$\mathbb{S}(T^{*}\mathbb{P}_{\mathbb{C}}(\mathsf{V}))$ of the two circle
bundles. The action on $\mathbb{S}(T^{*}\mathbb{P}_{\mathbb{C}}(\mathsf{V}))$
when viewed as a $\mathrm{Hom}$-bundle is by conjugation, that is, if
$f\in\mathbb{S}\mathrm{Hom}_{\mathbb{C}}(\mathsf{V}/[\mathbf{v}]_{\mathbb{C}},[\mathbf{v}]_{\mathbb{C}})$,
then

$M\cdot
f=MfM^{-1}\in\mathbb{S}\mathrm{Hom}_{\mathbb{C}}(\mathsf{V}/[M\mathbf{v}]_{\mathbb{C}},[M\mathbf{v}]_{\mathbb{C}})$

is the image of $f$ by the action of $A\in\mathsf{PSp}(2,\mathbb{C})$. The
isomorphism in Proposition 4.2 intertwines the two actions on the total
spaces.

######  Proposition 4.3.

The map
$F:\mathbb{P}_{\mathbb{R}}(\mathsf{V})\rightarrow\mathbb{S}(T^{*}\mathbb{P}_{\mathbb{C}}(\mathsf{V}))$
is an isomorphism of contact manifolds.

###### Proof.

We will use the same trivializations $\varphi,\psi$ as in the proof of
Proposition 4.2.

In the coordinate chart $\varphi$, a path $\gamma(t)=(z(t),e^{i\theta(t)})$ is
tangent to the contact distribution of
$\mathbb{P}_{\mathbb{R}}{\mathbb{C}^{2}}$ if and only if

$\omega_{\mathbb{R}}(\gamma(t),\gamma^{\prime}(t))=\omega_{\mathbb{R}}\left(\begin{pmatrix}z(t)e^{\frac{i\theta(t)}{2}}\\\
e^{\frac{i\theta(t)}{2}}\end{pmatrix},\begin{pmatrix}z^{\prime}(t)e^{\frac{i\theta(t)}{2}}+z(t)\frac{i\theta^{\prime}(t)}{2}e^{\frac{i\theta(t)}{2}}\\\
\frac{i\theta^{\prime}(t)}{2}e^{\frac{i\theta(t)}{2}}\end{pmatrix}\right)=0,$

which simplifies to

$\Re\left(e^{i\theta(t)}\omega_{\mathbb{C}}\left(\begin{pmatrix}z(t)\\\
1\end{pmatrix},\begin{pmatrix}z^{\prime}(t)\\\
0\end{pmatrix}\right)\right)=0.$

In the coordinate chart $\psi$ for $\mathbb{S}(T^{*}\mathbb{CP}^{1})$, the
path $\gamma(t)=(z(t),e^{i\theta(t)})$ is in the contact distribution of if
and only if

$\psi(\gamma(t))\left(d\pi\left(\left.\frac{\mathrm{d}}{\mathrm{d}s}\right|_{s=t}\psi(\gamma(s))\right)\right)=\Re\left(e^{i\theta(t)}\omega_{\mathbb{C}}\left(\begin{pmatrix}z^{\prime}(t)\\\
0\end{pmatrix},\begin{pmatrix}z(t)\\\ 1\end{pmatrix}\right)\right)=0.$

Since these equations define the same plane, and in these two charts the
isomorphism $F$ is the identity, we conclude that $F$ is a contactomorphism. ∎

Now that we have established the contact isomorphism between
$\mathbb{P}(\mathsf{V})$ and the space of oriented contact elements to the
sphere, we can begin to explore the geometric interpretations. The object
which corresponds to a Legendrian line (of Lagrangian subspace) is a co-
oriented circle.

######  Definition 4.4.

A _co-oriented circle_ in $\mathbb{P}_{\mathbb{C}}(\mathsf{V})$ is a round
circle $C$ of $\mathbb{P}_{\mathbb{C}}(\mathsf{V})$, possibly degenerated to a
point, together with a choice of disk bounded by $C$ when nondegenerate.

We say that two co-oriented circles are tangent if the circles are tangent and
the orientations match at the tangency point. If one of the circles is
degenerate, they are considered tangent if the point circle is contained in
the other circle.

######  Lemma 4.5.

Let $L\subset\mathsf{V}$ be a Lagrangian subspace for $\omega$. If $L$ is not
a complex subspace, then $\pi(L)$ is a circle. The two connected components of
$\mathbb{P}_{\mathbb{C}}(\mathsf{V})\setminus\pi(L)$ are distinguished by the
Maslov index :

$\\{\ell\in\mathbb{P}_{\mathbb{C}}(\mathsf{V})~{}|~{}\mathcal{M}(L,\ell,JL)=1\\}$

and

$\\{\ell\in\mathbb{P}_{\mathbb{C}}(\mathsf{V})~{}|~{}\mathcal{M}(L,\ell,JL)=-1\\}.$

###### Proof.

Since $L$ is not a complex subspace, $L\oplus JL=\mathsf{V}$ defines a
splitting of the vector space $\mathsf{V}$ into a pair of Lagrangians.

The anti-symplectic, anti-linear involution $\sigma_{L,JL}=I\oplus-I$ defines
a _real structure_ on the complex vector space $\mathsf{V}$. When endowed with
this real structure, the complex projectivization $\pi(L)$ identifies with the
real projective line
$\mathbb{P}_{\mathbb{R}}(\mathsf{V})\subset\mathbb{P}_{\mathbb{C}}(\mathsf{V})$,
which is a (generalized) circle.

The involution $\sigma_{L,JL}$ exchanges the two connected components of the
complement $\mathbb{P}_{\mathbb{C}}(\mathsf{V})\setminus\pi(L)$. The Maslov
index $\mathcal{M}(L,\ell,JL)$ is the signature of the symmetric bilinear form
$b_{L,\ell,JL}=\omega(\cdot,\sigma_{L,JL}(\cdot))$ restricted to $\ell$.

Note that $J$ is an automorphism of the bilinear form $b_{L,\ell,JL}$:

$\omega(J\mathbf{u},\sigma_{L,JL}J\mathbf{v})=\omega(J\mathbf{u},-J\sigma_{L,JL}\mathbf{v})=\omega(\mathbf{u},\sigma_{L,JL}\mathbf{v}).$

Moreover, $\mathbf{v}$ is always orthogonal to $J\mathbf{v}$ for the form
$b_{L},J_{L}$. Hence, if $\ell$ is the complex line spanned the vector
$\mathbf{v}$, $b_{L,\ell,JL}$ is either definite or identically zero. If it is
identically zero, then $\omega(\mathbf{v},\sigma_{L,JL}\mathbf{v})=0$ and
$\omega(J\mathbf{v},\sigma_{L,JL}\mathbf{v})=0$ and the fact that
$\ell=\mathrm{span}_{\mathbb{R}}(\mathbf{v},J\mathbf{v})$ is a Lagrangian
imply that $\sigma_{L,JL}(\mathbf{v})\in\ell$, and hence $\ell\in\pi(L)$.

By continuity of the bilinear form $b_{L,\ell,JL}$ we conclude that the two
complementary components of $\pi(L)$ consist of where it is positive definite
and negative definite. ∎

Let $L\subset\mathsf{V}$ be a Lagrangian subspace for the symplectic form
$\omega$. If $L\neq JL$, let $C_{L}$ denote the co-oriented circle $\pi(L)$
with the co-orientation given by the disk of complex lines
$U\subset\mathsf{V}$ such that $\mathcal{M}(L,U,JL)=-1$. If $L=JL$, then $L$
is a complex subspace and we let $C_{L}=\pi(L)$ which we interpret as a zero
radius circle. Then, $L\leftrightarrow C_{L}$ is a bijection between the set
of Lagrangians in $\mathsf{V}$ and the set of co-oriented circles in
$\mathbb{P}_{\mathbb{C}}(\mathsf{V})$.

###  4.1. In coordinates

Concretely, in a basis
$\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4}$ for which the
symplectic form is given by

$\Omega=\begin{pmatrix}0&0&0&1\\\ 0&0&-1&0\\\ 0&1&0&0\\\
-1&0&0&0\end{pmatrix},$

a dense open set of Lagrangians is parametrized by the affine patch

$L(a,b,c)=\begin{pmatrix}1&0\\\ 0&1\\\ a&b+c\\\ b-c&-a\end{pmatrix}.$

Choosing the compatible complex structure with matrix

$J=\begin{pmatrix}0&1&0&0\\\ -1&0&0&0\\\ 0&0&0&-1\\\ 0&0&1&0\end{pmatrix},$

the complex structure acts on Lagrangians $L(a,b,c)$ by reversing the sign of
$c$, and so the Lagrangians in the affine patch which are complex lines are
precisely $L(x,y,0)$. The pair $(\mathbf{e}_{1},\mathbf{e}_{3})$ forms a
complex basis for $\mathsf{V}$ in which the complex symplectic form
$\omega_{\mathbb{C}}$ has matrix $\begin{pmatrix}0&-i\\\ i&0\end{pmatrix}$. In
this basis, we have

$L(x,y,0)=\begin{pmatrix}1&0\\\ 0&1\\\ x&y\\\
y&-x\end{pmatrix}=\begin{pmatrix}1\\\ x+iy\end{pmatrix}.$

The Maslov index $\mathcal{M}(L(a,b,c),L(x,y,0),JL(a,b,c))$ is the index of
the bilinear form

$\begin{pmatrix}\frac{(x-a)^{2}+(y-b)^{2}-c^{2}}{c}&0\\\
0&\frac{(x-a)^{2}+(y-b)^{2}-c^{2}}{c}\end{pmatrix},$

which is negative definite if and only if either: $c<0$ and $(x,y)$ is inside
of the circle of radius $|c|$ centered at $a+bi$, or $c>0$ and $(x,y)$ is
outside of that circle. Thus, $L(a,b,c)$ represents the circle of radius $|c|$
centered at $a+bi$ co-oriented towards the inside if $c>0$ and towards the
outside if $c<0$.

######  Lemma 4.6.

Let $\mathbf{v}\in\mathsf{V}$ and $L\subset\mathsf{V}$ be a Lagrangian. Then,
$\mathbf{v}\in L$ if and only if $\pi(\mathbf{v})\in C_{L}$ and the oriented
contact element to $C_{L}$ at $\pi(\mathbf{v})$ is given by
$[\mathbf{v}]_{\mathbb{R}}$ through the isomorphism $F$ of Proposition 4.2.

###### Proof.

Using homogeneity under the $\mathsf{PSp}(2,\mathbb{C})$ action, it would be
enough to show this for a single $\mathbf{v}\in\mathsf{V}$. Since the
calculation is simple enough, we show it for any point which projects to a
standard affine patch in $\mathbb{CP}^{1}$.

In affine coordinates as above, we consider

$[\mathbf{v}]_{\mathbb{R}}=\begin{bmatrix}k\\\ l\\\ ka+l(b+c)\\\
k(b-c)-la\end{bmatrix}_{\mathbb{R}}\subset L(a,b,c).$

This point in real projective space projects to

$\pi([\mathbf{v}]_{\mathbb{R}})=[\mathbf{v}]_{\mathbb{C}}=\begin{bmatrix}k-il\\\
k(a+(b-c)i)+l(b+c)-ai\end{bmatrix}=\begin{bmatrix}1\\\ a+bi-
ic\frac{k+il}{k-il}\end{bmatrix}\in\mathbb{CP}^{1}.$

This is a parametrization by $[k,l]\in\mathbb{RP}^{1}$ of the circle
corresponding to $L(a,b,c)$ where $[\mathbf{v}]_{\mathbb{C}}$ is the unique
point on the circle of radius $|c|$ centered at $a+bi$ which is in the
direction of $-i\frac{(k+il)^{2}}{|k+il|^{2}}$ if $c>0$, and in the opposite
direction if $c<0$.

The tangent vector at $[\mathbf{v}]_{\mathbb{C}}$ corresponding to
$z\in\mathbb{C}$, as a linear map, is represented by

$X_{z}=\mathbf{v}_{\mathbb{C}}^{*}\otimes\begin{pmatrix}0\\\
(k-il)z\end{pmatrix}\in\mathrm{Hom}_{\mathbb{C}}([\mathbf{v}]_{\mathbb{C}},\mathbb{C}^{2}/[\mathbf{v}]).$

Evaluating the covector $F([\mathbf{v}]_{\mathbb{R}})$ at $X_{z}$ yields

$\omega_{\mathbb{C}}\left(\begin{pmatrix}0\\\
(k-il)z\end{pmatrix},\mathbf{v}\right)=i(k-il)^{2}z.$

We conclude that the halfspace of the tangent space at
$[\mathbf{v}]_{\mathbb{C}}$ defined by the covector
$F([\mathbf{v}]_{\mathbb{R}})$ is

$\\{z\in\mathbb{C}~{}|~{}\Re(zi(k-il)^{2})>0\\}=\\{z\in\mathbb{C}~{}|~{}\Re(z\overline{(-i(k+il)^{2})})>0\\}.$

This halfspace is bounded by the line spanned by
$(k+il)^{2}=|k+il|^{2}\frac{k+il}{k-il}$ and contains $-i(k+il)^{2}$.

Therefore, if $c>0$, the circle corresponding to $L(a,b,c)$ is oriented
outwards and the halfspace of the tangent space at $a+bi-ic\frac{k+il}{k-il}$
determined by $F([\mathbf{v}]_{\mathbb{R}})$ is also oriented outwards. If
$c<0$, the circle corresponding to $L(a,b,c)$ is oriented inwards and the
halfspace of the tangent space is also oriented inwards (see Fig. 2).

We conclude that the oriented contact elements represented by lines in
$\mathbb{R}^{4}$ contained in the Lagrangian $L(a,b,c)$ are precisely the
oriented contact elements of the co-oriented circle corresponding to
$L(a,b,c)$, proving the proposition. ∎

######  Corollary 4.7.

Let
$[\mathbf{u}]_{\mathbb{R}},[\mathbf{v}]_{\mathbb{R}}\in\mathbb{P}_{\mathbb{R}}(\mathsf{V})$.
Then, $\omega(\mathbf{u},\mathbf{v})=0$ if and only if
$F([\mathbf{u}]),F([\mathbf{v}])$ are oriented contact elements tangent to a
common co-oriented circle.

Figure 2. An oriented contact element to the circle corresponding to
$L(a,b,c)$ with $c>0$, in local coordinates.

In the coordinates we are using, the embedding
$\mathsf{PSL}(2,\mathbb{C})\cong\mathsf{PSp}(2,\mathbb{C})\hookrightarrow\mathsf{PSp}(4,\mathbb{R})$
induced by forgetting the imaginary part of the complex symplectic form
$\omega_{\mathbb{C}}$ is

$\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto\begin{pmatrix}\Re(a)&\Im(a)&\Re(b)&-\Im(b)\\\
-\Im(a)&\Re(a)&-\Im(b)&-\Re(b)\\\ \Re(c)&\Im(c)&\Re(d)&-\Im(d)\\\
\Im(c)&-\Re(c)&\Im(d)&\Re(d)\\\ \end{pmatrix}.$

These matrices act by Möbius transformations on the unit tangent bundle of
$S^{2}$.

Let $r\in\mathbb{R}$ and define

$T_{r}:=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&r&1&0\\\
-r&0&0&1\end{pmatrix}.$

Then, $T_{r}(L(a,b,c))=L(a,b,c+r)$. In geometric terms, the transformation
$T_{r}\in\mathsf{PSp}(4,\mathbb{R})$ maps any oriented circle in the affine
patch $L$ to the circle with the same center and radius increased by $r$. Its
action on piecewise circular curves is therefore the radial translation of
parameter $r$.

######  Proposition 4.8.

Möbius transformations and radial translations $T_{r}$ generate the Lie group
$\mathsf{PSp}(4,\mathbb{R})$.

###### Proof.

The Proposition follows by a Lie algebra computation using the matrix
expressions of the two subgroups and the fact that
$\mathsf{PSp}(4,\mathbb{R})$ is connected. ∎

Therefore, the classification problem of piecewise circular $n$-gons up to
Möbius transformations and radial translations reduces to the classification
of Legendrian $n$-gons in $\mathbb{RP}^{3}$ modulo the action of
$\mathsf{PSp}(4,\mathbb{R})$.

We can now justify the interpretation of the Maslov index as a generalization
of nestedness of circles mentioned in the introduction. The group
$\mathsf{PSp}(4,\mathbb{R})$ acts transitively on pairs of transverse
Lagrangians, and transitively on pairwise-transverse triples of Lagrangians
which have the same Maslov index. Therefore, any triple with Maslov index $+1$
is equivalent to the triple $L(0,0,0)$, $L(0,0,1)$, $L(0,0,2)$ corresponding
to three nested circles.

In other words, any configuration of three co-oriented circles which are
pairwise non tangent can be brought by a sequence of Möbius transformations
and radial translations to exactly one of the configurations depicted in
Figure 3, and Maslov index $1$ corresponds to nested circles with increasing
radius (or equivalently, decreasing curvature).

(a) $\mathcal{M}=1$

(b) $\mathcal{M}=-1$

(c) $\mathcal{M}=0$

Figure 3. The Maslov index of a triple of co-oriented circles.

######  Definition 4.9.

We say that a transverse Legendrian polygon and its corresponding piecewise
circular curve have _decreasing curvature_ if every triple of non-adjacent
segments is a triple of Lagrangians with Maslov index $1$.

Recall that we defined a _simple_ piecewise circular curve by the property
that every radial translate is simple.

######  Proposition 4.10.

A piecewise circular curve is simple if and only if the corresponding
Legendrian polygon is transverse.

###### Proof.

A self-intersection of a piecewise circular curve can be interpreted as a
zero-radius circle which is tangent to two oriented contact elements of the
curve lying on non-adjacent pieces. This means that there exists a Lagrangian
containing two points in $\mathbb{P}(\mathsf{V})$ lying on non-adjacent edges
of the corresponding Legendrian polygon. In other words, those two points are
incident, so the Legendrian polygon is not transverse.

Conversely, if a Legendrian polygon is non-transverse, then there exist two
points on non-adjacent edges which are incident. The Lagrangian spanned by
these two points corresponds to a circle tangent to two non-adjacent circular
pieces of the piecewise circular curve. Applying a Möbius transformation if
needed, we may assume that this circle is not a line. Then, applying a radial
translation by the opposite of its radius brings that circle to a zero radius
circle, producing a translate with a self-intersection. ∎

###  4.2. The dictionary

From the above analysis, we deduce the dictionary in Table 1 between the
geometry of the symplectic vector space $(\mathsf{V},\omega)$ and the geometry
of circles in the Riemann sphere $\mathbb{P}_{\mathbb{C}}(\mathsf{V})$.

$(\mathsf{V},\omega)$ | $\mathbb{P}_{\mathbb{C}}(\mathsf{V})$
---|---
Point in $\mathbb{P}_{\mathbb{R}}(\mathsf{V})$ | Oriented contact element
Lagrangian subspace of $\mathsf{V}$ | Co-oriented circle
Isotropic flag in $\mathsf{V}$ | Pointed co-oriented circle
Non-transverse Lagrangians | Tangent co-oriented circles
Incident points in $\mathbb{P}_{\mathbb{R}}(\mathsf{V})$ | Oriented contact elements
| to a common co-oriented circle
Legendrian polygon | Co-oriented
| piecewise circular curve
Moduli space of generic | Moduli space of generic
Legendrian $n$-gons | circular $n$-gons
Transverse Legendrian $n$-gon | Simple circular $n$-gon
Table 1. Dictionary between Legendrian polygons and piecewise circular curves

We can translate the results of the previous sections with this dictionary as
follows.

######  Corollary 4.11.

The moduli space of simple circular $n$-gons and the moduli space of generic
circular $n$-gons are a smooth manifolds of dimension $2(n-5)$ when $n\geq 5$.

######  Corollary 4.12.

The moduli space of simple circular quadrilaterals is a pair of points
(Proposition 3.11).

##  5\. Positivity

###  5.1. Flag positivity

In this section, we describe positivity in the oriented flag manifold
$G/B_{0}$ of the group
$G=\mathsf{Sp}(\mathsf{V},\omega)\cong\mathsf{Sp}(4,\mathbb{R})$. For a more
general perspective on positivity in oriented flag manifolds see [BT18].

Let $F$ be a flag in $\mathsf{V}$. We denote by $F^{(k)}$ the $k$-dimensional
part of $F$.

Note that the symplectic vector space $\mathsf{V}$ is canonically oriented by
the volume form $\omega\wedge\omega$. Whenever $U,W\subset\mathsf{V}$ are
oriented vector subspaces of $\mathsf{V}$, the equality $U=W$ will mean that
the subspaces are equal and the orientations agree.

######  Definition 5.1.

An _isotropic flag_ in $\mathsf{V}$ is a full flag $F$ such that
$F^{(4-k)}=F^{(k)\perp}$ for all $k$.

Let $F$ be an isotropic flag and choose orientations on each $F^{(k)}$. The
orientations on $F^{(1)}$ and $F^{(3)}$ induce an orientation on
$F^{(3)}/F^{(1)}$ in the following way: if
$(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3})$ is an oriented basis of
$F^{(3)}$ such that $\mathbf{e}_{1}$ is an oriented basis of $F^{(1)}$, then
$(\mathbf{e}_{2},\mathbf{e}_{3})$ gives a well-defined orientation on the
quotient. We say that the orientations on $F^{(1)}$ and $F^{(3)}$ are
_compatible_ if this orientation on $F^{(3)}/F^{(1)}$ matches the orientation
induced on this quotient by the symplectic form $\omega$.

######  Definition 5.2.

An _oriented isotropic flag_ in $\mathsf{V}$ is an isotropic flag $F$ together
with a choice of orientations satisfying:

  * •

The orientations on $F^{(1)}$ and $F^{(3)}=F^{(1)\perp}$ are compatible;

  * •

The orientation on $F^{(4)}$ is the same as that of $\mathsf{V}$.

######  Remark 5.3.

Because of the condition on compatible orientations, an oriented isotropic
flag $F$ is uniquely determined by $F^{(1)}$ and $F^{(2)}$.

We call a basis
$E=(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{4})$ of
$\mathsf{V}$ in which the symplectic form is $\Omega$ (defined in Equation
(1)) a _symplectic basis_.

A symplectic basis determines an oriented isotropic flag
$F_{E}=\mathrm{span}(\mathbf{e}_{1})\subset\mathrm{span}(\mathbf{e}_{1},\mathbf{e}_{2})\subset\mathrm{span}(\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3})\subset\mathsf{V}$.
We now fix such a symplectic basis $E$ and we denote an oriented isotropic
flag $F$ by a $4\times 4$ matrix such that the first $k$ columns form an
oriented basis of $F^{(k)}$. To shorten notation we will also sometimes use
the $4\times 2$ consisting of only the first two columns, which is equivalent
by Remark 5.3.

######  Definition 5.4.

A pair of oriented isotropic flags $F,G$ is called _oriented-transverse_ if
there exists a symplectic basis $E$ such that $F=F_{E}$ and
$G=F_{\widehat{E}}$.

The group $G=\mathsf{Sp}(\mathsf{V},\omega)$ acts transitively on oriented
isotropic flags and the stabilizer of the flag $F_{0}:=F_{E}$ is the subgroup
$B_{+}^{0}$ of upper triangular matrices with positive entries on the diagonal
(in the basis $E$). The space of oriented isotropic flags then identifies with
the homogeneous space $G/B_{+}^{0}$.

The group $G$ also acts transitively on pairs of oriented-transverse oriented
isotropic flags $F_{E},F_{\widehat{E}}$, with stabilizer diagonal matrices
with positive entries.

The unipotent radical of the Borel subgroup $B_{+}$ is the subgroup:

$U_{+}=\left\\{\left.\begin{pmatrix}1&a&b&c\\\ 0&1&d&ad-b\\\ 0&0&1&a\\\
0&0&0&1\\\ \end{pmatrix}~{}\right|~{}a,b,c,d\in\mathbb{R}\right\\}.$

We define the _opposite symplectic basis_ $\widehat{E}$ of $E$ to be
$(\mathbf{e}_{4},-\mathbf{e}_{3},\mathbf{e}_{2},-\mathbf{e}_{1})$.

The stabilizer of $F_{\infty}:=F_{\widehat{E}}$ is the subgroup $B_{-}^{0}$ of
lower triangular matrices with positive entries on the diagonal. The Borel
subgroup $B_{-}$ is opposite to $B_{+}$, and its unipotent radical is

$U_{-}=\left\\{\left.\begin{pmatrix}1&0&0&0\\\ a&1&0&0\\\ b&d&1&0\\\
c&ad-b&a&1\\\ \end{pmatrix}~{}\right|~{}a,b,c,d\in\mathbb{R}\right\\}.$

We say that two oriented isotropic flags $(F_{1},F_{2})$ are _transverse_ if
they are in the $\mathsf{Sp}(\mathsf{V},\omega)$-orbit of the pair
$(F_{0},F_{\infty})$, or equivalently if there is a symplectic basis $E$ such
that $F_{1}=F_{E}$ and $F_{2}=F_{\widehat{E}}$.

The _positive semigroup_ $U_{+}^{>0}$ is the following subsemigroup of
$U_{+}$:

$U_{+}^{>0}=\left\\{\left.\begin{pmatrix}1&a&b&c\\\ 0&1&d&ad-b\\\ 0&0&1&a\\\
0&0&0&1\\\ \end{pmatrix}~{}\right|~{}\begin{array}[]{c}a>0,~{}b>0,~{}c>0\\\
ad-b>0,~{}-b^{2}+abd-cd>0\end{array}\right\\},$

and similarly

$U_{-}^{>0}=\left\\{\left.\begin{pmatrix}1&0&0&0\\\ a&1&0&0\\\ b&d&1&0\\\
c&ad-b&a&1\\\ \end{pmatrix}~{}\right|~{}\begin{array}[]{c}a>0,~{}b>0,~{}c>0\\\
ad-b>0,~{}-b^{2}+abd-cd>0\end{array}\right\\}.$

The inequalities defining the positive semigroups are equivalent to the
statement that all minors of the matrix which aren’t zero by triangularity are
positive ([Pin10, Theorem 2.8]).

A simple computation shows that these semigroups satisfy
$\left(U_{\pm}^{>0}\right)^{-1}=KU_{\pm}^{>0}K$, where $K$ is the diagonal
matrix $K=\mathrm{diag}(1,-1,1,-1)$.

######  Definition 5.5.

A triple of pairwise transverse oriented isotropic flags
$(F_{E},F,F_{\widehat{E}})$ is _positive_ if $F=uF_{0}$ with $u\in
U_{-}^{>0}$.

A tuple of flags $(F_{1},\dots,F_{n})$ is _positive_ if every ordered sub-
triple $F_{i},F_{j},F_{k}$ with $1\leq i<j<k\leq n$ is positive.

In a positive triple, all pairs are oriented-transverse. Additionally,
positivity satisfies the following:

  1. (1)

If $(F_{1},F_{2},F_{3})$ is positive, then $(F_{2},F_{3},-F_{1})$ is positive.

  2. (2)

If $(F_{1},F_{2},F_{3})$ is positive, then $(F_{3},F_{2},F_{1})$ is not
positive.

  3. (3)

If $(F_{1},F_{2},F_{3})$ and $(F_{1},F_{2},F_{4})$ are positive, then
$(F_{1},F_{3},F_{4})$ is positive.

The first property implies that if $(F_{1},\dots,F_{n})$ is positive, then
$(-F_{1},\dots,-F_{n})$ also is. This motivates the introduction of the
following moduli space.

######  Definition 5.6.

The space $\mathscr{F}^{(n)}$ is the space of
$\mathsf{PSp}(\mathsf{V})$-orbits of $n$-tuples of pairwise transverse
oriented isotropic flags up to the diagonal action of $-I$. The subspace
consisting of positive $n$-tuples is denoted by $\mathscr{F}^{(n)}_{>0}$.

As a straightforward consequence of the definitions, we have:

######  Proposition 5.7.

A quadruple of oriented isotropic flags $F_{0},F_{1},F_{2},F_{\infty}$ is
positive if and only if $F_{1}=u_{1}F_{0}$ and $F_{2}=u_{1}u_{2}F_{0}$ with
$u_{1},u_{2}\in U_{-}^{>0}$.

A quadruple of oriented isotropic flags $F_{0},F_{+},F_{\infty},F_{-}$ is
positive if and only if $F_{+}=u_{+}F_{0}$ and $F_{-}=-u_{-}F_{0}$, where
$u_{+}\in U_{-}^{>0}$ and $u_{-}\in KU_{-}^{>0}K$.

The following simple lemma will be useful in the proof of Proposition 5.16.

######  Lemma 5.8.

Let $(F_{1},\dots,F_{n})\in\mathscr{F}^{(n)}$. If $(F_{i},F_{i+1},F_{j})$ is
positive for every $1\leq i+1<j\leq n$, then $(F_{1},\dots,F_{n})$ is
positive.

###### Proof.

Let $1\leq i<j<k\leq n$. Since $(F_{i},F_{i+1},F_{k})$ and
$(F_{i+1},F_{i+2},F_{k})$ are positive by hypothesis, by property $(1)$ above
we find that $(F_{k},-F_{i},-F_{i+1})$ and $(F_{k},-F_{i+1},-F_{i+2})$ are
positive. By properties $(3)$ and $(1)$, we conclude that
$(F_{i},F_{i+2},F_{k})$ is positive. Repeating this argument we find that
$(F_{i},F_{i+m},F_{k})$ is positive for all $0<m<k-i$ and so
$(F_{i},F_{j},F_{k})$ is positive. ∎

Positivity was originally defined for unoriented flags. Let us denote by
$\mathsf{Flag}$ the space of unoriented isotropic flags. An $n$-tuple of
isotropic flags is called positive if it can be written

$F_{0},u_{1}F_{0},u_{1}u_{2}F_{0},\dots,(u_{1}u_{2}\dots
u_{n-2})F_{0},F_{\infty},$

where $u_{i}\in U_{-}^{>0}$. Let $\mathsf{Flag}^{(n)}$ denote the moduli space
of $n$-tuples of pairwise transverse flags up to the action of
$\mathsf{PSp}(4,\mathbb{R})$.

We will use the following special case of a theorem of Fock and Goncharov to
prove our main result.

######  Theorem 5.9 ([FG06, Theorem 1.5]).

Positivity of an $n$-tuple of flags is invariant under the action of
$\mathsf{PSp}(V)$, and the subset
$\mathsf{Flag}^{(n)}_{>0}\subset\mathsf{Flag}^{(n)}$ consisting of positive
$n$-tuples is a connected component homeomorphic to a ball of dimension
$4n-10$.

For our oriented isotropic flag setting, we have the same result.

######  Proposition 5.10.

The subset $\mathscr{F}^{(n)}_{>0}\subset\mathscr{F}^{(n)}$ consisting of
positive $n$-tuples of oriented isotropic flags is a connected component
homeomorphic to a ball of dimension $4n-10$.

###### Proof.

The projection $\mathscr{F}^{(n)}_{>0}\rightarrow\mathsf{Flag}^{(n)}$
corresponding to removing the orientations is a covering map. By Theorem 5.9,
the preimage of the positive component, which is contractible, is a disjoint
union of homeomorphic copies.

The definition of a positive $n$-tuples of flags gives a prefered choice of
lift to oriented flags, namely the lift

$F_{0},u_{1}F_{0},u_{1}u_{2}F_{0},\dots(u_{1}u_{2}\dots
u_{n-2})F_{0},F_{\infty},$

now considered as oriented flags. Since the elements $u_{i}\in U_{-}$ are
uniquely determined, this lift is well-defined and is positive as a tuple of
oriented isotropic flags.

This is the only positive lift of this tuple, since we can assume that the
lift of $F_{0}$ is $F_{0}$ by acting with $\mathsf{PSp}(4,\mathbb{R})$, and
any pair of unoriented flags $F_{0},F_{i}$ has a unique oriented-transverse
lift. Since every positive tuple of oriented isotropic flags is pairwise
oriented-transverse, the positive lift is unique. ∎

The cited Fock-Goncharov result shows that positive $n$-tuples form a
connected component in the space of _generic_ $n$-tuples, rather than
pairwise-transverse $n$-tuples. However, since positivity is characterized by
triples, and a positive triple can only degenerate to a triple containing a
non-transverse pair ([Lus94, Prop. 8.14]), the theorem is true as stated.

The proof of Theorem 5.9 provides explicit coordinates for
$\mathsf{Flag}^{(n)}_{>0}$ and hence for $\mathscr{F}^{(n)}_{>0}$.

######  Example 5.11.

The following map is a parametrization of $\mathscr{F}_{>0}^{(3)}$:

$\displaystyle\mathbb{R}_{>0}\times\mathbb{R}_{>0}$
$\displaystyle\longrightarrow\mathscr{F}^{(3)}_{>0}$ $\displaystyle(x,y)$
$\displaystyle\longmapsto(F_{0},\left(\begin{array}[]{cccc}1&0\\\
x+y+\frac{1}{y}&1\\\ y&1\\\ 1&x+\frac{1}{y}\\\
\end{array}\right),F_{\infty}).$

The two-dimensional parts of a triple of oriented isotropic flags are
Lagrangians, and so given a triple of flags we obtain a triple of Lagrangians.
If the triple is positive, their Maslov index is always $+1$:

######  Lemma 5.12.

Let $F_{1},F_{2},F_{3}$ be a positive triple of oriented isotropic flags.
Then, the Maslov index
$\mathcal{M}\left(F_{1}^{(2)},F_{2}^{(2)},F_{3}^{(2)}\right)$ is $+1$.

###### Proof.

After acting by an element of $\mathsf{PSp}(4,\mathbb{R})$, we may assume

$F_{1}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix},\quad F_{2}=\begin{pmatrix}1&0&0&0\\\ a&1&0&0\\\
b&d&1&0\\\ c&ad-b&a&1\end{pmatrix},\quad F_{3}=\begin{pmatrix}0&0&0&-1\\\
0&0&1&0\\\ 0&-1&0&0\\\ 1&0&0&0\end{pmatrix}$

with the matrix for $F_{2}$ in the semigroup $U_{-}^{>0}$.

The Maslov bilinear form of the three Lagrangians
$F_{1}^{(2)},F_{2}^{(2)},F_{3}^{(2)}$ is then given by

$2\begin{pmatrix}ab-c&b\\\ b&d\end{pmatrix}.$

Since $d>0$ and the determinant $(ab-c)d-b^{2}=b(ad-b)-cd>0$, this matrix is
positive-definite. ∎

###  5.2. Positivity and Legendrian polygons

Given a generic, non-contractible Legendrian polygon $P$ with $2k$ vertices
$p_{1},\dots,p_{2k}$, we can associate to it a $k$-tuple of oriented isotropic
flags as follows. Choose a set of representatives so that
$P=P_{-}(\mathbf{v}_{1},\dots,\mathbf{v}_{2k})$, and let
$F_{1}=(\mathrm{span}(\mathbf{v}_{1})\subset\mathrm{span}(\mathbf{v}_{1},\mathbf{v}_{2}))$,
$F_{2}=(\mathrm{span}(\mathbf{v}_{3})\subset\mathrm{span}(\mathbf{v}_{3},\mathbf{v}_{4}))$,
…,
$F_{k}=(\mathrm{span}(\mathbf{v}_{2k-1})\subset\mathrm{span}(\mathbf{v}_{2k-1},\mathbf{v}_{2k}))$.
Since the representatives $\mathbf{v}_{i}$ are unique up to positive scalar
multiplication or replacing all $\mathbf{v}_{i}$ with $-\mathbf{v}_{i}$, this
procedure is well-defined on $P$. We denote the induced map by

$\mathcal{F}:\mathscr{P}^{-}_{2k}\rightarrow\mathscr{F}^{(k)}.$

######  Remark 5.13.

The configuration of oriented flags $\mathcal{F}(P)$ depends on the labeling
of the vertices of $P$. If we re-label the vertices starting at any odd vertex
$p_{2k+1}$ we obtain a cyclic permutation of the flags, but if we start at an
even vertex or reverse the orientation we obtain different flags. In total, up
to cyclic permutations and choice of cyclic ordering, there are four different
(unlabeled) $k$-tuples of flags that could be associated to the Legendrian
polygon $P$.

######  Proposition 5.14.

Let $P=P_{-}(\mathbf{v}_{1},\dots,\mathbf{v}_{6})$ be a positive-transverse
Legendrian polygon. Then, the associated triple
$\mathcal{F}(P)=(F_{1},F_{2},F_{3})$ is positive.

###### Proof.

We fix a basis $E$ in which the symplectic form is given by $\Omega$. Applying
an element of $\mathsf{PSp}(\mathsf{V},\omega)$, we may assume that
$F_{1}=F_{E}$ and $F_{3}=F_{\widehat{E}}$.

Then, we may scale by positive scalars so that $\mathbf{v}_{1}=\mathbf{e}_{1}$
and $\mathbf{v}_{5}=\mathbf{e}_{4}$. Moreover, since
$F_{1}^{(2)}=\mathrm{span}(\mathbf{v}_{1},\mathbf{v}_{2})$, we have
$\mathbf{v}_{2}=a\mathbf{e}_{1}+\mathbf{e}_{2}$ for some $a\in\mathbb{R}$.
Since $F_{3}^{(2)}=\mathrm{span}(\mathbf{v}_{5},\mathbf{v}_{6})$ and
$\omega(\mathbf{v}_{6},\mathbf{v}_{1})=0$ we have
$\mathbf{v}_{6}=-\mathbf{e}_{3}$.

Using the relations $\omega(\mathbf{v}_{3},\mathbf{v}_{5})>0$ and
$\omega(\mathbf{v}_{4},\mathbf{v}_{6})>0$ we can normalize $\mathbf{v}_{3}$
and $\mathbf{v}_{4}$ and represent the flags $F_{1},F_{2},F_{3}$ by the
following three matrices:

$\begin{pmatrix}1&a\\\ 0&1\\\ 0&0\\\ 0&0\\\
\end{pmatrix},~{}\begin{pmatrix}1&0\\\ b&1\\\ ad&c\\\ d&e\\\
\end{pmatrix},~{}\begin{pmatrix}0&0\\\ 0&0\\\ 0&-1\\\ 1&0\\\ \end{pmatrix},$

where $a,b,c,d,e\in\mathbb{R}$, and $e-bc+ad=0$. We must show that the matrix
representing $F_{2}$ is totally positive. It suffices to show the positivity
of $b$, $ad$, $d$, $bc-ad$, and $(ae-c)d$.

But $b=\omega(\mathbf{v}_{3},\mathbf{v}_{6})>0$,
$a=\omega(\mathbf{v}_{2},\mathbf{v}_{5})>0$,
$d=\omega(\mathbf{v}_{1},\mathbf{v}_{3})>0$, $e=bc-
ad=\omega(\mathbf{v}_{1},\mathbf{v}_{4})>0$, and
$ae-c=\omega(\mathbf{v}_{2},\mathbf{v}_{4})>0$, proving the claim. ∎

We get the following consequence for piecewise circular hexagons.

######  Corollary 5.15.

A simple circular hexagon either has decreasing curvature or it has increasing
curvature.

For Legendrian polygons with more than $6$ vertices however, transversality
does not suffice to guarantee the positivity of the associated tuple of flags.

An explicit example is the transverse Legendrian octagon with vertices
$(1,0,0,0)$, $(1,1,0,0)$, $(1,1,1,1)$, $(4,2,-1,1)$, $(4,8,-4,1)$,
$(0,1,1,3)$, $(0,0,0,1)$, and $(0,0,-1,0)$ in a symplectic basis. We show its
image in the space of oriented contact elements to $S^{2}$ in Figure 4.

Figure 4. A simple circular octagon which is not positive.

######  Proposition 5.16.

Suppose $P=P_{-}(\mathbf{v}_{1},\dots,\mathbf{v}_{2N})$ is positive-transverse
and has decreasing curvature. Then, the associated $N$-tuple
$\mathcal{F}(P)=(F_{1},\dots,F_{N})$ is positive.

###### Proof.

We first show that $F_{1},F_{2},F_{k}$ is positive for $k=3,\dots,N$.

Choose a symplectic basis such that $\mathbf{v}_{1}=\mathbf{e}_{1}$,
$\mathbf{v}_{2}=a\mathbf{e}_{1}+\mathbf{e}_{2}$,
$\mathbf{v}_{2k-1}=\mathbf{e}_{4}$ and
$\mathbf{v}_{2k}=-\mathbf{e}_{3}+b\mathbf{e}_{4}$. Then, normalizing each
vector we may write
$\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{2k-1},\mathbf{v}_{2k}$
as the columns of the following matrix:

$\begin{pmatrix}1&a&1&1&0&0\\\ 0&1&c&e&0&0\\\ 0&0&ad&f&0&-1\\\ 0&0&d&d+cf-
ade&1&b\\\ \end{pmatrix}.$

So we must show that the triple of flags

(2) $\begin{pmatrix}1&0\\\ 0&1\\\ 0&0\\\ 0&0\\\
\end{pmatrix},~{}\begin{pmatrix}1&0\\\ c&1\\\ ad&\frac{f-ad}{e-c}\\\
d&\frac{cf-ade}{e-c}\end{pmatrix},~{}\begin{pmatrix}0&0\\\ 0&0\\\ 0&-1\\\
1&0\\\ \end{pmatrix}$

is positive, and for this it suffices to show that the second matrix is
totally positive.

From oriented transversality we obtain the inequalities: $d>0$, $a>0$, $d+cf-
ade>0$,

(3) $a(d+cf-ade)-f>0.$

The decreasing curvature condition means that the Maslov index of the three
Lagrangians (2) is $1$. We find that the Maslov bilinear form is

$\begin{pmatrix}(ac-1)d&d(ae-1)\\\ d(ae-1)&d(ae-1)+(e-c)f\\\ \end{pmatrix}$

and so from Sylvester’s positivity criterion we obtain

(4) $(ac-1)d>0$

and

(5) $d(e-c)(a(d+cf-ade)-f)>0.$

Inequality (4) together with $a>0$ and $d>0$ immediately implies $c>0$.

Inequality (5), together with $d>0$ and inequality (3) implies that $e-c>0$.

Moreover, inequalities (3) and (4) as well as $d>0$ imply

$f>\frac{ad(ae-1)}{ac-1}.$

It follows that

$\displaystyle cf-ade$ $\displaystyle>\frac{c(ad)(ae-1)-ade(ac-1)}{ac-1}$
$\displaystyle=\frac{ad(e-c)}{ac-1}>0.$

Finally, using equation (3)

$ad(d+cf-ade)-df>df-df=0.$

This shows that the triple $F_{1},F_{2},F_{k}$ is positive.

Since transversality and monotonicity are invariant under cyclic permutations,
we might have chosen any odd index for $\mathbf{v}_{1}$ and so all triples of
the form $F_{i},F_{i+1},F_{k}$ with $i+1\neq k$ are positive. This implies, by
Lemma 5.8, that the $N$-tuple of flags is positive. ∎

We illustrate examples of simple circular $2k$-gons with decreasing curvature
in Figures 1 and 5.

Figure 5. A simple circular $20$-gon with decreasing curvature. Oriented
contact elements are indicated by arrows at the vertices.

Next we will define an left inverse for $\mathcal{F}$. Given oriented
subspaces $U,W\subset\mathsf{V}$, their intersection $U\cap W$ is endowed with
the unique orientation satisfying: whenever $e_{1},\dots,e_{N}$ is an oriented
basis of $U+W$ such that $e_{1},\dots,e_{k_{2}}$ is an oriented basis of $U$
and $e_{k_{1}},\dots,e_{N}$ is an oriented basis of $W$, then
$e_{k_{1}},\dots,e_{k_{2}}$ is an oriented basis of $U\cap W$.

Define a map $\mathcal{C}:\mathscr{F}_{>0}^{(k)}\rightarrow\mathscr{P}_{2k}$
by

$\displaystyle\mathcal{C}(F_{1},\dots,F_{k})$ $\displaystyle=$ $\displaystyle
P_{-}(F_{1}^{(1)},$ $\displaystyle~{}F_{1}^{(2)}\cap
F_{2}^{(3)},F_{2}^{(1)},\dots,F_{k-1}^{(2)}\cap
F_{k}^{(3)},F_{k}^{(1)},F_{k}^{(2)}\cap F_{1}^{(3)}),$

where we slightly abuse notation since the image of $P_{-}$ does not depend on
the choice of representative for each oriented $1$-dimensional subspace.

######  Example 5.17.

The image by $\mathcal{C}$ of the parametrized triple of flags from Example
5.11 is the Legendrian polygon $P_{-}(\mathbf{v}_{1},\dots,\mathbf{v}_{6})$,
where

$\displaystyle\mathbf{v}_{1}=\begin{pmatrix}1\\\ 0\\\ 0\\\
0\end{pmatrix},~{}\mathbf{v}_{2}=\begin{pmatrix}y\\\ 1\\\ 0\\\
0\end{pmatrix},~{}\mathbf{v}_{3}=\begin{pmatrix}1\\\ x+y+\frac{1}{y}\\\ y\\\
1\end{pmatrix},$ $\displaystyle\mathbf{v}_{4}=\begin{pmatrix}0\\\ 1\\\ 1\\\
x+\frac{1}{y}\end{pmatrix},~{}\mathbf{v}_{5}=\begin{pmatrix}0\\\ 0\\\ 0\\\
1\end{pmatrix},~{}\mathbf{v}_{6}=\begin{pmatrix}0\\\ 0\\\ -1\\\
0\end{pmatrix}.$

It is simple to check that it is positive-transverse and has decreasing
curvature.

######  Proposition 5.18.

The map $\mathcal{F}$ is a homeomorphism onto its image.

###### Proof.

Let $P=P_{-}(\mathbf{v}_{1},\dots,\mathbf{v}_{2k})$ be a generic,
noncontractible Legendrian polygon let $F_{1},\dots,F_{k}=\mathcal{F}(P)$.
Since $\omega(\mathbf{v_{2j-1}},\mathbf{v_{2k-1}})\neq 0$ for all $j\neq k$,
we have that $F_{j}^{(1)}$ is transverse to $F_{k}^{(3)}$ for all $j\neq k$.
Moreover, since the edges spanned by $\mathbf{v}_{2j-1},\mathbf{v}_{2j}$ and
$\mathbf{v}_{2k-1},\mathbf{v}_{2k}$ are linearly independent, $F_{j}^{(2)}$ is
transverse to $F_{k}^{(2)}$ for all $j\neq k$. We conclude that the image of
$\mathcal{F}$ is an $n$-tuple of pairwise transverse flags.

The map $\mathcal{C}$, when restricted to the image
$\mathcal{F}(\mathscr{P}^{-}_{2k})$, is an inverse of $\mathcal{F}$. Both maps
are continuous since they are $\mathsf{PSp}(\mathsf{V},\omega)$-equivariant
and the topologies on both moduli spaces are induced by that of the group, so
the claim follows. ∎

######  Proposition 5.19.

Let $F_{1},\dots,F_{k}$ be a positive $k$-tuple of oriented isotropic flags.
Then, $\mathcal{C}(F_{1},\dots,F_{k})$ is a positive-transverse Legendrian
$2k$-gon with decreasing curvature.

###### Proof.

Positivity of the tuple $F_{1},\dots,F_{k}$ means positivity of every
cyclically ordered sub-triple. Consider a sub-triple of the form
$F_{i},F_{i+1},F_{j}$ with $j>i+1$ and apply an element of
$\mathsf{PSp}(4,\mathbb{R})$ so that

$F_{i}=\begin{pmatrix}1&0\\\ 0&1\\\ 0&0\\\ 0&0\end{pmatrix},\quad
F_{i+1}=\begin{pmatrix}1&0\\\ a&1\\\ b&d\\\ c&ad-b\end{pmatrix},\quad
F_{j}=\begin{pmatrix}0&0\\\ 0&0\\\ 0&-1\\\ 1&0\end{pmatrix}.$

By positivity, $a,b,c>0$, $ad-b>0$, and $-b^{2}+abd-cd>0$. Then, the vertices
$\mathbf{v}_{2i-1},\mathbf{v}_{2i},\mathbf{v}_{2i+1},\mathbf{v}_{2i+2},\mathbf{v}_{2j-1},\mathbf{v}_{2j}$
of $\mathcal{C}(F_{1},\dots,F_{k})$ have coordinates of the form given in the
columns of the following matrix:

$\begin{pmatrix}1&\frac{b}{c}&1&k_{1}&0&0\\\ 0&1&a&1+k_{1}a&0&0\\\
0&0&b&d+k_{1}b&0&-1\\\ 0&0&c&ad-b+k_{1}c&1&k_{2}\\\ \end{pmatrix}$

for some $k_{1},k_{2}\in\mathbb{R}$.

We first show that $k_{1},k_{2}\geq 0$.

If $j=i+2$, then $k_{1}=0$. Otherwise, note that the flag $F_{i+2}$ has the
form

$F_{i+2}=\begin{pmatrix}1&0&0&0\\\ x&1&0&0\\\ y&w&1&0\\\
z&xw-y&x&1\end{pmatrix},$

and by positivity of the quadruple $F_{0},F_{i+1},F_{i+2},F_{j}$ and
Proposition 5.7,

$\displaystyle\begin{pmatrix}1&0&0&0\\\ a&1&0&0\\\ b&d&1&0\\\
c&ad-b&a&1\end{pmatrix}^{-1}\begin{pmatrix}1&0&0&0\\\ x&1&0&0\\\ y&w&1&0\\\
z&xw-y&x&1\end{pmatrix}=$ $\displaystyle\begin{pmatrix}1&0&0&0\\\ x-a&1&0&0\\\
ad-b+y-dx&w-d&1&0\\\ z-c-ay+bx&b-y+w(x-a)&x-a&1\end{pmatrix}\in U_{-}^{>0}.$

The fact that $\omega(\mathbf{v}_{2i+2},\mathbf{v}_{2i+3})=0$ implies that
$k_{1}=\frac{ad-b+y-dx}{-ay+bx+z-c}$, which is positive since the numerator
and denominator appear as entries in the first column of the above totally
positive lower triangular matrix.

Similarly, if $j=k$ and $i=1$ so that $F_{j}$ and $F_{i}$ are adjacent flags
in the cyclic tuple, then $k_{2}=0$. Otherwise, the flag $F_{j+1}$ is of the
form

$\begin{pmatrix}1&0&0&0\\\ -\alpha&-1&0&0\\\ \beta&\delta&1&0\\\
-\gamma&-\alpha\delta+\beta&-\alpha&-1\end{pmatrix}\in KU_{-}^{>0}K,$

and $k_{2}=\alpha>0$.

Now we compute the symplectic products:

$\displaystyle\omega(\mathbf{v}_{2i-1},\mathbf{v}_{2i+1})=c$
$\displaystyle>0;$
$\displaystyle\omega(\mathbf{v}_{2i-1},\mathbf{v}_{2i+2})=ad-b+kc$
$\displaystyle>0;$
$\displaystyle\omega(\mathbf{v}_{2i-1},\mathbf{v}_{2i+2})=ad-b+kc$
$\displaystyle>0;$
$\displaystyle\omega(\mathbf{v}_{2i-1},\mathbf{v}_{2j-1})=1$
$\displaystyle>0;$
$\displaystyle\omega(\mathbf{v}_{2i-1},\mathbf{v}_{2j})=k_{2}$
$\displaystyle\geq 0;$
$\displaystyle\omega(\mathbf{v}_{2i},\mathbf{v}_{2i+2})=\frac{-b^{2}+abd-
cd}{c}$ $\displaystyle>0;$
$\displaystyle\omega(\mathbf{v}_{2i},\mathbf{v}_{2j-1})=\frac{b}{c}$
$\displaystyle>0;$
$\displaystyle\omega(\mathbf{v}_{2i},\mathbf{v}_{2j})=\frac{bk_{2}}{c}+1$
$\displaystyle>0;$
$\displaystyle\omega(\mathbf{v}_{2i+1},\mathbf{v}_{2j-1})=1$
$\displaystyle>0;$
$\displaystyle\omega(\mathbf{v}_{2i+1},\mathbf{v}_{2j})=k_{2}+a$
$\displaystyle>0;$
$\displaystyle\omega(\mathbf{v}_{2i+2},\mathbf{v}_{2j-1})=k_{1}$
$\displaystyle\geq 0;$
$\displaystyle\omega(\mathbf{v}_{2i+2},\mathbf{v}_{2j})=k_{1}k_{2}+1+k_{1}a$
$\displaystyle>0.$

The two non-strict inequalities are strict as soon as they are not forced to
be zero which occurs precisely when $F_{j}$ is adjacent to $F_{i}$ or
$F_{i+1}$, in which case the vectors in the inequality represent adjacent
vertices.

Since $i,j$ are arbitrary, we conclude that
$\omega(\mathbf{v}_{i},\mathbf{v}_{j})>0$ whenever $i+1<j$ and
$\mathcal{C}(F_{1},\dots,F_{k})$ is positive-transverse.

Finally, we conclude from Lemma 5.12 that the Maslov indices are $+1$, and so
the Legendrian polygon has decreasing curvature. ∎

######  Theorem 5.20.

The set of positive-transverse Legendrian $2k$-gons with decreasing curvature
is a connected component of $\mathscr{P}_{2k}$ homeomorphic to a ball.

###### Proof.

By Proposition 5.18, the map
$\mathcal{F}:\mathscr{P}^{-}_{2k}\rightarrow\mathscr{F}^{(k)}$ is a
homeomorphism onto its image. By Propositions 5.16 and 5.19, the set of
positive-transverse polygons with decreasing curvature is mapped onto the set
of positive tuples of flags by $\mathcal{F}$. We conclude, by the Fock-
Goncharov theorem (Theorem 5.9, Proposition 5.10), that the former is a
connected component homeomorphic to a ball. ∎

Translating with the dictionary of Section 4, we find

######  Corollary 5.21.

The subspace of simple circular $2k$-gons with decreasing curvature is a
connected component homeomorphic to a ball in the moduli space of generic
circular $2k$-gons.

###  5.3. Flag curves

Let $\gamma:\mathbb{R}\rightarrow\mathbb{P}_{\mathbb{C}}(\mathsf{V})$ be a
smooth curve. A _Legendrian lift_ (or _contact lift_) of $\gamma$ to
$\mathbb{P}_{\mathbb{R}}(V)\cong\mathbb{S}(T^{*}\mathbb{P}_{\mathbb{C}}(\mathsf{V}))$
is a curve $\hat{\gamma}:\mathbb{R}\rightarrow\mathbb{P}_{\mathbb{R}}(V)$ such
that $\pi\circ\hat{\gamma}=\gamma$. There are exactly two contact lifts of
$\gamma$, given by a choice of transverse orientation.

######  Proposition 5.22.

Let $\gamma:\mathbb{R}\rightarrow\mathbb{P}_{\mathbb{C}}(\mathsf{V})$ be a
$C^{2}$ curve with non-vanishing curvature. Let $\hat{\gamma}$ be one of its
two contact lifts to $\mathbb{P}_{\mathbb{R}}(V)$. Then, the tangent lines to
$\hat{\gamma}$ are Lagrangians which project to the osculating circles to
$\gamma$.

###### Proof.

For this proof we use coordinates as in Section 4.1. Assume without loss of
generality that $\gamma$ is parametrized by arc length and write, in an affine
chart,

$\gamma(s)=\begin{bmatrix}1\\\ z(s)\end{bmatrix}.$

Then, the contact lifts of $\gamma$ are

$\hat{\gamma}(s)=\sqrt{\pm\overline{T(s)}}\begin{pmatrix}1\\\
z(s)\end{pmatrix},$

where $T(s):=z^{\prime}(s)$ is the unit tangent vector to $z(s)$. Denote
$T^{\prime}(s)=k(s)N(s)$, where $N=iT$ is the unit normal vector and $k(s)$
the signed curvature function. We have

$\hat{\gamma}^{\prime}(s)=\begin{pmatrix}\frac{\pm
k(s)\overline{N(s)}}{2\sqrt{\overline{\pm T(s)}}}\\\ \frac{\pm
k(s)\overline{N(s)}z(s)}{2\sqrt{\overline{\pm T(s)}}}+\sqrt{\overline{\pm
T(s)}}T(s)\end{pmatrix},$

and a simple computation shows that
$\omega_{\mathbb{C}}(\hat{\gamma}(s),\hat{\gamma}^{\prime}(s))=\pm i$,
verifying that $\hat{\gamma}$ is Legendrian for
$\omega=\Re(\omega_{\mathbb{C}})$.

We also compute, choosing the positive sign in the choice of contact lift:

$\displaystyle\pi(a\hat{\gamma}(s)+b\hat{\gamma}^{\prime}(s))$
$\displaystyle=\begin{bmatrix}2a\overline{T(s)}+bk(s)\overline{N(s)}\\\
z(s)(2a\overline{T(s)}+bk(s)\overline{N(s)})+2b\end{bmatrix}$
$\displaystyle=\begin{bmatrix}1\\\
z(s)+\frac{2b}{2a\overline{T(s)}+bk(s)\overline{N(s)}}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}1\\\
z(s)+\frac{1}{k(s)}N(s)-\frac{1}{k(s)}N(s)+\frac{4abT(s)+2b^{2}k(s)N(s)}{4a^{2}+b^{2}k(s)^{2}}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}1\\\
z(s)+\frac{1}{k(s)}N(s)+\frac{4abT(s)+2b^{2}k(s)N(s)}{k(s)(4a^{2}+b^{2}k(s)^{2})}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}1\\\
z(s)+\frac{1}{k(s)}N(s)+\frac{1}{k(s)}\frac{(bk(s)-2ai)^{2}N(s)}{(4a^{2}+b^{2}k(s)^{2})}\end{bmatrix}$

which, since $\left|\frac{(bk(s)-2ai)^{2}}{(4a^{2}+b^{2}k(s)^{2})}\right|=1$,
is a parametrization of the circle of radius $\frac{1}{k(s)}$ centered at
$z(s)+\frac{1}{k(s)}N(s)$ in the affine patch. This circle is the osculating
circle to the curve $z$ at $z(s)$. A similar computation with the negative
sign yields the same circle (but corresponds to the opposite co-orientation).
∎

Let $\gamma:S^{1}\rightarrow\mathbb{P}_{\mathbb{R}}(\mathsf{V})$ be a
Legendrian curve. Motivated by the previous proposition, we call the
projection to $\mathbb{P}_{\mathbb{C}}(\mathsf{V})$ of the tangent line at
$\gamma(t)$ the _osculating circle_ to $\pi\circ\gamma$ at $\pi(\gamma(t))$.

Accordingly, we say that $\gamma$ has _decreasing curvature_ if for every
cyclically ordered $t_{1},t_{2},t_{3}\in S^{1}$, the triple of Lagrangians
given by the tangent lines to $\gamma$ at $\gamma(t_{i})$ have Maslov index
$1$. See Figure 8 for an example of osculating circles to a decreasing
curvature curve.

Now consider a curve $\xi:S^{1}\rightarrow\mathscr{F}$ in the space of
isotropic flags. The curve $\xi$ is _Frenet_ if whenever
$n_{1}+n_{2}+n_{3}=p\leq 3$, $0\leq n_{i}\leq 2$ we have

$\lim_{\begin{subarray}{c}t_{1},t_{2},t_{3}\text{ distinct}\\\
t_{i}\rightarrow
t\end{subarray}}\xi(t_{1})^{(n_{1})}\oplus\xi(t_{2})^{(n_{2})}\oplus\xi(t_{3})^{(n_{3})}=\xi(t)^{(p)}.$

######  Proposition 5.23.

If $\xi$ is Frenet, the projection $\pi(\xi(t)^{(2)})$ is the osculating
circle to the curve $\gamma=\pi\circ\gamma^{(1)}$ at $\gamma(t)$.

###### Proof.

By definition of a Frenet flag curve, $\xi(t)^{(2)}$ is the tangent line to
$\xi^{(1)}$ at $\xi(t)$, so its projection to
$\mathbb{P}_{\mathbb{C}}(\mathsf{V})$ is the osculating circle to the
projection $\pi\circ\xi^{(1)}$. ∎

A closed curve $\xi:S^{1}\rightarrow\mathscr{F}$ is _positive_ if for every
cyclically ordered triple $t_{1},t_{2},t_{3}\in S^{1}$, the triple of flags
$\xi(t_{1}),\xi(t_{2}),\xi(t_{3})$ is positive. We obtain:

######  Theorem 5.24.

A positive Frenet curve in $\mathscr{F}$ is the tangent curve to the contact
lift of an decreasing curvature simple closed curve in
$\mathbb{P}_{\mathbb{C}}(\mathsf{V})$.

By theorems of Labourie [Lab06, Theorem 1.4] and Fock-Goncharov [FG06,
Theorems 1.14 and 1.15], we know that the limit curve of a Hitchin
representation is a positive Frenet curve.

We conclude

######  Corollary 5.25.

The limit curve of a Hitchin representation in $\mathsf{PSp}(\mathsf{V})$ is
the contact lift of a simple closed curve in
$\mathbb{P}_{\mathbb{C}}(\mathsf{V})$ with decreasing curvature.

For example, for a _Fuchsian_ representation, that is, one which factors
through the irreducible representation
$\mathsf{PSL}(2,\mathbb{R})\rightarrow\mathsf{PSp}(\mathsf{V})$, the limit
curve is the Veronese curve (or _twisted cubic_) and its projection to
$\mathbb{P}_{\mathbb{C}}(\mathsf{V})$ is depicted in Figure 6.

Figure 6. The projection to $\mathbb{P}_{\mathbb{C}}(\mathsf{V})$ of the
Veronese curve.

We show the collection of osculating circles to the Veronese curve in Figure
7, and highlight the decreasing curvature by showing only half of them in
Figure 8.

Figure 7. The osculating circles to the projected Veronese curve. Color
represents orientation. Figure 8. The osculating circles to half of the
projected Veronese curve.

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