# Constrained elastic curves and surfaces with spherical curvature lines

Joseph Cho  Vienna University of Technology, Wiedner Hauptstraße 8-10/104,
A-1040 Vienna. Austria<EMAIL_ADDRESS>,  Mason Pember
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi
24, I-10129 Torino, Italy<EMAIL_ADDRESS>and  Gudrun Szewieczek
Vienna University of Technology, Wiedner Hauptstraße 8-10/104, A-1040 Vienna.
Austria<EMAIL_ADDRESS>
###### Abstract.

In this paper we consider surfaces with one or two families of spherical
curvature lines. We show that every surface with a family of spherical
curvature lines can locally be generated by a pair of initial data: a suitable
curve of Lie sphere transformations and a spherical Legendre curve. We then
provide conditions on the initial data for which such a surface is Lie
applicable, an integrable class of surfaces that includes cmc and
pseudospherical surfaces. In particular we show that a Lie applicable surface
with exactly one family of spherical curvature lines must be generated by the
lift of a constrained elastic curve in some space form. In view of this goal,
we give a Lie sphere geometric characterisation of constrained elastic curves
via polynomial conserved quantities of a certain family of connections.

###### 2020 Mathematics Subject Classification:

Primary 53B25; Secondary 53A40, 53C12, 53E99

##  1\. Introduction

###  1.1. Background

In 1691 Jacob Bernoulli posed the problem of finding the shape of an elastic
rod; Daniel Bernoulli proposed that the shape taken can be modelled by curves
$\gamma$, called _elastic curves_ , that are critical for the bending energy
(also known as the total squared curvature)

(1.1) $\int_{\gamma}k^{2}ds,$

with respect to variations that fix length, where $s$ is the arc length
parameter of $\gamma$, and Euler [E1744] gave a complete classification of
these curves in the plane. Geometrically, such curves are characterised by the
property that their curvature is proportional to the distance of the curve
from a fixed line. These curves have since been generalised to Riemannian
2-dimensional space forms in [BG1986, LS1984i] and pseudo-Riemannian space
forms (see for example [FGJL2006, J2004, P2018]). Moreover, these curves
appear in various geometric contexts, such as in the construction of Willmore
tori using the Hopf fibration in [P1985ii] and channel Willmore surfaces in
[BG1986, H2003, LS1984ii, MN1999ii].

By considering the critical curves of (1.1) with respect to variations that
fix length and enclosed area, we obtain a larger class of curves called
constrained elastic curves. These were studied in [BPP2008] and are used in
[H2014] to create constrained Willmore tori via the Hopf fibration. The Euler-
Lagrange equation for this variational problem is the stationary modified
Korteweg-de-Vries (mKdV) equation. Thus constrained elastic curves are the
stationary curves of the mKdV flow. In this paper we shall see that these
curves appear in the study of Lie applicable surfaces with spherical curvature
lines.

On the other hand, our examination of surfaces with spherical curvature lines
is rooted in the works of classical geometers in the 19th century and early
20th century. Most of the work in the era focused on classifying surfaces with
certain curvature restrictions which admit one family of planar or spherical
curvature lines. Bonnet considered minimal surfaces with planar curvature
lines in [B1855] (see also [CO2017]). It was Enneper who first looked into
constant curvature surfaces with one family of spherical curvature lines in
his work [E1868], pioneering the way for examining these surfaces via
analytical methods. Dobriner then used these methods to classify all
pseudospherical surfaces and minimal surfaces with one family of spherical
curvature lines in [D1887i, D1887ii]. The early developments of these
analytical methods are well documented in the works of Enneper [E1878],
Darboux [D1889, D1896], or Eisenhart [E1909], and these works had a profound
impact in the modern surface theory scene.

A prototypical example of such modern development can be found in the study of
constant mean curvature (cmc) surfaces in the Euclidean space. A long standing
problem of Hopf was to determine whether any cmc surface that is both compact
and complete must necessarily be a sphere, and it was Wente in 1984 who was
influenced by the aforementioned classical works to find a counterexample to
this problem now known as the Wente torus in [W1986]. Abresch and Walter then
independently noticed that the simplifying ansatz to facilitate finding such
examples was precisely the geometric problem Enneper looked into: assuming
that one family of curvature lines is planar. Abresch in particular
rediscovered the analytical methods of Enneper and applied them to finding cmc
tori in [A1987]; Walter noticed that the ansatz of one family of curvature
lines being planar forces the other family of curvature lines to be spherical
in [W1987], leading to an explicit parametrization of the cmc tori. We note
here that the classical geometers had already found these surfaces, but their
compactness was not known until Wente’s rediscovery.

The ansatz of Enneper, requiring that one family of curvature lines be
spherical, has had pervasive modern influence, extending beyond the class of
cmc surfaces in the Euclidean space. Minimal tori in the $3$-sphere were
discovered via the spherical curvature lines condition in [Y1987]. Isothermic
tori were found by assuming that both families of curvature lines are
spherical in [B2001]. Examples of linear Weingarten surfaces with spherical
curvature lines were constructed in [T2002, CFT2003] via Ribaucour
transformations of the circular cylinder. Surfaces with planar curvature lines
were studied in the realm of Laguerre geometry in [MN1999i], and their
applications to architecture have been explored in [MDBL2018]. Monge surfaces
have one family of planar curvature lines, and their application was explored
in [BG2018]. Surfaces with two families of spherical curvature lines were
shown to admit a $3$-parameter family of Lie deformations in [F2002], and it
is demonstrated how these surfaces relate to commuting Schrödinger operators
with magnetic fields.

Many of the surfaces examined in the aforementioned works lie in the
intersection of two larger classes of surfaces: surfaces with spherical
curvature lines and Lie applicable surfaces [F2002, MN2006]. The class of Lie
applicable surfaces is comprised of exactly the $\Omega$\- and
$\Omega_{0}$-surfaces, first discovered by Demoulin in [D1911i, D1911ii,
D1911iii], and includes a wide range of well-studied surface classes:
isothermic surfaces such as minimal and cmc surfaces, Guichard surfaces such
as pseudospherical and non-cmc linear Weingarten surfaces, and $L$-isothermic
surfaces such as minimal surfaces and linear Weingarten surfaces of Bryant
type. Thus, the examination of Lie applicable surfaces often leads to a
uniform approach to considering these various surface classes, with their
transformation theory being one of such examples [BHPR2019].

Lie applicable surfaces are Lie sphere invariant, allowing Lie sphere geometry
to step in as the natural setting for examining these surfaces. The
comprehensive work of Blaschke [B1929] paved the way for using the sphere
geometries of Möbius, Laguerre, and Lie to study various surface classes.
Blaschke derived invariants of surfaces in Lie sphere geometry that
generically determine a surface up to Lie sphere transformations. The surfaces
that are not determined are precisely the Lie applicable surfaces. Moreover,
he showed how surfaces with a family of spherical curvature lines can be
studied in this geometry via the use of sphere complexes, in particular
showing that curves in this family are Lie sphere transformations of each
other.

The tools of Lie sphere geometry have also received multifarious modern
interest, with a comprehensive introduction provided in [C2008]. We now list
some of such results that are particularly relevant to our work. Ribaucour
transformations were given a Lie sphere geometric characterisation in
[BH2006]. Flat fronts in hyperbolic geometry and linear Weingarten surfaces in
space forms were characterised via Lie sphere geometry in [BHR2010, BHR2012].
Channel surfaces and their transformations were given a modern treatment in
the realm of Lie sphere geometry in [PS2018]. Lie applicable surfaces were
shown to constitute an integrable system in [F2000ii] and their
characterisation as the deformable surfaces of Lie sphere geometry was shown
in [MN2006]. A gauge-theoretic characterisation of Lie applicable surfaces and
their transformations was given in [C2012i, P2020], and the identification of
subclasses of Lie applicable surfaces via polynomial conserved quantities was
examined in [BHPR2019].

###  1.2. Objectives

Having reached an understanding of the historical and modern significance of
surfaces with spherical curvature lines and the class of Lie applicable
surfaces, we now state the aim of the paper. We seek to provide a modern
reinterpretation of Blaschke’s methods of examining surfaces with spherical
curvature lines, and use these methods to characterise all Lie applicable
surfaces with spherical curvature lines. In the pursuit of such a
characterisation, we will see that constrained elastic curves play a central
role. Therefore, using Lie sphere geometry, we will:

  * •

characterise constrained elastic curves in Lie sphere geometry via polynomial
conserved quantities of a certain $1$-parameter family of connections,

  * •

show that every surface with spherical curvature lines is generated by a
certain curve of Lie sphere transformations applied to an initial spherical
curve, and

  * •

characterise Lie applicable surfaces with spherical curvature lines by
restricting the curve of Lie sphere transformations and initial spherical
curve, namely to a constrained elastic curve in an appropriate space form.

Along the way we shall obtain additional results regarding surfaces with
spherical curvature lines, for example showing that any such surface is a
Ribaucour transform of a $3$-parameter family of channel surfaces.

###  1.3. Structure of the paper

After a preparatory Section 2, where we introduce the basics of Lie sphere
geometry and the gauge-theoretic description of Lie applicable surfaces, we
restrict our attention to the theory of curves in $2$-dimensional Riemannian
and Lorentzian space forms in Section 3. In Theorem 3.13 we prove that
constrained elastic curves can be characterised by a certain enveloped circle
congruence belonging to a circle complex. A gauge-theoretic approach to curve
theory was given in the realm of Möbius geometry in [BHMR2016], which we apply
to our situation of Lie sphere geometry, culminating in the characterisation
of constrained elastic curves in terms of polynomial conserved quantities in
Theorem 3.15, analogous to the approaches of [BS2012, BHPR2019].

We then switch our attention to surfaces with spherical curvature lines in
Section 4, where we mainly focus on reinterpreting Blaschke’s work in modern
context. After characterising surfaces with spherical curvature lines in Lie
sphere geometry in Theorem 4.3, we show that every surface with spherical
curvature lines is obtained via a certain $1$-parameter family of Lie sphere
transformations of a given spherical curve in Theorem 4.8, which we call
_spherical evolution map_ (see Definition 4.7). We also recover the
_osculating sphere complexes_ of Blaschke, obtaining another Lie sphere
geometric characterisation of surfaces with spherical curvature lines in
Proposition 4.19. Finally, we break symmetry to conformal geometry, Laguerre
geometry, and Euclidean geometry to characterise surfaces with orthogonally
intersecting spherical curvature lines, surfaces with planar curvature lines,
and Monge surfaces, respectively, in Section 4.4. This then enables us to
recover Blaschke’s classification of surfaces with two families of spherical
curvature lines.

In Section 5, we examine the relationship between surfaces with spherical
curvature lines and Ribaucour transformations. Blaschke showed that Ribaucour
transformations of a channel surface must have one family of spherical
curvature lines. We consider the converse statement, and show that every
surface with spherical curvature lines must be a Ribaucour transform of a
$3$-parameter family of channel surfaces in Theorem 5.6. We then provide an
alternative geometric proof of this result using the spherical evolution map.

Finally, in Section 6, we consider Lie applicable surfaces with spherical
curvature lines. We obtain a characterisation of such surfaces in Theorem 6.3,
showing that either both families of curvature lines are spherical or that the
surface is obtained by a certain evolution of a constrained elastic curve. We
then further examine the case that both families of curvature lines are
spherical, using the curved flat formalism of [BP2020]. In so doing we show
that Joachimsthal surfaces are characterised as the Darboux transforms of
Dupin cyclides in Theorem 6.6. Finally, we show that Calapso transforms of Lie
applicable surfaces preserve the property of spherical curvature lines.

###  1.4. Acknowledgements.

We would like to thank Shoichi Fujimori, Udo Hertrich-Jeromin, Emilio Musso,
Yuta Ogata and Álvaro Pámpano for pleasant and fruitful conversations about
the topics of this paper.

We gratefully acknowledge the financial support of the following research
grants: the JSPS/FWF Joint Project I3809-N32 “Geometric shape generation”; the
JSPS Grant-in-Aid for JSPS Fellows 19J10679; the FWF research project
P28427-N35 “Non-rigidity and symmetry breaking”; GNSAGA of INdAM and the MIUR
grant “Dipartimenti di Eccellenza” 2018 - 2022, CUP: E11G18000350001, DISMA,
Politecnico di Torino.

##  2\. Lie sphere geometry

Let $n\in\mathbb{N}$ and let $\mathbb{R}^{n+2,2}$ denote the
$(n+4)$-dimensional pseudo-Euclidean space equipped with a non-degenerate
symmetric bilinear form $(,)$ of signature $(n+2,2)$. Denoting by
$\mathcal{L}$ the light cone of $\mathbb{R}^{n+2,2}$, the Lie quadric
$\mathbb{P}(\mathcal{L})$ parametrises oriented hyperspheres of
$(n+1)$-dimensional space forms (see for example [C2008]). Two points in the
Lie quadric lie on a line in $\mathbb{P}(\mathcal{L})$ if and only if the
corresponding spheres are in oriented contact. Thereby, the Grassmannian of
isotropic $2$-planes in $\mathbb{R}^{n+2,2}$, denoted by $\mathcal{Z}$,
represents the manifold of contact elements.

In this setting, the orthogonal group $\operatorname{O}(n+2,2)$ acts
transitively on $\mathcal{L}$, and is a double cover for the group of Lie
sphere transformations. Throughout the paper, we identify the Lie algebra
$\mathfrak{o}(n+2,2)$ with the exterior algebra $\wedge^{2}\mathbb{R}^{n+2,2}$
via

$(a\wedge b)c=(a,c)b-(b,c)a$

for $a,b,c\in\mathbb{R}^{n+2,2}$.

Let $\Sigma$ be a simply-connected $n$-dimensional manifold. A smooth map
$f\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathcal{Z}$ can be considered
as a rank 2 null subbundle of the trivial bundle
$\underline{\mathbb{R}}^{n+2,2}\mathrel{\mathop{\mathchar
58\relax}}=\Sigma\times\mathbb{R}^{n+2,2}$. We shall denote by $f^{(1)}$ the
set of sections of $f$ and derivatives of sections of $f$. In Lie sphere
geometry, one studies hypersurfaces by considering their contact lifts which
in this setting amounts to the following definition:

######  Definition 2.1.

A map $f\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathcal{Z}$ is called a
_Legendre map_ if $f^{(1)}\leq f^{\perp}$ and, for any $x\in\Sigma$, if $X\in
T_{x}\Sigma$ such that $d\sigma(X)\in\Gamma f$ for all $\sigma\in\Gamma f$
then $X=0$.

For any rank 2 null subbundle $f$ of $\underline{\mathbb{R}}^{n+2,2}$ we have
that $f^{\perp}/f$ is a rank $n$ subbundle of
$\underline{\mathbb{R}}^{n+2,2}/f$ that inherits a positive definite metric
from $\mathbb{R}^{n+2,2}$. When $f$ is a Legendre map, we have that
$f^{(1)}=f^{\perp}$.

Suppose that $f\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathcal{Z}$ is a
Legendre map. Fix a point $x\in\Sigma$. We say that a 1-dimensional subspace
$s(x)\in f(x)$ is a curvature sphere of $f$ at $x$ if there exists a non-zero
subspace $T_{s(x)}\leq T_{x}\Sigma$ such that $d_{X}\sigma\in f(x)$ for any
$X\in T_{s(x)}$ and $\sigma\in\Gamma f$ such that $\sigma(x)\in s(x)$. We call
the maximal $T_{s(x)}$ the curvature space of $s(x)$. We say that a point
$x\in\Sigma$ is umbilic if there exists exactly one curvature sphere $s(x)\leq
f(x)$, in which case $T_{s(x)}=T_{x}\Sigma$.

###  2.1. Symmetry breaking

As seen in [BHPR2019, C2008] one can break the symmetry of Lie sphere geometry
to obtain conformal geometry and space form geometries. We briefly recall this
process in this subsection.

Fix $\mathfrak{p}\in\mathbb{R}^{n+2,2}$ with $\chi\mathrel{\mathop{\mathchar
58\relax}}=-(\mathfrak{p},\mathfrak{p})=\pm 1$. If $\chi=1$ then
$\langle\mathfrak{p}\rangle^{\perp}\cong\mathbb{R}^{n+2,1}$ defines a
Riemannian conformal geometry (see for example [H2003]), whereas if $\chi=-1$
then $\langle\mathfrak{p}\rangle^{\perp}\cong\mathbb{R}^{n+1,2}$ defines a
Lorentzian conformal geometry. We call $\mathfrak{p}$ the point sphere complex
of the conformal geometry defined by $\langle\mathfrak{p}\rangle^{\perp}$. Lie
sphere transformations $A\in\operatorname{O}(n+2,2)$ satisfying
$A\mathfrak{p}=\mathfrak{p}$ are then identified with conformal
transformations.

In the case that $\chi=1$, the conformal $(n+1)$-sphere $S^{n+1}$ is
identified with
$\mathbb{P}(\mathcal{L})\cap\langle\mathfrak{p}\rangle^{\perp}$. Let
$\pi_{\mathfrak{p}}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{n+2,2}\to\langle\mathfrak{p}\rangle^{\perp}$ be the
projection onto $\langle\mathfrak{p}\rangle^{\perp}$. A point
$s\in\mathbb{P}(\mathcal{L})\backslash\mathbb{P}(\mathcal{L})\cap\langle\mathfrak{p}\rangle^{\perp}$
projects to a spacelike $1$-dimensional subbundle $S=\pi_{\mathfrak{p}}(s)$,
and thus such points $s$ represent hyperspheres of this conformal geometry.
Note that $\pi_{\mathfrak{p}}^{-1}(S)=s\oplus\langle\mathfrak{p}\rangle$ is a
$2$-dimensional subspace of $\mathbb{R}^{n+2,2}$ with signature $(1,1)$. Thus
there are two points in
$\mathbb{P}(\mathcal{L})\backslash\mathbb{P}(\mathcal{L})\cap\langle\mathfrak{p}\rangle^{\perp}$
that project to $S$, giving rise to a notion of orientation for hyperspheres
(see [C2008]).

We may break symmetry further by choosing a non-zero vector
$\mathfrak{q}\in\langle\mathfrak{p}\rangle^{\perp}$ and defining

$\mathfrak{Q}^{n+1}\mathrel{\mathop{\mathchar
58\relax}}=\\{y\in\mathcal{L}\mathrel{\mathop{\mathchar
58\relax}}(y,\mathfrak{q})=-1,(y,\mathfrak{p})=0\\},$

which is isometric to a space form with constant sectional curvature
$\kappa=-(\mathfrak{q},\mathfrak{q})$. If $\chi=1$, then $\mathfrak{Q}^{n+1}$
is a Riemannian space form and if $\chi=-1$, then $\mathfrak{Q}^{n+1}$ is a
Lorentzian space form. We call $\mathfrak{q}$ the space form vector of
$\mathfrak{Q}^{n+1}$. The quadric

$\mathfrak{P}^{n+1}\mathrel{\mathop{\mathchar
58\relax}}=\\{y\in\mathcal{L}\mathrel{\mathop{\mathchar
58\relax}}(y,\mathfrak{p})=-1,(y,\mathfrak{q})=0\\}$

is then identified with the space of hyperplanes (complete, totally geodesic
hypersurfaces) in this space form. The Lie sphere transformations
$A\in\operatorname{O}(n+2,2)$ satisfying $A\mathfrak{p}=\mathfrak{p}$ and
$A\mathfrak{q}=\mathfrak{q}$ are identified with the isometries of
$\mathfrak{Q}^{n+1}$.

Given a Legendre map $f\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathcal{Z}$ we call the projection to a space form
$\mathfrak{f}\mathrel{\mathop{\mathchar 58\relax}}=f\cap\mathfrak{Q}^{n+1}$
the point sphere map and $\mathfrak{t}\mathrel{\mathop{\mathchar
58\relax}}=f\cap\mathfrak{P}^{n+1}$ the tangent plane congruence of $f$ with
respect to $\mathfrak{p},\mathfrak{q}$.

######  Remark 2.2.

In [BHPR2019] it is shown how a non-zero lightlike vector
$\mathfrak{q}\in\mathcal{L}$ defines a Laguerre subgeometry. One then has that
the Lie sphere transformations $A\in\operatorname{O}(n+2,2)$ for which
$\mathfrak{q}$ is an eigenvector represent Laguerre transformations.

###  2.2. Linear sphere complexes

A central concept to this paper is the notion of linear sphere complexes
developed in [L1872] (see also [B1929, H2003, RS2020]).

######  Definition 2.3.

A 1-dimensional subspace $L\leq\mathbb{R}^{n+2,2}$ defines a linear sphere
complex $E_{L}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{P}(\mathcal{L})\cap L^{\perp}$, a $(n+1)$-dimensional
family of hyperspheres. If $L$ is spacelike then we say that $E_{L}$ is an
_elliptic_ linear sphere complex.

Suppose that $L$ defines an elliptic linear sphere complex. After a choice of
a timelike point sphere complex $\mathfrak{p}$, one has that
$L\oplus\langle\mathfrak{p}\rangle$ is a $(1,1)$ plane. Thus there exist two
hyperspheres $L^{\pm}\in\mathbb{P}(\mathcal{L})$ such that $L^{+}\oplus
L^{-}=L\oplus\langle\mathfrak{p}\rangle$. These represent the same hypersphere
$S=\pi_{\mathfrak{p}}(L)$ in the conformal geometry
$\langle\mathfrak{p}\rangle^{\perp}$ with opposite orientations. The
hyperspheres belonging to $E_{L}$ are then the hyperspheres intersecting $S$
transversally at a fixed non-zero angle.

######  Remark 2.4.

In the case that $L\perp\mathfrak{p}$, one has that the hyperspheres belonging
to $E_{L}$ intersect $S$ orthogonally. In the presence of a space form vector
$\mathfrak{q}$, $L\perp\mathfrak{q}$ implies that the hypersphere defined by
$S$ represents a hyperplane in $\mathfrak{Q}^{n+1}$.

###  2.3. Lie applicable Legendre maps

Demoulin [D1911i, D1911ii, D1911iii] discovered a class of surfaces belonging
to Lie sphere geometry that constitute an integrable system. In [C2012i,
P2020] these hypersurfaces were given a gauge theoretic interpretation and we
shall recall that here.

######  Definition 2.5.

A Legendre map $f\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathcal{Z}$ is
Lie applicable if there exists $\eta\in\Omega^{1}(f\wedge f^{\perp})$
satisfying $d\eta=[\eta\wedge\eta]=0$ such that the quadratic differential
$\mathcal{Q}$ defined by

$\mathcal{Q}^{\eta}(X,Y)=\textrm{tr}(f\to f\mathrel{\mathop{\mathchar
58\relax}}\sigma\mapsto\eta(X)d_{Y}\sigma)$

is non-zero. We call such an $\eta$ a gauge potential of $f$.

Gauge potentials are not unique: given a gauge potential $\eta$ one has that
$\tilde{\eta}\mathrel{\mathop{\mathchar 58\relax}}=\eta-d\tau$ is a gauge
potential for any $\tau\in\Gamma(\wedge^{2}f)$. We say that $\tilde{\eta}$ is
gauge equivalent to $\eta$ and we denote by $[\eta]$ the equivalence class of
gauge potentials that are gauge equivalent to $\eta$, called the gauge orbit
of $\eta$. The quadratic differential $\mathcal{Q}^{\eta}$ is invariant of
gauge transformation, i.e., $\mathcal{Q}^{\tilde{\eta}}=\mathcal{Q}^{\eta}$.

Given a Lie applicable surface $f$ with gauge potential $\eta$, we have that
$\\{d+t\eta\\}_{t\in\mathbb{R}}$ is a 1-parameter family of flat metric
connections. It then follows that there exist local orthogonal trivialising
gauge transformations $T(t)\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\operatorname{O}(n+2,2)$, that is,

$T(t)\cdot(d+t\eta)=d.$

These gauge transformations give new Lie applicable Legendre maps via
$f^{t}\mathrel{\mathop{\mathchar 58\relax}}=T(t)f$, which are called Calapso
transforms of $f$.

Since $d+m\eta$ is a flat metric connection for any
$m\in\mathbb{R}\backslash\\{0\\}$, there exist many parallel sections. Suppose
that $\hat{s}$ is a parallel rank 1 null subbundle of $d+m\eta$ such that
$\hat{s}$ is nowhere orthogonal to $f$. Let $\hat{f}\mathrel{\mathop{\mathchar
58\relax}}=\hat{s}\oplus s_{0}\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathcal{Z}$, where $s_{0}\mathrel{\mathop{\mathchar
58\relax}}=f\cap\hat{s}^{\perp}$. These $\hat{f}$ are Lie applicable Legendre
maps, called $m$-Darboux transforms of $f$. One also has that $f$ is an
$m$-Darboux transform of $\hat{f}$. Thus, we refer to $(f,\hat{f})$ as an
$m$-Darboux pair.

##  3\. Curves in Lie sphere geometry

In this section we shall consider the case that $n=1$. Thus
$\mathbb{P}(\mathcal{L})$ parametrises circles in $2$-dimensional space forms
such as the Euclidean $2$-plane. This was utilised by Blaschke [B1929, Chapter
5] to study curves. See [BLPT2020] for a recent application of planar Lie
sphere geometry.

Assume that $I\mathrel{\mathop{\mathchar 58\relax}}=\Sigma$ is an open
interval. We then have that Legendre maps $C\mathrel{\mathop{\mathchar
58\relax}}I\to\mathcal{Z}$ project to fronts, that is, spacelike curves with
admissible singularities in Riemannian or Lorentzian space forms (see
[AGV1985]). We thus refer to Legendre maps in this setting as Legendre curves.
In this case the “hyperspheres” enveloped by $C$ are circles. At each point
$x\in I$, there exists exactly one curvature sphere $c(x)$ which corresponds
to the osculating circle. This gives rise to a rank 1 subbundle $c\leq C$
satisfying $d\sigma\in\Omega^{1}(C)$ for any section $\sigma\in\Gamma c$. If
at some point $x\in I$ we have that $d\sigma(T_{x}I)\leq c(x)$, for any
$\sigma\in\Gamma c$, then we say that $x$ is an inflection point of $C$. If
every point of $I$ is an inflection point then $c$ is constant and we say that
$C$ is circular.

######  Lemma 3.1.

$C\perp\mathfrak{q}$ for some non-zero vector
$\mathfrak{q}\in\mathbb{R}^{3,2}$ if and only if $C$ is circular and
$\mathfrak{q}\in\Gamma C$.

###### Proof.

If $C$ is circular then $c\leq C$ is constant and we may choose a constant
$\mathfrak{q}\in\Gamma c$.

Conversely, if $C\perp\mathfrak{q}$ for some constant $\mathfrak{q}$, then it
follows that $C^{(1)}\perp\mathfrak{q}$. Since $C$ is Legendre one has that
$C^{(1)}=C^{\perp}$. On the other hand $(C^{\perp})^{\perp}=C$. Thus
$\mathfrak{q}\in\Gamma C$. ∎

######  Remark 3.2.

Notice that by Lemma 3.1 we have that $C\perp\mathfrak{q}$ necessarily implies
that $\mathfrak{q}$ is lightlike. If $\mathfrak{q}$ is timelike, then
$C(x)\not\perp\mathfrak{q}$ for any $x\in I$, whereas if $\mathfrak{q}$ is
spacelike, then the set of points $x\in I$ where $C(x)\perp\mathfrak{q}$ is
discrete.

######  Lemma 3.3.

Let $W_{0}$ be a constant $3$-dimensional subspace of $\mathbb{R}^{3,2}$ and
suppose that $c_{0}\mathrel{\mathop{\mathchar 58\relax}}=C\cap W_{0}$ is a
rank 1 subbundle of $C$. Then $C$ is circular.

###### Proof.

If $C^{\perp}\cap W_{0}$ is rank $1$ then, since $C^{(1)}=C^{\perp}$, it
follows that $c_{0}$ is constant and thus $C$ is circular. If
$C^{\perp}=W_{0}$ then $C$ fails to be a Legendre curve. If $C^{\perp}\cap
W_{0}$ is rank $2$ then $\tilde{c}_{0}\mathrel{\mathop{\mathchar
58\relax}}=C\cap W_{0}^{\perp}$ has rank $1$ and it follows that
$\tilde{c}_{0}$ is constant. Hence, $C$ is circular. ∎

###  3.1. Polarisations and polynomial conserved quantities

Since $TI$ is rank $1$, any $\xi\in\Omega^{1}(C\wedge C^{\perp})$ satisfies
$d\xi=[\xi\wedge\xi]=0$. Therefore any $\xi\in\Omega^{1}(C\wedge C^{\perp})$
with non-zero quadratic differential111Note that, unlike the $n=2$ case,
$\mathcal{Q}^{\xi}$ does not determine $[\xi]$. $\mathcal{Q}^{\xi}$ is a gauge
potential. In analogy to [BHMR2016] we refer to gauge orbits $[\xi]$ in this
setting as polarisations of $C$.

######  Example 3.4.

Let $\mathfrak{p},\mathfrak{q}$ be a point sphere complex and space form
vector for a space form $\mathfrak{Q}^{2}$ and let
$\mathfrak{f}=C\cap\mathfrak{Q}^{2}$ denote the corresponding point sphere
map. We then call the gauge orbit $[\xi]$ with $\xi\mathrel{\mathop{\mathchar
58\relax}}=-\mathfrak{f}\wedge d\mathfrak{f}$ the arclength polarisation of
$C$ with respect to $\mathfrak{p},\mathfrak{q}$. The corresponding quadratic
differential is given by $\mathcal{Q}^{\xi}=(d\mathfrak{f},d\mathfrak{f})$.

In analog to [BS2012], we have the following definition:

######  Definition 3.5.

We say that
$p(t)=p_{0}+tp_{1}+...+t^{d}p_{d}\in(\Gamma\underline{\mathbb{R}}^{3,2})[t]$
is called a polynomial conserved quantity of $d+t\xi$ of degree $d$ if
$(d+t\xi)p(t)=0$ for all $t\in\mathbb{R}$ and $p_{d}\in\Gamma C$.

If $\tilde{\xi}=\xi-d\tau\in[\xi]$, then
$d+t\tilde{\xi}=\exp(t\tau)\cdot(d+t\xi)$. Thus if $p(t)$ is a a polynomial
conserved quantity of $d+t\xi$ of degree $d$, then
$\tilde{p}(t)\mathrel{\mathop{\mathchar 58\relax}}=\exp(t\tau)p(t)$ is a
polynomial conserved quantity of $d+t\tilde{\xi}$ of degree $d$. Note that
without the requirement that $p_{d}\in\Gamma C$, we would have that the degree
of polynomial conserved quantities is not invariant under gauge
transformation.

Evaluation of the coefficients of the polynomial $(d+t\xi)p(t)=0$ yields the
following lemma, analogous to [BS2012, Proposition 2.2]:

######  Lemma 3.6.

$p(t)=p_{0}+tp_{1}+...+t^{d}p_{d}$ is a polynomial conserved quantity of
$d+t\xi$ of degree $d$ if and only if

  * •

$p_{0}$ is constant,

  * •

$dp_{i}+\xi p_{i-1}=0$ for all $i\in\\{1,...,d\\}$, and

  * •

$p_{d}\in\Gamma C$.

Clearly constant linear combinations of polynomial conserved quantities of
degree $d$ are polynomial conserved quantities of degree $d$. Hence the space
of polynomial conserved quantities of $d+t\xi$ of degree $d$ forms a vector
space over $\mathbb{R}$. In this paper we will focus on spaces of linear
conserved quantities.

######  Lemma 3.7.

Suppose that $p$ and $\tilde{p}$ are linear conserved quantities of $C$ with
$p(0)=\tilde{p}(0)$. Then either $p=\tilde{p}$ or $C$ is circular.

###### Proof.

We may write $p=\mathfrak{p}+t\sigma$ and
$\tilde{p}=\mathfrak{p}+t\tilde{\sigma}$ for some constant
$\mathfrak{p}\in\mathbb{R}^{3,2}$ and sections $\sigma,\tilde{\sigma}\in\Gamma
C$. If $\sigma=\tilde{\sigma}$ then $p=\tilde{p}$. Otherwise we have that
$p-\tilde{p}=t(\sigma-\tilde{\sigma})$ is a conserved quantity of $d+t\xi$.
This then implies that $d(\sigma-\tilde{\sigma})=0$. Hence
$\sigma-\tilde{\sigma}\in\Gamma C$ is constant, implying that $C$ is circular.
∎

The next lemma shows that for any curve $C$, we can always find polarisations
$[\xi]$ such that $d+t\xi$ admits $2$-dimensional spaces of linear conserved
quantities.

######  Lemma 3.8.

Suppose that $W_{0}\leq\mathbb{R}^{4,2}$ is a $2$-dimensional subspace with
$W_{0}\cap C^{\perp}=\\{0\\}$. Then there exists a polarisation $[\xi]$ of $C$
such that the space $W$ of linear conserved quantities of $d+t\xi$ satisfies
$W_{0}\leq W(0)$.

###### Proof.

Fix $\mathfrak{q},\tilde{\mathfrak{q}}\in W_{0}$. Then, by the assumption
$W_{0}\cap C^{\perp}=\\{0\\}$, we can find sections
$\sigma,\tilde{\sigma}\in\Gamma C$ such that

$(\sigma,\mathfrak{q})=(\tilde{\sigma},\tilde{\mathfrak{q}})=0\quad\text{and}\quad(\tilde{\sigma},\mathfrak{q})=(\sigma,\tilde{\mathfrak{q}})=-1.$

Now define $\xi\mathrel{\mathop{\mathchar 58\relax}}=-\sigma\wedge
d\sigma-\tilde{\sigma}\wedge d\tilde{\sigma}$. Then
$\mathcal{Q}^{\xi}=(d\sigma,d\sigma)+(d\tilde{\sigma},d\tilde{\sigma})\neq 0$,
as otherwise $C$ would fail to be a Legendre curve. Hence $[\xi]$ is a
polarisation of $C$. Moreover,

$q(t)\mathrel{\mathop{\mathchar
58\relax}}=\mathfrak{q}+t\tilde{\sigma}\quad\text{and}\quad\tilde{q}(t)\mathrel{\mathop{\mathchar
58\relax}}=\tilde{\mathfrak{q}}+t\sigma$

are linear conserved quantities of $d+t\xi$ satisfying $q(0)=\mathfrak{q}$ and
$\tilde{q}(0)=\tilde{\mathfrak{q}}$. Hence $W\mathrel{\mathop{\mathchar
58\relax}}=\langle q,\tilde{q}\rangle$ satisfies $W_{0}\leq W(0)$. ∎

Let $W$ be a vector space of linear conserved quantities of $d+t\xi$. Since
$d+t\xi$ is a flat metric connection, we have that the coefficients of the
polynomial $(p(t),q(t))$ are constant for any $p,q\in W$. We may now equip $W$
with a pencil of metrics $\\{(,)_{t}\\}_{t\in\mathbb{R}\cup\\{\infty\\}}$
where

$(p,q)_{t}\mathrel{\mathop{\mathchar 58\relax}}=(p(t),q(t))$

for $t\in\mathbb{R}$ and $(,)_{\infty}\mathrel{\mathop{\mathchar
58\relax}}=\lim_{t\to\infty}\frac{1}{t}(,)_{t}$. Writing $p=p_{0}+tp_{1}$ and
$q=q_{0}+tq_{1}$ we have that

$(p,q)_{\infty}=(p_{0},q_{1})+(q_{0},p_{1}).$

Note that this pencil of metrics is invariant under gauge transformation.

######  Lemma 3.9.

Suppose that $C$ is non-circular and that $W$ is a $2$-dimensional space of
linear conserved quantities of $d+t\xi$. Then $(,)_{\infty}\neq 0$.

###### Proof.

Suppose that $(,)_{\infty}\equiv 0$ and $p,q\in W$ are linearly independent.
Writing $p(t)=p_{0}+tp_{1}$ and $q(t)=q_{0}+tq_{1}$ we have that $p_{0}$ and
$q_{0}$ are linearly independent by Lemma 3.7 and, since $(,)_{\infty}\equiv
0$, we have that

(3.1) $0=(p_{0},p_{1})=(q_{0},q_{1})=(p_{0},q_{1})+(q_{0},p_{1}).$

If $(p_{0},q_{1})=0$ then we can deduce that $C$ is circular from Lemma 3.3.
Hence, we must have that $(p_{0},q_{1})\neq 0$. Now using $p_{1},q_{1}$ as a
basis for $C$ we can compute the quadratic differential of $\xi$ to be

$\displaystyle\mathcal{Q}^{\xi}(X,X)$
$\displaystyle=\frac{1}{(p_{1},q_{0})}(\xi(X)d_{X}p_{1},q_{0})+\frac{1}{(q_{1},p_{0})}(\xi(X)d_{X}q_{1},p_{0})$
$\displaystyle=\frac{1}{(p_{1},q_{0})}((\xi(X)d_{X}p_{1},q_{0})-(\xi(X)d_{X}q_{1},p_{0})).$

On the other hand, since $p,q$ are linear conserved quantities of $d+t\xi$, we
have that

$dp_{1}+\xi p_{0}=dq_{1}+\xi q_{0}=0,$

from which one deduces that $\mathcal{Q}^{\xi}(X,X)=0$, contradicting that
$\xi$ is a polarisation of $C$. ∎

Let $\xi=-\mathfrak{f}\wedge d\mathfrak{f}$ be the arclength polarisation of a
Legendre curve $C$ into a space form $\mathfrak{Q}^{2}$. Then $d+t\xi$ admits
a two dimensional space of linear conserved quantities $W=\langle p,q\rangle$
where $p(t)=\mathfrak{p}$ and $q(t)=\mathfrak{q}-t\mathfrak{f}$. Moreover,
$(p,p)_{0}\neq 0$, $(q,q)_{\infty}=2$ and
$(p,p)_{\infty}=(p,q)_{\infty}=(p,q)_{0}=0$.

Conversely, suppose that $W=\langle p,q\rangle$ is a two dimensional space of
linear conserved quantities of $d+t\xi$ satisfying $(p,p)_{0}\neq 0$,
$(q,q)_{\infty}\neq 0$ and $(p,p)_{\infty}=(p,q)_{\infty}=(p,q)_{0}=0$. By
reparameterising $t$, we may assume that $(q,q)_{\infty}=2$. Let
$\mathfrak{Q}^{2}$ be the space form with point sphere complex
$\mathfrak{p}\mathrel{\mathop{\mathchar 58\relax}}=p(0)$ and space form vector
$\mathfrak{q}\mathrel{\mathop{\mathchar 58\relax}}=q(0)$ and let
$\mathfrak{f}=C\cap\mathfrak{Q}^{2}$ and $\mathfrak{t}=C\cap\mathfrak{P}^{2}$
be the corresponding point sphere map and tangent plane congruence. Define
$\tau=\mathfrak{t}\wedge p_{1}$. Then $\tilde{p}(t)\mathrel{\mathop{\mathchar
58\relax}}=\exp(t\tau)p(t)=\mathfrak{p}$. With
$\tilde{q}(t)\mathrel{\mathop{\mathchar
58\relax}}=\exp(t\tau)q(t)=\mathfrak{q}+t\tilde{q}_{1}$, the conditions
$(\tilde{q},\tilde{q})_{\infty}=2$ and $(\tilde{p},\tilde{q})_{\infty}=0$
imply that $\tilde{q}_{1}=-\mathfrak{f}$. The conditions

$d\tilde{p}_{1}+\xi\mathfrak{p}=d\tilde{q}_{1}+\xi\mathfrak{q}=0$

then imply that $\xi=-\mathfrak{f}\wedge d\mathfrak{f}$.

We thus arrive at the following lemma:

######  Lemma 3.10.

Suppose that $W=\langle p,q\rangle$ is a $2$-dimensional space of linear
conserved quantities of $d+t\xi$ such that $(p,p)_{0}\neq 0$,
$(q,q)_{\infty}\neq 0$ and $(p,p)_{\infty}=(p,q)_{\infty}=(p,q)_{0}=0$. Then,
after possibly rescaling $\xi$, we have that $[\xi]$ is the arclength
polarisation of $C$ with respect to $\mathfrak{p},\mathfrak{q}$ where
$\mathfrak{p}\mathrel{\mathop{\mathchar 58\relax}}=p(0)$ and
$\mathfrak{q}\mathrel{\mathop{\mathchar 58\relax}}=q(0)$.

######  Lemma 3.11.

Suppose that $C$ is non-circular and that $W$ is a $3$-dimensional space of
linear conserved quantities of $d+t\xi$. Then $[\xi]$ is the arclength
polarisation of $C$ with respect to some $\mathfrak{p},\mathfrak{q}\in W(0)$.

###### Proof.

By Lemma 3.7 we have that $W(0)$ is a $3$-dimensional subspace of
$\mathbb{R}^{3,2}$. On the other hand $C^{\perp}$ is a $3$-dimensional
subbundle of $\underline{\mathbb{R}}^{3,2}$. Thus with
$\nu\mathrel{\mathop{\mathchar 58\relax}}=C^{\perp}\cap W(0)$ we have that
$\nu(x)\neq\\{0\\}$ for all $x\in I$. By Lemma 3.3 we must have for some $x\in
I$ that $\nu(x)\not\leq C(x)$. Thus there exists a vector $\mathfrak{p}\in
W(0)$ with $\mathfrak{p}\in(C(x))^{\perp}$ and
$(\mathfrak{p},\mathfrak{p})\neq 0$. We can then find $p\in W$ such that
$p(0)=\mathfrak{p}$. Since the top term of $p$ lies in $C$ and
$\mathfrak{p}\in(C(x))^{\perp}$ it follows that $(p,p)_{\infty}=0$. Denote by
$p^{\perp_{\infty}}$ and $p^{\perp_{0}}$ the subspaces of $W$ that are
perpendicular to $p$ with respect to $(,)_{\infty}$ and $(,)_{0}$. These are
$2$-dimensional subspaces of $W$, thus there exists a non-zero $q\in
p^{\perp_{\infty}}\cap p^{\perp_{0}}$. By Lemma 3.9 we must have that
$(q,q)_{\infty}\neq 0$, as otherwise $(,)_{\infty}$ would vanish on $\langle
p,q\rangle$. Thus, by Lemma 3.10, $[\xi]$ is the arclength polarisation of $C$
with respect to $\mathfrak{p}$ and $\mathfrak{q}\mathrel{\mathop{\mathchar
58\relax}}=q(0)$. ∎

###  3.2. Constrained elastic curves

Fix a point sphere complex $\mathfrak{p}$ and space form vector $\mathfrak{q}$
for a space form $\mathfrak{Q}^{2}$ with sectional curvature
$\kappa=-(\mathfrak{q},\mathfrak{q})$ and $\chi=-(\mathfrak{p},\mathfrak{p})$.
The corresponding point sphere map $\mathfrak{f}$ and tangent plane congruence
$\mathfrak{t}$ of a Legendre curve $C\mathrel{\mathop{\mathchar
58\relax}}I\to\mathcal{Z}$ satisfy the relation
$d\mathfrak{t}=-kd\mathfrak{f}$, where $k$ is the curvature of $\mathfrak{f}$.
The curvature sphere of $f$ is thus given by
$c=\langle\mathfrak{t}+k\mathfrak{f}\rangle$. Hence, $C$ is circular if and
only if $k$ is constant.

Let ′ denote differentiation with respect to the arclength parameter of
$\mathfrak{f}$. Then according to [BPP2008, H2014] (the Lorentzian case can be
deduced from [P2018] and the references therein), $\mathfrak{f}$ is a
constrained elastic curve in $\mathfrak{Q}^{2}$ if and only if

(3.2) $k^{\prime\prime}+\chi\frac{k^{3}}{2}+(\mu+\kappa)k+\lambda=0,$

for some $\mu,\lambda\in\mathbb{R}$. If $\lambda=0$ then $\mathfrak{f}$ is an
elastic curve and if $\lambda=\mu=0$ then $\mathfrak{f}$ is a free elastic
curve. Clearly circular curves, that is when $k$ is constant, are constrained
elastic curves.

Define a vector

(3.3) $\mathfrak{r}\mathrel{\mathop{\mathchar
58\relax}}=-(k\kappa+\lambda)\mathfrak{f}+(\tfrac{k^{2}}{2}\chi+\mu)\mathfrak{t}-k^{\prime}\mathfrak{f}^{\prime}+k\mathfrak{q}-\tfrac{k^{2}}{2}\mathfrak{p},$

for some constants $\lambda,\mu\in\mathbb{R}$. Noting that

$\mathfrak{f}^{\prime\prime}=-\kappa\mathfrak{f}+\chi
k\mathfrak{t}+\mathfrak{q}-k\mathfrak{p},$

we can calculate that

$\mathfrak{r}^{\prime}=(-k^{\prime\prime}-k\kappa-\lambda-\tfrac{k^{3}}{2}\chi-\mu
k)\mathfrak{f}^{\prime}.$

It then follows that $\mathfrak{r}^{\prime}=0$ if and only if $\mathfrak{f}$
is a constrained elastic curve satisfying (3.2). Moreover, one has that
$(\mathfrak{t}+\frac{k}{2}\mathfrak{f},\mathfrak{r})=0$.

Conversely, suppose that
$(\mathfrak{t}+\frac{k}{2}\mathfrak{f},\mathfrak{r})=0$, for some non-zero
constant $\mathfrak{r}\in\mathbb{R}^{3,2}$. We may write $\mathfrak{r}$ in
terms of the basis
$\\{\mathfrak{f},\mathfrak{t},\mathfrak{f}^{\prime},\mathfrak{q},\mathfrak{p}\\}$
as

$\mathfrak{r}=\alpha\mathfrak{f}+\beta\mathfrak{t}+\gamma\mathfrak{f}^{\prime}+\delta\mathfrak{q}+\epsilon\mathfrak{p},$

for some functions $\alpha,\beta,\gamma,\delta,\epsilon$.
$\mathfrak{r}^{\prime}=0$ implies that

$\displaystyle\alpha^{\prime}-\gamma\kappa=0,\quad\beta^{\prime}+\gamma\chi
k=0,\quad\alpha-\beta k+\gamma^{\prime}=0,$
$\displaystyle\gamma+\delta^{\prime}=0,\quad\gamma k-\epsilon^{\prime}=0.$

The condition $(\mathfrak{t}+\frac{k}{2}\mathfrak{f},\mathfrak{r})=0$ implies
that $\epsilon+\frac{k}{2}\delta=0$. It thus follows that
$k\delta^{\prime}=k^{\prime}\delta$. Hence, if $k\not\equiv 0$, we may write
$\delta=Ak$ for some constant $A$. If $A=0$ then it follows that
$\gamma=\delta=\epsilon=0$ and thus $\mathfrak{r}\in\Gamma f$. Hence, $C$ is
circular. Otherwise we may assume, without loss of generality, that $A=1$. It
then follows that $\epsilon=-\frac{k^{2}}{2}$, $\gamma=-\kappa^{\prime}$ and
that $\alpha=-k\kappa-\lambda$ and $\beta=\frac{k^{2}}{2}\chi+\mu$ for some
constants $\lambda$ and $\mu$. Thus $\mathfrak{r}$ has the form (3.3) and it
follows that $\mathfrak{f}$ is a constrained elastic curve satisfying (3.2).

######  Proposition 3.12.

$\mathfrak{f}$ is a constrained elastic curve if and only if
$(\mathfrak{t}+\frac{k}{2}\mathfrak{f},\mathfrak{r})=0$ for some non-zero
constant $\mathfrak{r}\in\mathbb{R}^{3,2}$.

Hence, we obtain the following geometric characterisation for constrained
elastic curves in space forms.

######  Theorem 3.13.

A curve with curvature $k$ is constrained elastic if and only if the enveloped
congruence of circles with geodesic curvature $\frac{k}{2}$ takes values in a
fixed linear circle complex.

######  Remark 3.14.

Notice from (3.3) that when $\lambda=0$ one has that
$\mathfrak{r}\perp\mathfrak{q}$, whereas when $\mu=0$ one has that
$\mathfrak{r}\perp\mathfrak{p}$. Thus, by Remark 2.4, for elastic curves in
Euclidean 2-space with $k^{\prime 2}+\frac{k^{4}}{4}+\mu\,k^{2}>0$ (so that
$\mathfrak{r}$ is spacelike), we have that the circle congruence
$\langle\mathfrak{t}+\frac{k}{2}\mathfrak{f}\rangle$ intersects a fixed line
at a constant angle $\theta$. Moreover for free elastic curves
$\theta=\frac{\pi}{2}$. See Figure 1 for some examples of this principle.

Figure 1. Various constrained elastic curves (in blue) and the enveloped
congruence of circles with geodesic curvature $\frac{k}{2}$ (in grey)
intersecting a fixed circle or line (in red) at a fixed angle.

Using the result of Proposition 3.12, we are now able to prove another
characterisation of constrained elastic curves in terms of linear conserved
quantities.

######  Theorem 3.15.

$d+t\xi$ admits a $3$-dimensional space of linear conserved quantities $W$ for
some polarisation $[\xi]$ of $C$ if and only if $C$ projects to a constrained
elastic curve in an appropriate space form. The point sphere complex
$\mathfrak{p}$ and space form vector $\mathfrak{q}$ of this space form satisfy
$\mathfrak{p},\mathfrak{q}\in W(0)$.

###### Proof.

Suppose that $C$ projects to a constrained elastic curve in a space form
$\mathfrak{Q}^{2}$ with point sphere complex $\mathfrak{p}$ and space form
vector $\mathfrak{q}$. Then by Proposition 3.12 we have that
$(\mathfrak{t}+\frac{k}{2}\mathfrak{f},\mathfrak{r})=0$ for some non-zero
constant $\mathfrak{r}\in\mathbb{R}^{3,2}$. Using the form of (3.3), we have
that $(\mathfrak{f},\mathfrak{r})=-k$ and
$(\mathfrak{t},\mathfrak{r})=\frac{k^{2}}{2}$. Now consider the arclength
polarisation $[\xi]$ with $\xi=-\mathfrak{f}\wedge d\mathfrak{f}$. Define
$r(t)\mathrel{\mathop{\mathchar
58\relax}}=\mathfrak{r}+t(2\mathfrak{t}+k\mathfrak{f})$. Then one can check
that $r(t)$ is a linear conserved quantity of $d+t\xi$. Thus we have a
3-dimensional space of linear conserved quantities $W=\langle p,q,r\rangle$ of
$d+t\xi$, where $p(t)=\mathfrak{p}$ and $q(t)=\mathfrak{q}-t\mathfrak{f}$.

Conversely, suppose that $d+t\xi$ admits a $3$-dimensional space of linear
conserved quantities $W$. Since circular curves are constrained elastic curves
we shall assume, without loss of generality, that $C$ is non-circular. By
Lemma 3.11, $[\xi]$ is the arclength polarisation of $C$ with respect to
$\mathfrak{p}\mathrel{\mathop{\mathchar 58\relax}}=p(0)$ and
$\mathfrak{q}\mathrel{\mathop{\mathchar 58\relax}}=q(0)$, for some $p,q\in W$.
Denoting by $\mathfrak{f}$ and $\mathfrak{t}$ the point sphere map and tangent
plane congruence of $C$ with respect to $\mathfrak{p},\mathfrak{q}$, we have
that $\xi=-\mathfrak{f}\wedge d\mathfrak{f}$. Now let $r\in W$ such that
$r\not\in\langle p,q\rangle$ and write
$r(t)=\mathfrak{r}+t(a\mathfrak{f}+b\mathfrak{t})$ for some
$\mathfrak{r}\in\mathbb{R}^{3,2}$ and functions $a,b$. The condition that $r$
is a linear conserved quantity amounts to

$0=da\mathfrak{f}+ad\mathfrak{f}+db\mathfrak{t}+bd\mathfrak{t}-(\mathfrak{f},\mathfrak{r})d\mathfrak{f}+(d\mathfrak{f},\mathfrak{r})\mathfrak{f}.$

Thus $db=0$, $da+(d\mathfrak{f},\mathfrak{r})=0$ and
$a-bk-(\mathfrak{f},\mathfrak{r})=0$. Hence

$(\mathfrak{f},\mathfrak{r})=\lambda k+\mu$

for some constants $\lambda,\mu$. It then follows that
$(d\mathfrak{t},\mathfrak{r})=-\lambda kdk$ and thus
$(\mathfrak{t},\mathfrak{r})=-\frac{\lambda}{2}k^{2}+\zeta$ for some constant
$\zeta$. Thus

$(\mathfrak{t}+\frac{k}{2}\mathfrak{f},\mathfrak{r})=\zeta+\mu\frac{k}{2}.$

Defining $\tilde{\mathfrak{r}}\mathrel{\mathop{\mathchar
58\relax}}=\mathfrak{r}+\zeta\mathfrak{p}+\mu\mathfrak{q}$, we have that

$(\mathfrak{t}+\frac{k}{2}\mathfrak{f},\tilde{\mathfrak{r}})=0.$

Thus by Proposition 3.12 we have that $\mathfrak{f}$ is a constrained elastic
curve. ∎

In light of Theorem 3.15 we make the following definition:

######  Definition 3.16.

Let $W_{0}$ be a $3$-dimensional subspace of $\mathbb{R}^{3,2}$. We say that
$C\mathrel{\mathop{\mathchar 58\relax}}I\to\mathcal{Z}$ is a constrained
elastic curve with respect to $W_{0}$ if there exists a polorarisation $[\xi]$
of $C$ such that $d+t\xi$ admits a $3$-dimensional space $W$ of linear
conserved quantities with $W(0)=W_{0}$.

The following proposition shows that, for non-circular curves, the spaces of
linear conserved quantities of $d+t\xi$ has maximal dimension $3$:

######  Proposition 3.17.

If $d+t\xi$ admits a $4$-dimensional space of linear conserved quantities then
$C$ is circular.

###### Proof.

Suppose that $C$ admits a $4$-dimensional space of linear conserved
quantities. If $C$ is non-circular, then as in the proof of Theorem 3.15, we
can show that there exists two linearly independent vectors
$\mathfrak{r}_{1},\mathfrak{r}_{2}\in\mathbb{R}^{3,2}$ such that

$(\mathfrak{t}+\frac{k}{2}\mathfrak{f},\mathfrak{r}_{1})=(\mathfrak{t}+\frac{k}{2}\mathfrak{f},\mathfrak{r}_{2})=0$

for some space form projection $\mathfrak{f}$ of $C$. Thus
$\tilde{c}\mathrel{\mathop{\mathchar
58\relax}}=\langle\mathfrak{t}+\frac{k}{2}\mathfrak{f}\rangle$ lives in a
constant $3$-dimensional space $W_{0}\mathrel{\mathop{\mathchar
58\relax}}=\langle\mathfrak{r}_{1},\mathfrak{r}_{2}\rangle^{\perp}$. But then
by Lemma 3.3, $C$ is circular, contradicting our assumption that $C$ is non-
circular. ∎

##  4\. Surfaces with spherical curvature lines

In this section we shall be considering the case that $n=2$, and thus Legendre
maps $f\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathcal{Z}$ project to
fronts in space forms, that is, surfaces with admissible singularities. We
will investigate the case that $f$ projects to fronts with one or two families
of spherical curvature lines, following the approach of Blaschke [B1929, §88].
We shall begin this section by recalling some facts about curvature lines of
Legendre maps.

Let $f\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathcal{Z}$ be a Legendre
map. Then, since $f^{(1)}\leq f^{\perp}$, we can define a tensor

$\beta\mathrel{\mathop{\mathchar
58\relax}}T\Sigma\to\operatorname{Hom}(f,f^{\perp}/f)\quad
X\mapsto(\sigma\mapsto d_{X}\sigma\,mod\,f).$

Moreover, since $f$ is a Legendre map, we have the condition
$\ker\beta=\\{0\\}$. We can then define a tensor
$\mathcal{B}\in\Gamma(S^{2}T\Sigma\otimes(\wedge^{2}f)^{*}\otimes\wedge^{2}(f^{\perp}/f))$
by

$\mathcal{B}(X,Y)\xi_{1}\wedge\xi_{2}=\frac{1}{2}(\beta(X)\xi_{1}\wedge\beta(Y)\xi_{2}+\beta(Y)\xi_{1}\wedge\beta(X)\xi_{2}),$

where $X,Y\in\Gamma T\Sigma$ and $\xi_{1},\xi_{2}\in\Gamma f$. Away from
points where $\mathcal{B}$ vanishes, we can identify $\mathcal{B}$ with an
indefinite conformal structure on $\Sigma$ whose null vectors are curvature
directions of $f$ (see [P2020, Section 2.4.1]). Thus, if $T\leq T\Sigma$ is a
rank $1$ distribution satisfying $\mathcal{B}(T,T)=0$, we have that the leaves
of $T$ are curvature lines of $f$. We shall denote by $f_{T}$ the set of
sections of $f$ and derivatives of sections of $f$ along $T$.

######  Lemma 4.1.

Let $T\leq T\Sigma$ be a rank $1$ distribution. Then:

  1. (1)

$\mathcal{B}(T,T)=0$ if and only if $f_{T}$ is a rank $3$ subbundle of
$f^{(1)}$.

  2. (2)

If $\mathcal{B}(T,T)=0$ then there exists a complementary rank $1$
distribution $\tilde{T}\leq T\Sigma$ such that
$\mathcal{B}(\tilde{T},\tilde{T})=0$ and $f_{T}\perp f_{\tilde{T}}$.

###### Proof.

Since $f$ is a Legendre map, one has that $\dim f_{T}(x)\geq 3$ for all
$x\in\Sigma$. On the other hand, by definition of $\mathcal{B}$, it is
straightforward to see that $\mathcal{B}(T,T)=0$ if and only if
$(f_{T}/f)\wedge(f_{T}/f)=\\{0\\}$, i.e., $f_{T}/f$ has rank $1$. This proves
(1).

Suppose that $\mathcal{B}(T,T)=0$ and thus $f_{T}$ has rank $3$, by (1). Then
$s_{T}\mathrel{\mathop{\mathchar 58\relax}}=\ker\beta(T)$ is a rank $1$
subbundle of $f$. Fix a nowhere-zero section $\sigma_{T}\in\Gamma s_{T}$.

Since $f$ is a Legendre map, we may find a section $\sigma\in\Gamma f$ such
that $\beta\sigma$ is an immersion. It then follows that $(d\sigma,d\sigma)$
is a positive definite metric on $\Sigma$. Let $\tilde{T}\leq T\Sigma$ be
orthogonal to $T$ with respect to this metric. We have that

$f_{\tilde{T}}=f+\langle d_{\tilde{X}}\sigma,d_{\tilde{X}}\sigma_{T}\rangle$

for some nowhere-zero section $\tilde{X}\in\Gamma\tilde{T}$. On the other
hand, since $f_{T}$ is rank $3$, we have that

$f_{T}=f\oplus\langle d_{X}\sigma\rangle,$

for some nowhere-zero section $X\in\Gamma T$. By construction of $\tilde{T}$,
we have that $d_{\tilde{X}}\sigma\perp f_{T}$. Using that $f^{(1)}\leq
f^{\perp}$, we have that

$(d_{X}\sigma,d_{\tilde{X}}\sigma_{T})=(d_{\tilde{X}}\sigma,d_{X}\sigma_{T})=0,$

since $\sigma_{T}\in\Gamma\ker\beta(T)$. Hence, $f_{\tilde{T}}\perp f_{T}$ and
it follows that $f_{\tilde{T}}$ has rank $3$. Applying (1) again implies that
$\mathcal{B}(\tilde{T},\tilde{T})=0$. ∎

###  4.1. Spherical curvature lines

In this paper we are interested in surfaces in space forms with one or two
families of spherical curvature lines. In order to express this notion, we
break symmetry to a conformal subgeometry. Let
$\mathfrak{p}\in\mathbb{R}^{4,2}$ be a timelike point sphere complex for a
conformal Riemannian geometry
$\langle\mathfrak{p}\rangle^{\perp}\cong\mathbb{R}^{4,1}$. Let
$\Lambda\mathrel{\mathop{\mathchar
58\relax}}=f\cap\langle\mathfrak{p}\rangle^{\perp}$ be the point sphere map of
$f$ with respect to this conformal geometry. Suppose that $T\leq T\Sigma$ is a
rank $1$ distribution satisfying $\mathcal{B}(T,T)=0$ and thus the leaves of
$T$ are a family of curvature lines of $\Lambda$. The condition that the
leaves of $T$ be spherical curvature lines then amounts to $\Lambda\perp\nu$
for some smooth map $\nu\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathbb{P}(\mathcal{L})$ that is constant along the leaves
of $T$. Recall from Subsection 2.1 that we obtain the conformal representative
of $\nu$ from $S\mathrel{\mathop{\mathchar
58\relax}}=\pi_{\mathfrak{p}}(\nu)$, where $\pi_{\mathfrak{p}}$ denotes the
projection of $\mathbb{R}^{4,2}$ onto $\langle\mathfrak{p}\rangle^{\perp}$.

######  Remark 4.2.

Throughout this paper, we will only be considering the case that the spheres
defined by $\nu$ intersect $\Lambda$ transversally. In particular, this
implies that $\nu\cap f^{\perp}=\\{0\\}$.

######  Theorem 4.3.

There exists a spacelike rank $1$ subbundle $L$ of $f^{\perp}$ that is
constant along the leaves of a rank $1$ distribution $T\leq T\Sigma$ if and
only if, where the projection immerses, $f$ projects to a surface with
spherical curvature lines along the leaves of $T$ in some (and in fact, any)
conformal Riemannian geometry.

###### Proof.

Suppose that a spacelike rank $1$ subbundle $L\leq f^{\perp}$ is constant
along the leaves of $T\leq T\Sigma$. Then $f_{T}\perp L$, and thus $f_{T}$ is
a rank $3$ subbundle of $f^{(1)}$. From Lemma 4.1 we have that the leaves of
$T$ are curvature lines of $f$. Choose a timelike point sphere complex
$\mathfrak{p}\in\mathbb{R}^{4,2}$. Then $W\mathrel{\mathop{\mathchar
58\relax}}=L\oplus\langle\mathfrak{p}\rangle$ is rank $2$ bundle with
signature $(1,1)$. Let $\nu^{\pm}$ denote the two rank $1$ null subbundles of
$W$. Then $\nu^{\pm}$ are constant along the leaves of $T$. Moreover, defining
$\Lambda\mathrel{\mathop{\mathchar
58\relax}}=f\cap\langle\mathfrak{p}\rangle^{\perp}$ to be the point sphere map
of $f$ with respect to $\mathfrak{p}$, we have that $\Lambda\perp\nu^{\pm}$.
Thus $\Lambda$ has spherical curvature lines along the leaves of $T$.

Conversely, assume that the point sphere map $\Lambda$ with respect to some
conformal geometry defined by timelike point sphere complex
$\mathfrak{p}\in\mathbb{R}^{4,2}$ immerses and has spherical curvature lines
along the leaves of a rank $1$ distribution $T\leq T\Sigma$. Thus
$\Lambda\perp\nu$ for some $\nu\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathbb{P}(\mathcal{L})$ that is constant along the leaves
of $T$. Then let $W\mathrel{\mathop{\mathchar
58\relax}}=\nu\oplus\langle\mathfrak{p}\rangle$. Now $W\perp\Lambda$ and,
since $\mathfrak{p}$ is timelike, we have that $L\mathrel{\mathop{\mathchar
58\relax}}=f^{\perp}\cap W$ is a rank $1$ spacelike subbundle of $f^{\perp}$.
Since the leaves of $T$ are curvature lines and $\Lambda$ is an immersion, we
have that $f_{T}=f+\Lambda_{T}$, where $\Lambda_{T}$ denotes the set of
sections of $\Lambda$ and derivatives of sections of $\Lambda$ along $T$.
Since $\Lambda\perp W$ and $W$ is constant along $T$, we learn that
$\Lambda_{T}\perp W$. Hence $L_{T}\perp f$ and, since $L_{T}\leq W$, we deduce
that $L$ is constant along the leaves of $T$. ∎

######  Remark 4.4.

Suppose that $\mathfrak{p}$ is a timelike point sphere complex for a
Riemannian conformal geometry $\langle\mathfrak{p}\rangle^{\perp}$. Then from
Subsection 2.2, we have that $S\mathrel{\mathop{\mathchar
58\relax}}=\pi_{\mathfrak{p}}(L)$ yields the conformal representative of the
spheres containing the curvature lines of $f$ in Theorem 4.3.

######  Remark 4.5.

Notice that Theorem 4.3 implies that even though we broke symmetry to a
conformal geometry to express the condition of having spherical curvature
lines, it is in fact a Lie invariant notion.

###  4.2. Surfaces with spherical curvature lines via Lie spherical evolutions

Denote by $\mathcal{S}$ the Grassmannian of $1$-dimensional spacelike
subspaces of $\mathbb{R}^{4,2}$. Then rank $1$ spacelike subbundles of
$\underline{\mathbb{R}}^{4,2}$ can be viewed as maps $\Sigma\to\mathcal{S}$.
Given a map $L\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathcal{S}$, we
may split the trivial bundle as

$\Sigma\times\mathbb{R}^{4,2}=L\oplus L^{\perp},$

inducing a splitting of the trivial connection

$d=\mathcal{D}^{L}+\mathcal{N}^{L}$

where $\mathcal{D}^{L}$ is the sum of the induced connections on $L$ and
$L^{\perp}$, and $\mathcal{N}^{L}\in\Omega^{1}(L\wedge L^{\perp})$.

Suppose that $f\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathcal{Z}$
admits spherical curvature lines along the leaves of a rank $1$ distribution
$T\leq T\Sigma$. Thus, by Theorem 4.3, there exists a rank $1$ spacelike
subbundle $L\leq f^{\perp}$ that is constant along the leaves of $T$. We then
have that $\mathcal{N}^{L}(T)=0$, from which one deduces that
$\mathcal{D}^{L}$ is a flat metric connection. By Lemma 4.1 there exists a
complementary rank $1$ null subbundle $\tilde{T}$ of the conformal structure
$\mathcal{B}$ with $f_{T}\perp f_{\tilde{T}}$. Since $f\perp L$, we have that
$f_{T}\perp L$ and thus $f_{\tilde{T}}=f\oplus L$. Hence, $f$ is a parallel
subbundle of $\mathcal{D}^{L}|_{\tilde{T}}$. We thus arrive at the following
lemma:

######  Lemma 4.6.

$\mathcal{D}^{L}$ is a flat metric connection and $f$ is a parallel subbundle
of $\mathcal{D}^{L}|_{\tilde{T}}$.

Locally we may restrict $f$ to a rectangular domain $\Sigma=I\times\tilde{I}$
where the parameter lines along $I$ are the leaves of $T$ and the parameter
lines of $\tilde{I}$ are the leaves of $\tilde{T}$. Thus, $f$ is parametrised
by curvature lines $(u,v)$ with $u\in I$ and $v\in\tilde{I}$. Since $L$ is
constant along the leaves of $T$, we may view it as a map from
$\tilde{I}\to\mathcal{S}$.

Fixing a point $v_{0}\in\tilde{I}$, there exists an orthogonal trivialising
gauge transformation $A\mathrel{\mathop{\mathchar
58\relax}}\tilde{I}\to\operatorname{O}(4,2)$ of $\mathcal{D}^{L}$ such that

(4.1) $A\cdot\mathcal{D}^{L}=d\quad\text{and}\quad A(v_{0})=id.$

Note that the construction of $A$ is completely independent of $f$ and we can
make the following definition for an arbitrary map
$L\mathrel{\mathop{\mathchar 58\relax}}\tilde{I}\to\mathcal{S}$:

######  Definition 4.7.

Given a map $L\mathrel{\mathop{\mathchar 58\relax}}\tilde{I}\to\mathcal{S}$
and $v_{0}\in\tilde{I}$, we call $A\mathrel{\mathop{\mathchar
58\relax}}\tilde{I}\to\operatorname{O}(4,2)$ satisfying (4.1) the spherical
evolution map along $L$ based at $v_{0}$.

Clearly $L$ is a parallel subbundle of $\mathcal{D}^{L}$. Thus, with
$L_{0}\mathrel{\mathop{\mathchar 58\relax}}=L(v_{0})$, we have that
$L=A^{-1}L_{0}$. Since $f$ is a parallel subbundle of
$\mathcal{D}^{L}|_{\tilde{T}}$ and $f\perp L$, there exists
$C\mathrel{\mathop{\mathchar 58\relax}}I\to\mathcal{Z}$ with $C\perp L_{0}$
such that $f=A^{-1}\,C$. Since $f$ is a Legendre map, we have that $C$ is a
Legendre curve when viewed as a map into $L_{0}^{\perp}\cong\mathbb{R}^{3,2}$.

Conversely, suppose that $C\mathrel{\mathop{\mathchar
58\relax}}I\to\mathcal{Z}$ is a Legendre curve in $L_{0}^{\perp}$ and define
$f\mathrel{\mathop{\mathchar 58\relax}}=A^{-1}C\mathrel{\mathop{\mathchar
58\relax}}I\times\tilde{I}\to\mathcal{Z}$. We have that $f_{T}=A^{-1}C^{(1)}$
and thus $f_{T}$ is a rank $3$ subbundle of $f^{\perp}$. Now $f\perp L$ and
$f$ is parallel for $D^{L}|_{\tilde{T}}$, thus $f_{\tilde{T}}\leq f\oplus
L\leq f^{\perp}$. In order that $f$ be a Legendre map, we need to have that
$f_{\tilde{T}}=f\oplus L$. Since $A\cdot\mathcal{D}^{L}=d$ and
$\mathcal{D}^{L}=d-\mathcal{N}^{L}$, it follows that

(4.2) $A^{-1}\cdot(d+Ad_{A}\mathcal{N}^{L})=d.$

One then deduces that

$f_{\tilde{T}}=f+A^{-1}(Ad_{A}\mathcal{N}^{L}(\tilde{T}))C.$

Therefore, in order that $f_{\tilde{T}}$ has rank $3$, we require that
$(Ad_{A}\mathcal{N}^{L})|_{C}$ nowhere vanishes222This condition is akin to
the regularity condition required to construct Monge surfaces, see for example
[BG2018].. Once this condition is established, it follows from Lemma 4.1 that
the parameter lines of $I$ and $\tilde{I}$ are curvature lines of $f$, and
from Theorem 4.3 it follows that the $I$ parameter lines are spherical
curvature lines.

We have thus arrived at the following theorem, illustrated in Figure 2:

######  Theorem 4.8.

Let $L\mathrel{\mathop{\mathchar 58\relax}}\tilde{I}\to\mathcal{S}$ be an
immersion, set $L_{0}\mathrel{\mathop{\mathchar 58\relax}}=L(v_{0})$ for some
$v_{0}\in\tilde{I}$ and denote by $A$ the spherical evolution map along $L$
based at $v_{0}$. Let $C\mathrel{\mathop{\mathchar 58\relax}}I\to\mathcal{Z}$
be a Legendre curve in $L_{0}^{\perp}\cong\mathbb{R}^{3,2}$ such that
$(Ad_{A}\mathcal{N}^{L})|_{C}$ is nowhere zero. Then

$f\mathrel{\mathop{\mathchar 58\relax}}I\times\tilde{I}\to\mathcal{Z},\quad
f(u,v)=A^{-1}(v)C(u)$

is a Legendre map parametrised by curvature lines for which the $u$-parameter
lines are spherical.

Conversely, any surface with one family of spherical curvature lines may
locally be written in this way.

Figure 2. Given a curve of spheres with prescribed angles of intersection,
i.e., a curve of elliptic sphere complexes, one can obtain the spherical
evolution map and evolve an initial appropriate Legendre curve. In these
examples, one obtains either a well-known constant negative Gaussian curvature
torus (top) (see, for example, [E1868, melko_integrable_1994,
ura_constant_2018]), or the Wente torus (bottom) (see, for example, [A1987,
W1987, W1986]).

######  Remark 4.9.

For a different choice of initial point $\tilde{v}_{0}\in\tilde{I}$, the
spherical evolution map $\tilde{A}$ along $L$ based at $\tilde{v}_{0}$ differs
from $A$ by $\tilde{A}=BA$ for some constant $B\in\operatorname{O}(4,2)$. We
then have that $\tilde{C}\mathrel{\mathop{\mathchar
58\relax}}=BC\perp\tilde{L}_{0}$ with $\tilde{L}_{0}=L(\tilde{v}_{0})$ and the
resulting Legendre map

$\tilde{f}=\tilde{A}^{-1}\tilde{C}=A^{-1}B^{-1}BC=A^{-1}C=f.$

In this sense, the construction in Theorem 4.8 is independent of the choice of
$v_{0}$,

######  Remark 4.10.

As noted by Blaschke [B1929], Theorem 4.8 implies that if one family of
curvature lines is spherical then any two members of this family are related
by a Lie sphere transformation.

######  Remark 4.11.

Recall from Remark 4.2 that we excluded the case that the spheres containing
curvature lines are tangential to the surface. Without this restriction, $L$
could become lightlike, resulting in singular points of the corresponding
spherical evolution maps $A$.

######  Remark 4.12.

Suppose that $c$ is the curvature circle congruence of $C$ (viewed as a
Legendre curve in $L_{0}^{\perp}\cong\mathbb{R}^{3,2}$). As a map into
$\mathbb{R}^{4,2}$, $c$ yields a $1$-parameter family of curvature spheres
along the initial spherical curve of $f$. One then obtains a curvature sphere
congruence by parallel transporting $c$ along $L$, i.e.,
$s\mathrel{\mathop{\mathchar 58\relax}}=A^{-1}c$ is a curvature sphere
congruence of $f=A^{-1}C$.

###  4.3. Osculating sphere complexes

In this subsection we shall consider the case that
$f\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathcal{Z}$ is an umbilic-
free Legendre map. We then recover the results of [B1929, §88], defining the
osculating sphere complexes of such an $f$ and characterising the presence of
spherical curvature lines in terms of these.

Assume that $f\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathcal{Z}$ is an
umbilic-free Legendre map i.e., there are two curvature sphere congruences
$s_{1},s_{2}\leq f$ with $s_{1}\cap s_{2}=\\{0\\}$. Writing the corresponding
curvature subbundles as $T_{1}\mathrel{\mathop{\mathchar 58\relax}}=T_{s_{1}}$
and $T_{2}\mathrel{\mathop{\mathchar 58\relax}}=T_{s_{2}}$, we have that

$d_{X}\sigma_{1},d_{Y}\sigma_{2}\in\Gamma f$

for any $\sigma_{1}\in\Gamma s_{1}$, $\sigma_{2}\in\Gamma s_{2}$, $X\in\Gamma
T_{1}$ and $Y\in\Gamma T_{2}$.

If for some $i\in\\{1,2\\}$, $s_{i}$ is constant along the leaves of $T_{i}$,
i.e., $d_{X_{i}}\sigma_{i}\in\Gamma s_{i}$, then $f$ is a channel surface and
the curvature lines along $T_{i}$ are circular (see [PS2018]). Whereas if both
of the curvature sphere congruences immerse then we say that $f$ is regular.

######  Remark 4.13.

If $s_{i}\perp\mathfrak{q}$ for some non-zero
$\mathfrak{q}\in\mathbb{R}^{4,2}$ then $f$ is a channel surface and in fact
projects to a tube over a curve in space forms (see [BHR2012]).

For $f_{i}\mathrel{\mathop{\mathchar 58\relax}}=f_{T_{i}}$ we have that

$f_{1}=f\oplus\langle d_{X}\sigma_{2}\rangle\quad\text{and}\quad
f_{2}=f\oplus\langle d_{Y}\sigma_{1}\rangle$

for non-zero sections $\sigma_{1}\in\Gamma s_{1}$, $\sigma_{2}\in\Gamma
s_{2}$, $X\in\Gamma T_{1}$ and $Y\in\Gamma T_{2}$. These form rank 3
subbundles with $f_{1}\perp f_{2}$ and we may write the derived bundle as
$f^{(1)}=f_{1}+f_{2}$. Moreover the rank 1 bundles $f_{i}/f$ inherit a
positive definite metric from $\mathbb{R}^{4,2}$.

We may split the trivial bundle as

$\underline{\mathbb{R}}^{4,2}=S_{1}\oplus_{\perp}S_{2}$

where

(4.3)
$S_{1}=\langle\sigma_{1},d_{Y}\sigma_{1},d_{Y}d_{Y}\sigma_{1}\rangle\quad\text{and}\quad
S_{2}=\langle\sigma_{2},d_{X}\sigma_{2},d_{X}d_{X}\sigma_{2}\rangle$

for $\sigma_{1}\in\Gamma s_{1}$, $\sigma_{2}\in\Gamma s_{2}$, $X\in\Gamma
T_{1}$, and $Y\in\Gamma T_{2}$. This is called the Lie cyclide splitting.
Geometrically this represents a congruence of Dupin cyclides that make most
contact with $f$ along its curvature lines. We may then split the trivial
connection as

$d=\mathcal{D}+\mathcal{N}$

where $\mathcal{D}$ is the sum of the induced connections on $S_{1}$ and
$S_{2}$ and $\mathcal{N}\in\Omega^{1}(S_{1}\wedge S_{2})$. By [PS2018,
Corollary 3.6], a Legendre map parametrises a channel surface if and only if
the Lie cyclides are constant along the leaves of one of the curvature
subbundles $T_{i}$, i.e., $\mathcal{N}(T_{i})=0$.

If $f$ is regular then, since the curvature spheres immerse, we have that
$\mathcal{N}(T_{1})s_{1}=s_{2}$ and $\mathcal{N}(T_{2})s_{2}=s_{1}$. By (4.3)
we have that

$d_{Y}\sigma_{1},d_{Y}d_{Y}\sigma_{1}\in\Gamma S_{1}\quad\text{and}\quad
d_{X}\sigma_{2},d_{X}d_{X}\sigma_{2}\in\Gamma S_{2}.$

Since $S_{1}\cap f^{\perp}=\langle\sigma_{1},d_{Y}\sigma_{1}\rangle$ and
$S_{2}\cap f^{\perp}=\langle\sigma_{2},d_{X}\sigma_{2}\rangle$, it follows
that

(4.4) $\ker(\mathcal{N}(T_{2})|_{S_{1}})=S_{1}\cap
f^{\perp}\quad\text{and}\quad\ker(\mathcal{N}(T_{1})|_{S_{2}})=S_{2}\cap
f^{\perp}.$

Hence $\mathcal{N}(T_{1})$ and $\mathcal{N}(T_{2})$ are decomposable
subbundles of $S_{1}\wedge S_{2}$. We can then define rank 1 subbundles
$L_{i}\leq S_{i}$ by

(4.5) $L_{1}\mathrel{\mathop{\mathchar 58\relax}}=\ker(\mathcal{N}(T_{1}))\cap
S_{1}\cap f^{\perp}\quad\text{and}\quad L_{2}\mathrel{\mathop{\mathchar
58\relax}}=\ker(\mathcal{N}(T_{2}))\cap S_{2}\cap f^{\perp}.$

Since $\mathcal{N}(T_{1})s_{2}=s_{1}$ and $\mathcal{N}(T_{2})s_{1}=s_{2}$, we
must have that $L_{i}\cap s_{i}=\\{0\\}$. Thus the $L_{i}$ are spacelike.
(4.4) and (4.5) together imply that the $L_{i}$ are the unique333If $f$ is a
channel surface enveloping the sphere curve $s_{i}$ then the uniqueness of
$L_{i}$ fails. However, one of the $L_{i}$ may still be uniquely determined as
long as the corresponding curvature sphere $s_{i}$ is an immersion. subbundles
of $S_{i}$ satisfying $\mathcal{N}L_{i}\equiv 0$. We have thus arrived at the
following lemma:

######  Lemma 4.14.

Suppose that $f$ is regular. Then for each $i\in\\{1,2\\}$, there exists a
unique spacelike rank 1 subbundle $L_{i}\leq S_{i}$ such that
$\mathcal{N}L_{i}\equiv 0$. Moreover, $f\perp L_{i}$ and $f_{1}=f\oplus L_{2}$
and $f_{2}=f\oplus L_{1}$.

######  Remark 4.15.

Suppose that $(u,v)$ are curvature line coordinates for $f$ with
$\frac{\partial}{\partial u}\in\Gamma T_{1}$ and $\frac{\partial}{\partial
v}\in\Gamma T_{2}$. We may choose lifts $\sigma_{1}\in\Gamma s_{1}$,
$\sigma_{2}\in\Gamma s_{2}$ such that

$\sigma_{1,u}=\beta\sigma_{2}\quad\text{and}\quad\sigma_{2,v}=\gamma\sigma_{2}$

for some smooth functions $\beta$ and $\gamma$. The condition that $f$ is
regular is equivalent to assuming that $\beta$ and $\gamma$ are nowhere zero.
We then have that

$L_{1}=\langle\beta\sigma_{1,v}-\beta_{v}\sigma_{1}\rangle\quad\text{and}\quad
L_{2}=\langle\gamma\sigma_{2,u}-\gamma_{u}\sigma_{2}\rangle.$

Since the $L_{i}$ are spacelike, they form two congruences of elliptic linear
sphere complexes $E_{L_{i}}$ and, since $f\perp L_{i}$, we have that $f\subset
E_{L_{i}}$, that is, at each point $x\in\Sigma$, the spheres contained in
$f(x)$ belong to the linear sphere complex $E_{L_{i}(x)}$. This leads us to
the following definition:

######  Definition 4.16.

$L_{1}$ and $L_{2}$ are called the osculating sphere complexes of $f$.

######  Remark 4.17.

One can show that $L_{i}$ is perpendicular to $f$ and derivatives of sections
of $f$ along $T_{i}$ to the third order. We can thus interpret $L_{i}$ as the
unique linear sphere complex containing the spheres of $4$ infinitesimally
close contact elements of $f$ along the curvature lines $T_{i}$.

######  Lemma 4.18.

Suppose that $L\leq f^{\perp}$ is a rank $1$ spacelike subbundle of
$\underline{\mathbb{R}}^{4,2}$ that is constant along the leaves of a rank $1$
distribution $T\leq T\Sigma$. Then either $L=L_{1}$ and $T=T_{1}$ or $L=L_{2}$
and $T=T_{2}$.

###### Proof.

Let $f_{T}$ denote the set of sections of $f$ and derivatives of sections of
$f$ along $T$. Since $L$ is constant along $T$, it follows that $f_{T}\perp
L$. Thus $f_{T}$ is a rank $3$ subbundle of $f^{(1)}$. By Lemma 4.1 we have
that $T$ is either $T_{1}$ or $T_{2}$. Without loss of generality, assume that
$T=T_{1}$. By differentiating $s_{2}$ along $T_{1}$, we deduce that $L\perp
S_{2}$. Thus $L\leq S_{1}\cap f^{\perp}$ and, since $L$ is constant along
$T_{1}$, Lemma 4.14 implies that $L=L_{1}$. ∎

Using Lemma 4.18 and Theorem 4.3, we can recover the characterisation of
Blaschke for umbilic-free surfaces that have spherical curvature lines in
terms of the osculating sphere complexes:

######  Proposition 4.19.

The leaves of $T_{i}$ are spherical curvature lines of $f$ if and only if the
osculating complex $L_{i}$ is constant along the leaves of $T_{i}$.

###### Proof.

Suppose that $L_{i}$ is constant along the leaves of $T_{i}$. Then by Theorem
4.3 $f$ has spherical curvature lines along the leaves of $T_{i}$.

Conversely assume, without loss of generality, that the leaves of $T_{i}$ are
spherical curvature lines of $f$. Then by Theorem 4.3, there exists a rank $1$
spacelike subbundle $L\leq f^{\perp}$ that is constant along the leaves of
$T_{i}$. By Lemma 4.18, it then follows that $L=L_{i}$. ∎

An immediate corollary to Proposition 4.19 is the following:

######  Corollary 4.20.

All of the curvature lines of $f$ are spherical if and only if the osculating
complexes $L_{1}$ and $L_{2}$ are constant along the leaves of $T_{1}$ and
$T_{2}$, respectively.

######  Remark 4.21.

In terms of the special lifts $\sigma_{1},\sigma_{2}$ of the curvature spheres
given in Remark 4.15, one can deduce that $f$ has spherical curvature lines
along the leaves of $T_{1}$ or $T_{2}$ if and only

$(\ln\beta)_{uv}-\beta\gamma=0\quad\text{or}\quad(\ln\gamma)_{uv}-\beta\gamma=0,$

respectively.

Let $l_{i}\in\Gamma L_{i}$ be unit spacelike sections of $L_{i}$. By Lemma
4.14 we have that $l_{i}\perp f_{i}$ and $\mathcal{N}L_{i}\equiv 0$. It thus
follows that

(4.6) $d|_{T_{i}}l_{i}\in\Gamma(T_{i}^{*}\otimes s_{i}).$

From Proposition 4.19 we then have that the leaves of $T_{i}$ are spherical
curvature lines if and only if $d|_{T_{i}}l_{i}=0$.

Fix nowhere-zero sections $X\in\Gamma T_{1}$ and $Y\in\Gamma T_{2}$ and define

$\mathcal{H}_{1}\mathrel{\mathop{\mathchar 58\relax}}=\langle
l_{1},d_{Y}l_{1},d_{Y}d_{Y}l_{1}\rangle\quad\text{and}\quad\mathcal{H}_{2}\mathrel{\mathop{\mathchar
58\relax}}=\langle l_{2},d_{X}l_{2},d_{X}d_{X}l_{2}\rangle.$

Note that $\mathcal{H}_{i}$ are independent of choices. Blaschke [B1929]
showed that these are rank $3$ subbundles of $\underline{\mathbb{R}}^{4,2}$
and $\mathcal{H}_{1}\perp\mathcal{H}_{2}$. It is not always true that
$\mathcal{H}_{1}\cap\mathcal{H}_{2}\neq\\{0\\}$. For example, we shall see in
the next subsection that if $f$ has two families of planar curvature lines in
some Euclidean geometry with space form vector $\mathfrak{q}$, then we have
that $\mathfrak{q}\in\mathcal{H}_{1}\cap\mathcal{H}_{2}$.

######  Definition 4.22.

We call $\mathcal{H}_{i}$ the osculating bundles of $f$.

Clearly if $L_{i}$ is constant along $T_{i}$ then $\mathcal{H}_{i}$ is
constant along $T_{i}$. Conversely, if $\mathcal{H}_{i}$ is constant along
$T_{i}$ then $d|_{T_{i}}l_{i}\in\Gamma T_{i}^{*}\otimes\mathcal{H}_{i}$. On
the other hand, (4.6) implies that $d|_{T_{i}}l_{i}\in\Gamma(T_{i}^{*}\otimes
s_{i})$. One deduces that we must then have that $d|_{T_{i}}l_{i}=0$, as
otherwise $\mathcal{H}_{i}=S_{i}$ is constant along $T_{i}$ implying that $f$
is a channel surface. We thus recover the following result:

######  Proposition 4.23.

Suppose that $f$ is regular. Then the leaves of $T_{i}$ are spherical
curvature lines of $f$ if and only if $\mathcal{H}_{i}$ is constant along
$T_{i}$. In particular, both families of curvature lines of $f$ are spherical
if and only if $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are constant.

In the next subsection we shall see that surfaces with two families of
spherical curvature lines can be characterised in Euclidean space forms by
choosing appropriate point sphere complexes and space form vectors in the
constant spaces $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$.

###  4.4. Symmetry breaking

In this subsection we shall examine some specific cases of spherical curvature
lines that belong to subgeometries of Lie sphere geometry.

Consider the case that $L_{i}\perp\mathfrak{q}$ for some non-zero
$\mathfrak{q}\in\mathbb{R}^{4,2}$. It then follows that
$d|_{T_{i}}l_{i}\perp\mathfrak{q}$. Since $f$ is regular, Remark 4.13 implies
that $d|_{T_{i}}l_{i}=0$, and thus the leaves of $T_{i}$ are spherical
curvature lines of $f$. Using Remark 2.4 one can deduce the following
proposition:

######  Proposition 4.24.

The leaves of $T_{i}$ project to planar curvature lines in any space form with
space form vector $\mathfrak{q}$ if and only if $L_{i}\perp\mathfrak{q}$ .

The leaves of $T_{i}$ are orthogonally intersected spherical curvature lines
in a Riemannian conformal geometry with timelike point sphere complex
$\mathfrak{p}$ if and only if $L_{i}\perp\mathfrak{p}$.

Recall from Theorem 4.8 that we may locally write a surface with spherical
curvature lines as $f=A^{-1}C\mathrel{\mathop{\mathchar
58\relax}}I\times\tilde{I}\to\mathcal{Z}$ where $A$ is the spherical evolution
map along $L\mathrel{\mathop{\mathchar 58\relax}}\tilde{I}\to\mathcal{S}$
based at $v_{0}\in\tilde{I}$. In the case that $L\perp\mathfrak{q}$ for some
non-zero $\mathfrak{q}\in\mathbb{R}^{4,2}$, we have that
$A\mathfrak{q}=\mathfrak{q}$. Using Subsection 2.1 we deduce the following
corollary:

######  Corollary 4.25.

If $f$ has one family of planar curvature lines, then we may locally write
$f=A^{-1}C$ where $A$ is a $1$-parameter family of Laguerre transformations.

If $f$ has one family of orthogonally intersected spherical curvature lines,
then we may locally write $f=A^{-1}C$ where $A$ is a $1$-parameter family of
Möbius transformations.

Let $\mathfrak{q},\mathfrak{p}$ be the space form vector and point sphere
complex for a Euclidean geometry $\mathfrak{Q}^{3}$. From Proposition 4.24 we
deduce that $L_{i}\perp\mathfrak{q},\mathfrak{p}$ if and only if $T_{i}$ are a
family of orthogonally intersected planar curvature lines in
$\mathfrak{Q}^{3}$. According to [BG2018] such surfaces are locally Monge
surfaces, that is a surface formed by taking a curve in $\mathbb{R}^{3}$ and
parallel transporting an initial curve lying in the normal plane along that
curve. This is reflected by the fact that, locally, spherical evolution maps
$A$ along $L_{i}$ satisfy $A\mathfrak{p}=\mathfrak{p}$ and
$A\mathfrak{q}=\mathfrak{q}$, and are thus isometries of $\mathfrak{Q}^{3}$.

######  Proposition 4.26.

$f$ locally projects to a Monge surface in a Euclidean space form
$\mathfrak{Q}^{3}$ defined by point sphere complex $\mathfrak{p}$ and space
form vector $\mathfrak{q}$ if and only if one of the osculating complexes
satisfies $L_{i}\perp\mathfrak{p},\mathfrak{q}$.

Blaschke [B1929, §88] characterised surfaces with two families of spherical
curvature lines up to Lie sphere transformation and showed that three cases
exist depending on the signature of the constant bundles $\mathcal{H}_{1}$ and
$\mathcal{H}_{2}$. These characterisations can be recovered by making
appropriate choices of timelike point sphere complex $\mathfrak{p}$ and
lightlike space form vector $\mathfrak{q}$ for a Euclidean geometry
$\mathfrak{Q}^{3}$ and evaluating the corresponding point sphere map
$\mathfrak{f}=f\cap\mathfrak{Q}^{3}$:

  * •

If $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ both have signature $(2,1)$ then we
may choose $\mathfrak{p}\in\mathcal{H}_{1}$ and
$\mathfrak{q}\in\mathcal{H}_{2}$. By Proposition 4.24 it follows that
$\mathfrak{f}$ has planar curvature lines along $T_{1}$ and orthogonally
intersected spherical curvature lines along $T_{2}$. This characterises
$\mathfrak{f}$ as a Joachimsthal surface.

  * •

If $\mathcal{H}_{1}$ has signature $(3,0)$ and $\mathcal{H}_{2}$ has signature
$(1,2)$, then we may choose $\mathfrak{q},\mathfrak{p}\in\mathcal{H}_{2}$. By
Proposition 4.26, we then have that $\mathfrak{f}$ is a Monge surface with
planar curvature lines along $T_{1}$. Moreover, there exists a lightlike
vector $\mathfrak{o}\in\mathcal{H}_{2}$ that is perpendicular to
$\mathfrak{p}$ with $(\mathfrak{o},\mathfrak{q})=-1$. Thus $\mathfrak{o}$ is a
point in $\mathfrak{Q}^{3}$ and the planes defined by $L_{1}$ all contain this
point. Hence the planes containing the curvature lines along $T_{1}$ envelop a
cone.

  * •

If $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are degenerate, then we may choose
$\mathfrak{q}\in\mathcal{H}_{1}\cap\mathcal{H}_{2}$. Thus, by Proposition
4.24, we have that $\mathfrak{f}$ has planar curvature lines along $T_{1}$ and
$T_{2}$.

######  Remark 4.27.

Using the formulation in Remark 4.21, we have that surfaces with two families
of spherical curvature lines satisfy

$(\ln\beta)_{uv}=(\ln\gamma)_{uv}=\beta\gamma.$

In [F2002] this is used to show that such surfaces are Lie applicable. In
Section 6 we shall recover this result using a curved flat formulation of Lie
applicable surfaces.

##  5\. Ribaucour transforms of surfaces with spherical curvature lines

Classically a pair of surfaces is called a Ribaucour pair if they envelop a
common sphere congruence and the curvature lines of both surfaces correspond.
We then say that the two surfaces are Ribaucour transforms of each other. In
[BH2006] a Lie geometric characterisation of Ribaucour transforms was given in
terms of the flatness of a certain rank $2$ bundle. In [B1929] it is shown
that Ribaucour transforms of channel surfaces have a family of spherical
curvature lines. In this section we shall recover this result and show that
the converse is true.

Suppose that $f,\hat{f}\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathcal{Z}$ are Legendre maps such that
$s_{0}\mathrel{\mathop{\mathchar 58\relax}}=f\cap\hat{f}$ is a rank $1$
subbundle of $f$ and $\hat{f}$. Geometrically, $s_{0}$ is a common sphere
congruence enveloped by both $f$ and $\hat{f}$. From [P2020, Corollary 2.11]
we have the following characterisation of when $f$ and $\hat{f}$ are a
Ribaucour pair:

######  Lemma 5.1.

Suppose that $s\leq f$ and $\hat{s}\leq\hat{f}$ such that $s\cap
s_{0}=\hat{s}\cap s_{0}=\\{0\\}$. Then $f$ and $\hat{f}$ are Ribaucour
transforms of each other if and only if the induced connection on
$s\oplus\hat{s}$ is flat.

Suppose now that $f$ and $\hat{f}$ are umbilic-free. Let $s_{1}$ and $s_{2}$
denote the curvature sphere congruences of $f$ with respective curvature
subbundles $T_{1}$ and $T_{2}$, and likewise let $\hat{s}_{1}$, $\hat{s}_{2}$
denote the curvature sphere congruences of $\hat{f}$ with respective curvature
subbundles $\hat{T}_{1}$ and $\hat{T}_{2}$. We can then say that $f$ and
$\hat{f}$ are Ribaucour transforms of each other if and only if
$\hat{T}_{1}=T_{1}$ and $\hat{T}_{2}=T_{2}$. Moreover, from [PS2018,
Proposition 4.3] we have the following result:

######  Lemma 5.2.

Suppose that $s_{0}$ nowhere coincides with the curvature sphere congruences
of $f$ or $\hat{f}$. Then $f$ and $\hat{f}$ are Ribaucour transforms of each
other if and only if

$d\hat{\sigma}_{1}(\hat{T}_{2})\leq s_{1}\oplus\hat{s}_{1}\oplus
d\sigma_{1}(T_{2})\quad\text{and}\quad d\hat{\sigma}_{2}(\hat{T}_{1})\leq
s_{2}\oplus\hat{s}_{2}\oplus d\sigma_{2}(T_{1}),$

where $\sigma_{i}\in\Gamma s_{i}$ and $\hat{\sigma}_{i}\in\Gamma\hat{s}_{i}$.

For later use we have the following technical lemma:

######  Lemma 5.3.

Suppose that $f$ is a regular umbilic-free Legendre map and suppose that
$\hat{f}\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathcal{Z}$ with
$\hat{f}^{(1)}\perp\hat{f}$ and $f\cap\hat{f}=s_{0}$ for some rank $1$ null
subbundle $s_{0}$. Then $\hat{f}$ is a Legendre map and $s_{0}$ nowhere
coincides with the curvature sphere congruences of $f$ or $\hat{f}$.

###### Proof.

Fix a non-zero section $\sigma_{0}\in\Gamma s_{0}$. Then $d\sigma_{0}\perp
f+\hat{f}$. Since $f$ is regular, if $d_{X}\sigma_{0}\in f(x)$ for some $X\in
T_{x}\Sigma$, then we would have that $f=s_{0}(x)\oplus\langle
d_{X}\sigma_{0}\rangle$. This would imply that $f(x)\leq(\hat{f}(x))^{\perp}$
and thus $f(x)=\hat{f}(x)$, contradicting our assumption that $f\cap\hat{f}$
is rank $1$. It thus follows that $d\sigma_{0}(T\Sigma)$ is a spacelike rank
$2$ subbundle of $f^{\perp}$. Since $s_{0}\leq\hat{f}$, together with the
condition $\hat{f}^{(1)}\leq\hat{f}^{\perp}$, one deduces that $\hat{f}$ is a
Legendre map. ∎

###  5.1. Ribaucour transforms of surfaces with spherical curvature lines

Suppose that $f$ is an umbilic-free channel surface. Without loss of
generality, assume that $f$ has circular curvature directions along the leaves
of $T_{1}$. From [PS2018, Proposition 3.4] we have that the curvature sphere
congruence $s_{1}$ is constant along the leaves of $T_{1}$. It thus follows
that $s_{1}\oplus d\sigma_{1}(T_{2})$ is constant along the leaves of $T_{1}$.

Now suppose that $\hat{f}$ is a regular umbilic-free Ribaucour transform of
$f$ with enveloping sphere congruence $s_{0}=f\cap\hat{f}$. By Lemma 5.3 we
have that $s_{0}$ nowhere coincides with the curvature sphere congruences of
$\hat{f}$. We may then apply Lemma 5.2 to learn that

(5.1) $\hat{L}\mathrel{\mathop{\mathchar 58\relax}}=(s_{1}\oplus
d\sigma_{1}(T_{2}))\cap(\hat{s}_{1}\oplus
d\hat{\sigma}_{1}(T_{2}))=(s_{1}\oplus d\sigma_{1}(T_{2}))\cap\hat{f}^{\perp}$

has rank $1$. Since $s_{1}\oplus d\sigma_{1}(T_{2})$ is constant along $T_{1}$
and sections of $\hat{s}_{1}\oplus d\hat{\sigma}_{1}(T_{2})$ differentiate
into $\hat{f}^{\perp}$ along $T_{1}$, we have that $\hat{L}$ is constant along
$T_{1}$. From Lemma 4.18 we deduce that the osculating complex
$\hat{L}_{1}=\hat{L}$ and Proposition 4.19 implies that $\hat{f}$ has
spherical curvature lines along the leaves of $T_{1}$. We have thus recovered
the following result:

######  Proposition 5.4 ([B1929]).

Ribaucour transforms of channel surfaces have one family of spherical
curvature lines.

Dupin cyclides are those surfaces that are doubly channel, i.e., both families
of curvature lines are circular. From Proposition 5.4, we immediately recover
the result of [B1929] that Ribaucour transforms of Dupin cyclides have two
families of spherical curvature lines. However, we can in fact say more. By
(5.1), we have that $\hat{L}_{1}\leq S_{1}$ and $\hat{L}_{2}\leq S_{2}$, where
$S_{1}$ and $S_{2}$ are the Lie cyclides of $f$. Since $f$ is a Dupin cyclide,
the Lie cylides are the Dupin cylide itself. Hence, $S_{1}$ and $S_{2}$ are
constant $(2,1)$ bundles. Thus the osculating bundles $\hat{\mathcal{H}}_{1}$
and $\hat{\mathcal{H}}_{2}$ of $\hat{f}$ are both constant $(2,1)$ bundles. It
thus follows from Subsection 4.4 that $\hat{f}$ projects to a Joachimsthal
surface in an appropriate Euclidean space form. We thus arrive at the
following proposition:

######  Proposition 5.5.

Ribaucour transforms of Dupin cyclides project to Joachimsthal surfaces in
appropriate Euclidean space forms.

We now seek a converse to Proposition 5.4. Suppose that
$f\mathrel{\mathop{\mathchar 58\relax}}\Sigma\to\mathcal{Z}$ is a regular
umbilic-free Legendre map, and without loss of generality, assume that the
leaves of $T_{1}$ are spherical curvature lines. By Proposition 4.19 we have
that $L\mathrel{\mathop{\mathchar 58\relax}}=L_{1}$ is constant along $T_{1}$.
By Lemma 4.6 we have that $\mathcal{D}^{L}$ is flat. Since $L$ is parallel for
$\mathcal{D}^{L}$, there locally exists a $3$-parameter family of parallel
null rank $1$ subbundles $\hat{s}$ of $\mathcal{D}^{L}$ with $\hat{s}\perp L$
and $\hat{s}\cap f=\\{0\\}$. Define $s_{0}\mathrel{\mathop{\mathchar
58\relax}}=f\cap\hat{s}^{\perp}$ and $\hat{f}\mathrel{\mathop{\mathchar
58\relax}}=s_{0}\oplus\hat{s}$. Since $\hat{s}$ is parallel for
$\mathcal{D}^{L}$, it follows that $\hat{f}^{(1)}\perp\hat{f}$. Then by Lemma
5.3, we have that $\hat{f}$ is a Legendre map. Now fix a sphere congruence
$s\leq f$ with $s\cap s_{0}=\\{0\\}$. Then it follows from the fact that
$\hat{s}$ is parallel for $\mathcal{D}^{L}$ that the induced connection on
$s\oplus\hat{s}$ if flat. By Lemma 5.1 we then have that $\hat{f}$ is a
Ribaucour transform of $f$. Moreover, since $\mathcal{N}^{L}(T_{1})=0$, we
have that $\hat{s}$ is constant along $T_{1}$. Thus $\hat{f}$ is a channel
surface with circular curvature lines along the leaves of $T_{1}$. We thus
arrive at the following theorem:

######  Theorem 5.6.

Any regular umbilic-free Legendre map with spherical curvature lines is a
Ribaucour transform of a $3$-parameter family of channel surfaces.

######  Remark 5.7.

Theorem 5.6 is a Lie geometric version of the projective geometric result
found in [F2000i] that states that a surface with one family of asymptotic
lines in linear complexes admits a $3$-parameter family of $W$-transformations
to ruled surfaces.

In Subsection 6.2 we shall prove a converse to Proposition 5.5 using Darboux
transforms.

###  5.2. Ribaucour transforms via the spherical evolution map

In this subsection we shall see how Ribaucour pairs of surfaces with spherical
curvature lines can be constructed using spherical evolution of Ribaucour
pairs of Legendre curves. This leads to a geometric proof of Theorem 5.6.

Let $L\mathrel{\mathop{\mathchar 58\relax}}\tilde{I}\to\mathcal{S}$ be an
immersion and let $A$ be the spherical evolution map based at $v_{0}$. Suppose
that $C,\hat{C}\mathrel{\mathop{\mathchar 58\relax}}I\to\mathcal{Z}$ are two
Legendre curves in $L_{0}^{\perp}$, with $L_{0}\mathrel{\mathop{\mathchar
58\relax}}=L(v_{0})$, such that $Ad_{A}\mathcal{N}^{L}|_{C}$ and
$Ad_{A}\mathcal{N}^{L}|_{\hat{C}}$ are nowhere zero. Suppose further that
$C\cap\hat{C}=c_{0}$ is a rank $1$ subbundle of $C$ and $\hat{C}$.
Geometrically this means that $C$ and $\hat{C}$ envelop a common circle
congruence $c_{0}$. Thus $C$ and $\hat{C}$ are a Ribaucour pair of curves (see
[BHMR2016]). From Theorem 4.8 we have that

$f(u,v)=A^{-1}(v)C(u)\quad\text{and}\quad\hat{f}(u,v)=A^{-1}(v)\hat{C}(u)$

are Legendre maps parametrised by curvature line coordinates $(u,v)$ such that
the $u$-parameter lines are spherical curvature lines. Since $f,\hat{f}\perp
L$, we have that the $u$-parameter lines of $f$ and $\hat{f}$ lie on the same
$1$-parameter family of spheres. Now $s_{0}\mathrel{\mathop{\mathchar
58\relax}}=A^{-1}c_{0}$ is a rank $1$ subbundle of $f$ and $\hat{f}$ and thus
$f$ and $\hat{f}$ envelop a common sphere congruence $s_{0}$. Moreover, since
$(u,v)$ are curvature line coordinates of $f$ and $\hat{f}$, it follows that
$f$ and $\hat{f}$ are a Ribaucour pair of surfaces.

######  Proposition 5.8.

Spherical evolution of a Ribaucour pair of Legendre curves yields a Ribaucour
pair of Legendre maps.

######  Remark 5.9.

Following from Subsection 4.4, we see that Proposition 5.8 yields a
construction method for Ribaucour pairs of, for example, surfaces with planar
curvature lines or Monge surfaces.

By choosing a circular Legendre curve $\hat{C}\mathrel{\mathop{\mathchar
58\relax}}I\to\mathcal{Z}$, one has that the $u$-curvature lines of
$\hat{f}(u,v)=A^{-1}(v)\hat{C}(u)$ are circular and thus $\hat{f}$ is a
channel surface. In fact, such a circular $\hat{C}$ contains a constant circle
$\hat{c}\leq\hat{C}$, and this evolves to yield the sphere curve
$\hat{s}\mathrel{\mathop{\mathchar 58\relax}}=A^{-1}\hat{c}\leq\hat{f}$
enveloped by $\hat{f}$.

Given a Legendre curve $C\mathrel{\mathop{\mathchar 58\relax}}I\to\mathcal{Z}$
in $L_{0}^{\perp}$, there generically exists a $3$-parameter family of
circular Ribaucour transforms $\hat{C}\mathrel{\mathop{\mathchar
58\relax}}I\to\mathcal{Z}$ of $C$ in $L_{0}^{\perp}$. Any such transform can
be constructed by choosing a constant $\hat{c}\in E_{L_{0}}$ such that
$\hat{c}\cap C=\\{0\\}$ and defining $\hat{C}=\hat{c}\oplus c_{0}$ where
$c_{0}=C\cap\hat{c}^{\perp}$. By evolving such $\hat{C}$ we recover the three
parameter family of channel surfaces of Theorem 5.6 that are Ribaucour
transforms of $f=A^{-1}C$ (see Figures 3, 4 and 5).

Figure 3. Ribaucour pairs of Dupin cyclides. For Dupin cyclides, one can find
an evolution map that consists of point sphere preserving Lie inversions;
hence, the evolution map indeed transports the (point spheres of the) circles
of the Dupin cyclide. Figure 4. Compact channel surface (on the right) that is
a Ribaucour transform of the cmc surface known as a bubbleton (on the left),
with a few of the corresponding curvature lines highlighted. The channel
surface is obtained using the spherical evolution map induced by the planar
curvature lines of the cmc surface. Figure 5. Surface of revolution that is a
Ribaucour transform of the constant negative Gaussian curvature torus of
Figure 2, with two of the corresponding curvature lines highlighted. The
surface of revolution is obtained using the spherical evolution map induced by
the spherical curvature lines of the $K$-surface (see right figure).

##  6\. Lie applicable surfaces with spherical curvature lines

Lie applicable surfaces are the deformable surfaces of Lie sphere geometry
[MN2006]. Examples include isothermic surfaces and Guichard surfaces of
conformal geometry, $L$-isothermic surfaces of Laguerre geometry and linear
Weingarten surfaces of space form geometries (see [BHR2012, BHPR2019]).

From Definition 2.5, we have that $f\mathrel{\mathop{\mathchar
58\relax}}\Sigma\to\mathcal{Z}$ is a Lie applicable surface if and only if
there exists $\eta\in\Omega^{1}(f\wedge f^{\perp})$ satisfying
$d\eta=[\eta\wedge\eta]=0$, whose associated quadratic differential
$\mathcal{Q}^{\eta}$ is non-zero. In [P2020, Proposition 3.4] it is shown
that, when $f$ is umbilic-free, the closure of $\eta$ implies
$[\eta\wedge\eta]=0$ and $\eta(T_{i})\leq f\wedge f_{i}$. We then have that
the quadratic differential $\mathcal{Q}^{\eta}$ may be written as

$\mathcal{Q}^{\eta}=\mathcal{Q}^{\eta}_{1}+\mathcal{Q}^{\eta}_{2}$

with $\mathcal{Q}^{\eta}_{1}\in\Gamma(T^{*}_{1})^{2}$ and
$\mathcal{Q}^{\eta}_{2}\in\Gamma(T^{*}_{2})^{2}$.

Lie applicable surfaces have the property that they envelop isothermic sphere
congruences $s\leq f$. One characterisation of such sphere congruences is via
the existence of a Moutard lift, that is, a section $\sigma\in\Gamma s$ such
that $\sigma_{uv}\in\Gamma s$ where $(u,v)$ are curvature line coordinates of
$f$ with $\frac{\partial}{\partial u}\in\Gamma T_{1}$ and
$\frac{\partial}{\partial v}\in\Gamma T_{2}$. The class of Lie applicable
surfaces then splits into two classes:

  * •

if $\mathcal{Q}^{\eta}_{1}$ and $\mathcal{Q}^{\eta}_{2}$ are both non-zero
then $f$ envelops a pair of isothermic sphere congruences that separate the
curvature spheres harmonically. Such surfaces are called $\Omega$-surfaces.

  * •

if $\mathcal{Q}^{\eta}_{i}$ vanishes for some $i\in\\{1,2\\}$, then the
curvature sphere congruence $s_{i}$ is isothermic. Such surfaces are called
$\Omega_{0}$-surfaces.

In this section we shall consider the case of an umbilic-free Legendre map
having at least one family of spherical curvature lines, whilst additionally
being a Lie applicable surface. In [B2001], isothermic surfaces with one or
two families of spherical curvature lines are characterised in terms of
differential equations on the Calapso potential. In [MN1999i] it is shown that
surfaces with two families of planar curvature lines are $L$-isothermic and in
[F2002] it is shown that surfaces with two families of spherical curvature
lines are multiply Lie applicable. In [PS2018] it shown that channel surfaces
are examples of $\Omega_{0}$-surfaces and in [HPPwip] it is shown that channel
surfaces that are additionally $\Omega$-surfaces are rotational surfaces.

###  6.1. Lie applicable surfaces with one family of spherical curvature lines

Suppose that $f\mathrel{\mathop{\mathchar 58\relax}}I_{1}\times
I_{2}\to\mathcal{Z}$ is a regular umbilic-free Legendre map parametrised by
curvature line coordinates $(u,v)$ and, without loss of generality, assume
that the $u$-parameter lines are spherical. By Proposition 4.19 this implies
that the osculating sphere complex $L\mathrel{\mathop{\mathchar
58\relax}}=L_{1}$ is constant along the $u$-parameter lines, and thus we may
consider it as a map $I_{2}\to S$. Using Theorem 4.8 we may write $f$ as

$f=A^{-1}C\mathrel{\mathop{\mathchar 58\relax}}I_{1}\times
I_{2}\to\mathcal{Z}$

where $A$ is the spherical evolution map along $L$ based at $v_{0}\in I_{2}$
and $C\mathrel{\mathop{\mathchar 58\relax}}I_{1}\to\mathcal{Z}$ is a Legendre
curve in $L_{0}^{\perp}$ with $L_{0}\mathrel{\mathop{\mathchar
58\relax}}=L(v_{0})$.

Consider $\eta\in\Omega^{1}(f\wedge f^{\perp})$. We may write

$\eta=\eta_{f\wedge L}+\eta_{f\wedge L^{\perp}}$

where $\eta_{f\wedge L}\in\Omega^{1}(f\wedge L)$ and $\eta_{f\wedge
L^{\perp}}\in\Omega^{1}(f\wedge(f^{\perp}\cap L^{\perp}))$. By Lemma 4.14 we
have that $f_{1}=f\oplus L_{2}$ and $f_{2}=f\oplus L_{1}$. Thus the conditions
$\eta(T_{i})\leq f\wedge f_{i}$ imply that $\eta_{f\wedge L}(T_{1})=0$ and
$\eta_{f\wedge L^{\perp}}(T_{2})\leq\wedge^{2}f$. One then deduces that

$\displaystyle\mathcal{Q}^{\eta}_{1}(X,Y)$ $\displaystyle=\textrm{tr}(f\to
f\mathrel{\mathop{\mathchar 58\relax}}\sigma\mapsto\eta_{f\wedge
L^{\perp}}(X)d_{Y}\sigma),$ $\displaystyle\mathcal{Q}^{\eta}_{2}(X,Y)$
$\displaystyle=\textrm{tr}(f\to f\mathrel{\mathop{\mathchar
58\relax}}\sigma\mapsto\eta_{f\wedge L}(X)d_{Y}\sigma).$

The following result shows that if a regular Legendre map $f$ has spherical
curvature lines along the leaves of $T_{i}$, then the curvature sphere
congruence $s_{i}$ cannot be isothermic:

######  Lemma 6.1.

Suppose that $\eta$ is closed with $\mathcal{Q}^{\eta}\neq 0$. Then
$\mathcal{Q}^{\eta}_{1}\neq 0$.

###### Proof.

If $\mathcal{Q}^{\eta}_{1}=0$, then $s_{1}$ is isothermic and there exists a
Moutard lift $\sigma_{1}\in\Gamma s_{1}$, i.e., $\sigma_{1,uv}\in\Gamma
s_{1}$. Now $(\sigma_{1,v})_{u},(\sigma_{1,v})_{v}\in\Gamma S_{1}$ and one
deduces that $L_{1}=\langle\sigma_{1,v}\rangle$. Since $L_{1}$ is constant
along $T_{1}$, it follows that $\sigma_{1,uv}=(\sigma_{1,v})_{u}=0$. Fixing
some non-zero section $\sigma_{2}\in\Gamma s_{2}$, we have that
$\sigma_{1,u}=\alpha\sigma_{1}+\beta\sigma_{2}$ for some functions $\alpha$,
$\beta$. The condition that $\sigma_{1,uv}=0$ implies that $\alpha=0$ and
either $\beta=0$ or $\sigma_{2,v}=0$. This implies that one of the curvature
spheres of $f$ does not immerse, contradicting our regularity assumption on
$f$. ∎

Using the spherical evolution map $A$ we may write

$\eta=Ad_{A^{-1}}\cdot(\zeta_{C\wedge L_{0}}+\zeta_{C\wedge L_{0}^{\perp}}),$

where

$\zeta_{C\wedge L_{0}}\mathrel{\mathop{\mathchar
58\relax}}=Ad_{A}\cdot\eta_{f\wedge L}\in\Omega^{1}(C\wedge L_{0})\>\text{ and
}\>\zeta_{C\wedge L_{0}^{\perp}}\mathrel{\mathop{\mathchar
58\relax}}=Ad_{A}\cdot\eta_{f\wedge
L^{\perp}}\in\Omega^{1}(C\wedge(C^{\perp}\cap L_{0}^{\perp})).$

Since $\eta_{f\wedge L}(T_{1})=0$ and $\eta_{f\wedge
L^{\perp}}(T_{2})\leq\wedge^{2}f$, it follows that $\zeta_{C\wedge
L_{0}}(T_{1})=0$, and $\zeta_{C\wedge L_{0}^{\perp}}(T_{2})\leq\wedge^{2}C$.
Moreover, we have that

$\displaystyle\mathcal{Q}^{\eta}_{1}(X,Y)$ $\displaystyle=\textrm{tr}(C\to
C\mathrel{\mathop{\mathchar 58\relax}}\theta\mapsto\zeta_{C\wedge
L_{0}^{\perp}}(X)d_{Y}\theta),$ $\displaystyle\mathcal{Q}^{\eta}_{2}(X,Y)$
$\displaystyle=\textrm{tr}(C\to C\mathrel{\mathop{\mathchar
58\relax}}\theta\mapsto\zeta_{C\wedge L_{0}}(X)d_{Y}\theta).$

Towards a geometric classification of Lie applicable surfaces with one family
of spherical curvature lines (see Theorem 6.3), our strategy for the rest of
this subsection will be to show that when $\eta$ is closed, we may gauge
$\eta$ so that $\zeta_{C\wedge L_{0}^{\perp}}$ yields a polarisation for $C$.
The other consequences of $\eta$ being closed then give rise to linear
conserved quantities for this polarisation.

Using (4.2), the closure of $\eta$ is given by

$d\eta=Ad_{A^{-1}}\cdot((d+Ad_{A}\mathcal{N}^{L})(\zeta_{C\wedge
L_{0}}+\zeta_{C\wedge L_{0}^{\perp}})).$

Thus $\eta$ is closed if and only if

$(d+Ad_{A}\mathcal{N}^{L})(\zeta_{C\wedge L_{0}}+\zeta_{C\wedge
L_{0}^{\perp}})=0.$

By splitting this into $C\wedge L_{0}$ and $C\wedge L_{0}^{\perp}$ parts, we
see that this is equivalent to

(6.1) $\displaystyle d\zeta_{C\wedge
L_{0}}+[Ad_{A}\mathcal{N}^{L}\wedge\zeta_{C\wedge L_{0}^{\perp}}]$
$\displaystyle=0$ (6.2) $\displaystyle d\zeta_{C\wedge
L_{0}^{\perp}}+[Ad_{A}\mathcal{N}^{L}\wedge\zeta_{C\wedge L_{0}}]$
$\displaystyle=0.$

Since $Ad_{A}\mathcal{N}^{L}(T_{1})=0$ and $\zeta_{C\wedge L_{0}}(T_{1})=0$,
(6.2) becomes

(6.3) $d\zeta_{C\wedge L_{0}^{\perp}}=0.$

Consider a gauge transformation

$\tilde{\eta}=\eta-d\tau=Ad_{A^{-1}}\cdot(\tilde{\zeta}_{C\wedge
L_{0}}+\tilde{\zeta}_{C\wedge L_{0}^{\perp}})$

of $\eta$. We may write $\tau=A^{-1}\cdot\rho$ for some
$\rho\in\Gamma(\wedge^{2}C)$. It is then straightforward to see that

$\tilde{\zeta}_{C\wedge L_{0}}=\zeta_{C\wedge
L_{0}}-[Ad_{A^{-1}}\mathcal{N}^{L},\rho]\quad\text{and}\quad\tilde{\zeta}_{C\wedge
L^{\perp}_{0}}=\zeta_{C\wedge L^{\perp}_{0}}-d\rho.$

Now, $\zeta_{C\wedge L_{0}^{\perp}}(T_{2})\leq\wedge^{2}C$, and since $C$
doesn’t depend on $I_{2}$, it follows that $d\rho(T_{2})\leq\wedge^{2}C$.
Thus, we may choose $\rho$ so that $\tilde{\zeta}_{C\wedge
L_{0}^{\perp}}(T_{2})=0$. From (6.3) and Lemma 6.1, we can then
deduce444$\tilde{\xi}(T_{2})=0$ implies that $\tilde{\xi}=g\,du$ for some
smooth function $g$. Equation (6.3) then implies that $g_{v}=0$, and thus $g$
is a function of $u$ only. that $[\tilde{\xi}]$ with
$\tilde{\xi}\mathrel{\mathop{\mathchar 58\relax}}=\tilde{\zeta}_{C\wedge
L_{0}^{\perp}}$ is a polarisation of $C$ with quadratic differential
$\mathcal{Q}^{\tilde{\xi}}=\mathcal{Q}^{\eta}_{1}$.

On the other hand, (6.1) can be reformulated as

$(d+t\zeta_{C\wedge L_{0}^{\perp}})(Ad_{A}\mathcal{N}^{L}+t\zeta_{C\wedge
L_{0}})=0.$

Therefore we arrive at the following result:

######  Proposition 6.2.

$\eta=Ad_{A^{-1}}\cdot(\zeta_{C\wedge L_{0}}+\zeta_{C\wedge L_{0}^{\perp}})$
is closed with non-zero quadratic differential $\mathcal{Q}^{\eta}$ if and
only if, after an appropriate gauge transformation, $[\xi]$ with
$\xi\mathrel{\mathop{\mathchar 58\relax}}=\zeta_{C\wedge L_{0}^{\perp}}$ is a
polarisation of $C$ and

(6.4) $(d+t\xi)(Ad_{A}\mathcal{N}^{L}+t\zeta_{C\wedge L_{0}})=0.$

Fix a unit length vector $l_{0}\in L_{0}$. Then

$(Ad_{A}\mathcal{N}^{L}+t\zeta_{C\wedge L_{0}})(\tfrac{\partial}{\partial
v})=l_{0}\wedge\nu(t)$

for some linear $\nu(t)\in(\Gamma L_{0}^{\perp})[t]$. Then (6.4) implies that

(6.5) $(d+t\xi)|_{T_{1}}\nu(t)=0.$

Hence, $\nu\in\Gamma W$, where $W$ is the space of linear conserved quantities
of $(d+t\xi)|_{T_{1}}$. By differentiating (6.5) repeatedly with respect to
$v$, we find that

$\nu,\nu_{v},\nu_{vv},...\in\Gamma W.$

We thus have that

$\nu(0),(\nu(0))_{v},(\nu(0))_{vv},...\in\Gamma W(0),$

with $\nu(0)=Ad_{A}\mathcal{N}^{L}(\tfrac{\partial}{\partial v})l_{0}$. Now

$\displaystyle L$ $\displaystyle=A^{-1}L_{0},$ $\displaystyle\langle
l,l_{v}\rangle$ $\displaystyle=A^{-1}\langle l_{0},\nu(0)\rangle,$
$\displaystyle\mathcal{H}_{1}=\langle l,l_{v},l_{vv}\rangle$
$\displaystyle=A^{-1}\langle l_{0},\nu(0),(\nu(0))_{v}\rangle,$

and so on. In particular, since $\mathcal{H}_{1}$ has rank $3$, we must have
that $\dim W(0)\geq 2$. Hence $\dim W\geq 2$. On the other hand, by
Proposition 3.17, since $C$ is non-circular, we have that $\dim W\leq 3$.
Therefore, we arrive at two possibilities:

  * •

$\dim W=2$: then $\mathcal{H}_{1}$ is constant, implying that $f$ has two
families of spherical curvature lines by Proposition 4.23.

  * •

$\dim W=3$: then $L$ lives in the constant $4$-dimensional space $L_{0}\oplus
W(0)$ and $C$ is constrained elastic curve with respect to $W(0)$ (see
Definition 3.16).

Conversely, suppose that $L$ lives in a constant subspace
$L_{0}\oplus_{\perp}W_{0}$, with $W_{0}$ either $2$\- or $3$-dimensional. We
assume that there exists a polarisation $[\xi]$ of $C$ such that $d+t\xi$
admits a space $W$ of linear conserved quantities with $W_{0}\leq W(0)$.
Depending on the dimension of $W_{0}$ we have two cases:

  * •

If $W_{0}$ is $2$-dimensional, then by Lemma 3.8 there always exists such a
polarisation $[\xi]$.

  * •

If $W_{0}$ is $3$-dimensional, then $C$ is a constrained elastic curve with
respect to $W_{0}$.

Now $Ad_{A}\mathcal{N}^{L}\in\Gamma T_{2}^{*}\otimes(L_{0}\wedge W_{0})$ and
we can find a $1$-form $\zeta_{C\wedge L_{0}}\in\Gamma
T_{2}^{*}\otimes(L_{0}\wedge C)$ such that

(6.6) $(d+t\xi)(Ad_{A}\mathcal{N}^{L}+t\zeta_{C\wedge L_{0}}).$

Define

$\eta=Ad_{A^{-1}}\cdot(\zeta_{C\wedge L_{0}}+\xi)\in\Omega^{1}(f\wedge
f^{\perp}).$

Since $\xi$ is a polarisation of $C$ and (6.6) holds, we have by Proposition
6.2 that $\eta$ is closed and $\mathcal{Q}^{\eta}\neq 0$. Hence, $f$ is a Lie
applicable surface. We have thus arrived at the following theorem:

######  Theorem 6.3.

Suppose that $f=A^{-1}C\mathrel{\mathop{\mathchar 58\relax}}I_{1}\times
I_{2}\to\mathcal{Z}$ is a regular umbilic-free Legendre map with one family of
spherical curvature lines, where $A$ is the spherical evolution map along $L$
based at $v_{0}\in I_{2}$ and $C$ is a Legendre curve in $L_{0}^{\perp}$. Then
$f$ is Lie applicable if and only if either the other family of curvature
lines are also spherical or $L$ lies in a constant $4$-dimensional space
$L_{0}\oplus_{\perp}W_{0}$ and $C$ is a constrained elastic curve with respect
to $W_{0}$.

Figure 6. An example of a Lie applicable torus with spherical curvature lines
(on the right) lying on spheres whose centres all belong to the same line. The
initial curve is in bicycle correspondence with a circle (on the left). Figure
7. An example of a Lie applicable torus with planar curvature lines (on the
right) lying on planes that are orthogonal to a fixed plane. The initial curve
is an elastic lemniscate (on the left).

The condition that $L$ lies in a constant $4$-dimensional space
$L_{0}\oplus_{\perp}W_{0}$ has different interpretations depending on the
signature of $W_{0}$.

If $W_{0}$ contains a unit timelike vector $\mathfrak{p}$, then
$S=\pi_{\mathfrak{p}}(L)$ lies in a constant $3$-dimensional subspace of
$\langle\mathfrak{p}\rangle^{\perp}$. Using [H2003, Section 1.8.5], one then
determines the nature of $S$ after projecting to an appropriate Euclidean
geometry:

  * •

if $W_{0}$ has signature $(1,2)$ then $S$ is a family of spheres whose centres
lie on a fixed line (see Figure 6);

  * •

if $W_{0}$ has signature $(2,1)$ then $S$ is a family of planes intersecting a
fixed point;

  * •

if $W_{0}$ has signature $(1,1,1)$ then $S$ is a family of planes that
intersect a fixed plane orthogonally (see Figure 7).

If $(L_{0}\oplus W_{0})^{\perp}$ contains a unit timelike vector
$\mathfrak{p}$ then Proposition 4.24 tells us that $S=\pi_{\mathfrak{p}}(L)$
intersects the point sphere map $f\cap\langle\mathfrak{p}\rangle^{\perp}$
orthogonally. After projecting to an appropriate Euclidean geometry we have
the following additional cases:

  * •

If $W_{0}$ has signature $(3,0)$, then $S$ belongs to a hyperbolic sphere
complex.

  * •

If $W_{0}$ has signature $(2,0,1)$, then we may choose a space form vector
$\mathfrak{q}\in(L_{0}\oplus W_{0})^{\perp}$. It then follows from Proposition
4.26 that $f$ projects to a Monge surface in $\mathfrak{Q}^{3}$.

The only other possibility that remains is that $W_{0}$ has signature
$(1,0,2)$. One then has that the sphere complexes defined by $L$ contain a
fixed contact element, namely, the null $2$-dimensional subspace contained in
$W_{0}$.

A straightforward implication of Theorem 6.3 is:

######  Theorem 6.4.

If a Lie applicable surface has exactly one family of spherical curvature
lines then all members of this family are Lie sphere transforms of a certain
constrained elastic curve.

###  6.2. Surfaces with two families of spherical curvature lines

As we saw in Theorem 6.3, surfaces with two families of spherical curvature
lines are Lie applicable. In this subsection we shall give an alternative
proof of this using a curved flat characterisation developed in [BP2020].

Suppose that $f$ is a regular umbilic-free Legendre map and suppose that both
families of curvature lines of $f$ are spherical. Then by Proposition 4.19 we
have that both osculating complexes $L_{i}$ are constant along $T_{i}$. Now we
can define a bundle of $(2,2)$-planes $W\mathrel{\mathop{\mathchar
58\relax}}=(L_{1}\oplus L_{2})^{\perp}$ with $f\leq W$. This yields a
splitting of the trivial bundle $\underline{\mathbb{R}}^{4,2}=W\oplus
W^{\perp}$ and a splitting of the trivial connection

$d=\mathcal{D}^{W}+\mathcal{N}^{W},$

where $\mathcal{D}^{W}$ is the sum of the induced connections on $W$ and
$W^{\perp}$ and $\mathcal{N}^{W}\in\Omega^{1}(W\wedge W^{\perp})$. Now fix
unit length sections $l_{i}\in\Gamma L_{i}$. Then

$\mathcal{N}^{W}=l_{1}\wedge dl_{1}+l_{2}\wedge dl_{2},$

and since $d|_{T_{i}}l_{i}=0$, we have that $d\mathcal{N}^{W}=0$. Hence, $W$
is a curved flat in $G_{2,2}(\mathbb{R}^{4,2})$, where
$G_{2,2}(\mathbb{R}^{4,2})$ denotes the Grassmannian of $(2,2)$ planes in
$\mathbb{R}^{4,2}$. It then follows from [BP2020, Corollary 3.8] that $f$ is
Lie applicable. To summarise:

######  Proposition 6.5.

Suppose that both families of curvature lines of a regular umbilic-free
Legendre map $f$ are spherical. Then $f$ is Lie applicable and $(L_{1}\oplus
L_{2})^{\perp}$ is a curved flat in $G_{2,2}(\mathbb{R}^{4,2})$.

From [BP2020, Theorem 3.9] we have that the $\mathcal{D}^{W}$-parallel null
rank 2 subbundles of $W$ form two $1$-parameter families of Lie applicable
Legendre maps $\\{f_{\alpha}\\}$,$\\{\hat{f}_{\beta}\\}$ parametrised by
$\alpha,\beta\in\mathbb{R}P^{1}$, such that for some $m\in\mathbb{R}^{\times}$
any pair $(f_{\alpha},\hat{f}_{\beta})$ is an $m$-Darboux pair. Without loss
of generality, assume that $f\in\\{f_{\alpha}\\}$. Note that members of the
same family are complementary, i.e.,

$f_{\alpha_{1}}\cap
f_{\alpha_{2}}=\\{0\\}=\hat{f}_{\beta_{1}}\cap\hat{f}_{\beta_{2}},$

for all $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\in\mathbb{R}P^{1}$. From
Theorem 4.3, we can deduce that all members of these families have two
families of spherical curvature lines.

Suppose that $f$ projects to a Joachimsthal surface in an appropriate
Euclidean geometry. Then from Subsection 4.4, we have that the osculating
bundles $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are constant $(2,1)$ spaces.
Thus $V_{i}\mathrel{\mathop{\mathchar 58\relax}}=\mathcal{H}_{i}\cap W$ are
$(1,1)$ subbundles of $W$ and we denote by $\hat{s}^{j}_{i}\leq V_{i}$ the
corresponding null line bundles, for $j\in\\{1,2\\}$. Since $\mathcal{H}_{i}$
is constant it follows that each $\hat{s}_{i}^{j}$ is
$\mathcal{D}^{W}$-parallel and, since $L_{i}$ is constant along $T_{i}$, it
follows that $\hat{s}_{i}^{j}$ is constant along $T_{i}$. Note that we cannot
have that any of $\hat{s}_{i}^{j}$ are constant since this would contradict
that $\mathcal{H}_{i}$ are rank $3$. By taking various combinations
$\hat{f}^{j_{1}j_{2}}=\hat{s}_{1}^{j_{1}}\oplus\hat{s}_{2}^{j_{2}}$ with
$j_{1},j_{2}\in\\{1,2\\}$, we obtain four Legendre maps contained in $W$ that
are $\mathcal{D}^{W}$-parallel, and since $\hat{s}_{i}^{j_{1}}$ is constant
along $T_{i}$, it follows that $\hat{f}^{j_{1}j_{2}}$ are Dupin cyclides. We
also have that the pair $\hat{f}^{11}$, $\hat{f}^{22}$ are complementary and
both intersect the complementary pair $\hat{f}^{12}$, $\hat{f}^{21}$. Without
loss of generality, assume that $\hat{f}^{11},\hat{f}^{22}\in\\{f_{\alpha}\\}$
and $\hat{f}^{12},\hat{f}^{21}\in\\{\hat{f}_{\beta}\\}$. Thus, $\hat{f}^{12}$
and $\hat{f}^{21}$ are Darboux transforms of $f$. We thus arrive at the
following theorem that gives a converse to Proposition 5.5:

######  Theorem 6.6.

Joachimsthal surfaces are Darboux transforms of two Dupin cyclides.

######  Remark 6.7.

Joachimsthal surfaces have one family of curvature lines that lie on spheres
whose centres lie on a fixed line. Geometrically the two Dupin cyclides in
Theorem 6.6 are the fixed line, parametrised as the intersection points of the
spheres with this fixed line.

######  Remark 6.8.

Remark 6.7 immediately allows one to deduce that cmc tori with planar
curvature lines, including the Wente torus, can locally be obtained as a
Darboux transform of the circular cylinder using the fact that the parallel
constant Gaussian curvature tori are Joachimsthal surfaces. We will defer a
more detailed discussion of this fact to a future work. For an illustration of
this, see Figure 8.

Figure 8. Wente torus obtained as a Darboux transform of the circular
cylinder, with a few of the corresponding curvature lines highlighted. The
circular cylinder was obtained using the spherical evolution map induced by
the spherical curvature lines of the Wente torus.

###  6.3. Calapso transforms

Suppose that $f$ is a regular umbilic-free Lie applicable Legendre map with
closed $1$-form $\eta$. The Lie cyclide splitting of the trivial bundle gives
rise to a splitting
$\wedge^{2}\underline{\mathbb{R}}^{4,2}=\mathfrak{h}\oplus\mathfrak{m}$, where

$\mathfrak{h}\mathrel{\mathop{\mathchar 58\relax}}=S_{1}\wedge S_{1}\oplus
S_{2}\wedge S_{2}\quad\text{and}\quad\mathfrak{m}\mathrel{\mathop{\mathchar
58\relax}}=S_{1}\wedge S_{2}.$

We may then write $\eta=\eta_{\mathfrak{h}}+\eta_{\mathfrak{m}}$ where
$\eta_{\mathfrak{h}}\in\Omega^{1}(\mathfrak{h})$ and
$\eta_{\mathfrak{m}}\in\Omega^{1}(\mathfrak{m})$. In [P2020] it is shown that
within the gauge orbit of $[\eta]$ there is a distinguished gauge potential
$\eta^{mid}$ called the middle potential with the property
$\eta^{mid}_{\mathfrak{m}}\in\Omega^{1}(\wedge^{2}f)$. Now suppose that
$f^{t}\mathrel{\mathop{\mathchar 58\relax}}=T(t)f$ is a Calapso transform of
$f$ where $T(t)$ satisfies

$T(t)\cdot(d+t\eta^{mid})=d.$

In [P2020, Lemma 4.6] it is shown that the Lie cyclides of $f^{t}$ are given
by

$S_{1}^{t}=T(t)S_{1}\quad\text{and}\quad S_{2}^{t}=T(t)S_{2}$

and the subsequent splitting of the trivial bundle
$d=\mathcal{D}^{t}+\mathcal{N}^{t}$ satisfies

$\mathcal{D}^{t}=T(t)\cdot(\mathcal{D}+t\eta^{mid}_{\mathfrak{h}})\quad\text{and}\quad\mathcal{N}^{t}=T(t)\cdot(\mathcal{N}+t\eta^{mid}_{\mathfrak{m}}).$

Since $\eta^{mid}_{\mathfrak{m}}\in\Omega^{1}(\wedge^{2}f)$ it follows that

$\mathcal{N}^{t}(T(t)L_{i})=T(t)(\mathcal{N}+t\eta^{mid}_{\mathfrak{m}})L_{i}=0.$

Thus $L_{i}^{t}\mathrel{\mathop{\mathchar 58\relax}}=T(t)L_{i}$ are the
osculating complexes of $f^{t}$. Moreover, since $\eta(T_{i})\leq f\wedge
f_{i}$ and $L_{1}\leq f_{2}$ and $L_{2}\leq f_{1}$, it follows that

$d_{X_{i}}l_{i}^{t}=T(t)(d_{X_{i}}+t\eta^{mid}(X_{i}))l_{i}=T(t)d_{X_{i}}l_{i},$

for sections $l_{i}\in\Gamma L_{i}$ and $X_{i}\in\Gamma T_{i}$. Applying
Proposition 4.19, we arrive at the following result:

######  Proposition 6.9.

A Lie applicable surface $f$ has spherical curvature lines along the leaves of
$T_{i}$ if and only if any Calapso transform $f^{t}$ has spherical curvature
lines along the leaves of $T_{i}$.

##  7\. Final remarks

In this paper, we have discussed how every surface with spherical curvature
lines is generated by a pair of _initial data_ : a Legendre curve and a curve
of suitable Lie sphere transformations. Such observation yields a
characterisation of Lie applicable surfaces with spherical curvature lines in
terms of constrained elastic curves. The generality of the theory developed in
this paper leads to additional future questions, some of which we list below:

  1. (1)

The Lie sphere geometric characterisation of constrained elastic curves
depends on suitable choices of point sphere complex $\mathfrak{p}$ and space
form vector $\mathfrak{q}$ in $W_{0}$, raising the question of finding the
characteristics of such curves when projected with respect to different
choices of $\mathfrak{p},\mathfrak{q}\in W_{0}$.

  2. (2)

A closer look at the figures in this paper reveals that many Lie applicable
surfaces with spherical curvature lines have admissible singularities such as
cuspidal edges or swallowtails. This is not so surprising since many of the
well-known Lie applicable surfaces such as pseudospherical surfaces and linear
Weingarten surfaces are known to admit certain types of singularities. The
theory developed here assumes that the contact lifts are immersed, and hence
are fronts, allowing one to contemplate the relationship between the initial
data of the Lie applicable surfaces with spherical curvature lines and
recognising the types of singularities appearing on them.

  3. (3)

A recent work [ogata_ribaucour_nodate] reveals that the contact lifts of
Ribaucour transforms of surface of revolutions can fail to immerse, and
develop frontal singularities. The Ribaucour transforms treated in this work
should also have spherical curvature lines; therefore, one could also consider
a similar problem of finding the conditions on the initial data for the
resulting surface to have frontal singularities.

  4. (4)

Within the class of Lie applicable surfaces, many of the well-known surface
classes can be determined via the existence of polynomial conserved quantities
[BHR2010, BHR2012, BHPR2019]. One could consider the additional conditions on
the initial data so that the resulting surface admits polynomial conserved
quantities, thereby obtaining a characterisation of the surfaces in the
various subclasses with spherical curvature lines.

  5. (5)

Many of the examples included in this paper are compact: A closed initial
spherical Legendre curve and a suitable periodic spherical evolution map will
result in a closed compact surface with spherical curvature lines. Therefore,
finding the conditions on the family of elliptic linear sphere complexes
$L\mathrel{\mathop{\mathchar 58\relax}}\tilde{I}\to\mathbb{R}^{4,2}$ so that
the corresponding spherical evolution map becomes periodic is key to obtaining
(Lie applicable) tori with spherical curvature lines.

## References