Transformations and singularities of polarized curves

We study the limiting behaviour of Darboux and Calapso transforms of polarized curves in the conformal n-dimensional sphere when the polarization has a pole of first or second order at some point. We prove that for a pole of first order, as the singularity is approached, all Darboux transforms converge to the original curve and all Calapso transforms converge. For a pole of second order, a generic Darboux transform converges to the original curve while a Calapso transform has a limit point or a limit circle, depending on the value of the transformation parameter. In particular, our results apply to Darboux and Calapso transforms of isothermic surfaces when a singular umbilic with index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2}$$\end{document}12 or 1 is approached along a curvature line.


Introduction
Transformations of surfaces play a central role in our present understanding of smooth and discrete differential geometry. Not only do they allow the construction of new surfaces of a given class from existing ones, but their existence reveals a great deal about the underlying (integrable) structure of the corresponding classes of surfaces, c.f. [5]. For example, given a pseudospherical surface (K = − 1), the Bäcklund transformation yields a two-parameter family of new pseudospherical surfaces by solving an integrable, first-order partial differential equation (see [11,Sect. 120]). In [22], these Bäcklund transformations are shown to be generators of an infinite-dimensional transformation group of the space of pseudospherical surfaces and the relations among these generators are given by a permutability theorem discovered by Bianchi [1,Sect. 257] in the nineteenth century. In particular, it is shown that pseudospherical surfaces fall into the class of integrable systems. Similar developments have been achieved for other classes of surfaces such as minimal surfaces, constant mean curvature B Andreas Fuchs afuchs@geometrie.tuwien.ac.at 1 Vienna, Austria surfaces, constant Gaussian curvature surfaces (c.f. [2]) and curved flats in symmetric spaces such as Darboux pairs of isothermic surfaces, c.f. [7,9,12].
In the discrete theory, the importance of transformations becomes even more apparent, as articulated in [3] (see also [4]): "In this setting, discrete surfaces appear as two-dimensional layers of multidimensional discrete nets, and their transformations correspond to shifts in the transversal lattice directions. A characteristic feature of the theory is that all lattice directions are on equal footing with respect to the defining geometric properties. Due to this symmetry, discrete surfaces and their transformations become indistinguishable." The interplay between aspects of discrete and smooth differential geometry was explored in [6] with the study of semi-discrete isothermic surfaces, introduced in [18]. In analogy to the transformation theory of smooth isothermic surfaces (see [15,Chap. 8.6], [5] or [8]), the authors develop a notion of Christoffel, Darboux and Calapso transformations of polarized curves, that is, smooth curves equipped with a nowhere zero quadratic differential, called a polarization. They then show that semi-discrete isothermic surfaces are sequences of Darboux transforms of polarized curves. In line with the ideas of discrete differential geometry, permutability theorems of the transformations of polarized curves are shown to yield a corresponding transformation theory for semi-discrete isothermic surfaces.
We are interested in the class of smooth isothermic surfaces, classically characterized by the local existence of conformal curvature line coordinates, away from umbilics. For these surfaces, the transformation theory is defined only locally, that is, on simply connected surface patches on which regular nets of conformal curvature line coordinates exist (c.f. [5,15,20]). A global transformation theory is still missing. In particular, it seems necessary to reconsider the definition of an isothermic surface. The classical definition makes no restriction whatsoever on umbilic points. But certain configurations of umbilics are an obstacle for a global definition of the transformations (see [15,Sect. 5.2.20]). One candidate for an alternative definition of an isothermic surface is to require the existence of a globally defined holomorphic quadratic differential whose trajectories (see [21,Sect. 5.5]) agree with the curvature lines of the surface on the complement of its umbilic set. But due to the Poincaré-Hopf theorem [16], this would exclude all surfaces homeomorphic to a sphere, for example the ellipsoid, c.f. [20]. If we merely require the existence of a meromorphic quadratic differential, that includes topological spheres. Moreover, according to the local Carathéodory conjecture, the poles of such a meromorphic quadratic differential are at most of order two (see [14]). This paper makes a first step towards a global transformation theory of isothermic surfaces for which an underlying meromorphic quadratic differential exists. A curvature line of an isothermic surface together with the restriction of such a meromorphic quadratic differential yields a polarized curve in the sense of [6,Sect. 2]. Moreover, the transformations for isothermic surfaces descend to the corresponding transformations of the polarized curvature lines. Thus, in order to understand how the transforms of an isothermic surface behave when a pole of the underlying meromorphic quadratic differential is approached along a curvature line, we investigate how Darboux and Calapso transforms of polarized curves with singular polarizations behave when the singularity is approached. In particular, we investigate the limiting behaviour of Darboux and Calapso transforms of polarized curves where the polarization has a pole of first or second order. According to the local Carathéodory conjecture, these are the only poles that can occur on a curvature line of an isothermic surface.
In Sect. 2, we give Möbius geometric definitions of the Darboux and Calapso transformations of polarized curves in the conformal n-dimensional sphere S n . Our definitions are formulated with a projective model of Möbius geometry: we identify S n with the projectivization P(L n+1 ) of the light cone L n+1 in (n + 2)-dimensional Minkowski space. The Möbius group can then be identified with the projective Lorentz group PO R n+2 1 . Curves are described as maps into the projective light cone. For computations and in order to prove that our definitions agree with those of [6], we relate the projective model to its linearization based on the action of O R n+2 1 on L n+1 , where curves are described via their light cone lifts. We then give a definition of a polarization with a pole of first or second order.
The central object in our definition of the transformations (Definition 1) is the primitives Γ p (λω) of a family (λω) λ∈R of 1-forms associated with a polarized curve. Here, the primitive of a 1-form ψ with values in the Lie algebra po R n+2 c.f. [19]. When the polarization has a pole of first or second order at some point, the 1-forms λω associated with the polarized curve also have a pole at that point and hence the primitives are not defined there. Nevertheless, as we prove in Corollary 2 and Proposition 2 in Sect. 3, the primitives of a certain class of po R n+2 In Sect. 5, we consider the case of a polarization with a pole of second order. We cannot apply the results of Sect. 3 here directly because the 1-forms λω associated with the polarized curve have poles of second order. We first have to do a singular gauge transformation to transform the 1-forms with poles of second order to 1-forms with a pole of first order. The behaviour of the Darboux and Calapso transforms in this case is more diverse. A generic Darboux transform still converges to the original curve as the singularity is approached, but there are Darboux transforms which do not converge. Moreover, the Calapso transform has a limit point or a limit circle, depending on the value of the transformation parameter λ.

Darboux and Calapso transforms of polarized curves
In this section, we define Darboux and Calapso transforms of polarized curves, show that our definitions agree with those of [6] and specify the goal of this paper: the study of the limiting behaviour of Darboux and Calapso transforms at points where the polarization has a pole of first or second order.
We use the projective model of Möbius geometry (c.f. at the identity. The differential of · at the identity then restricts to an isomorphism of Lie algebras where we further identify Λ 2 R n+2 Here, ⟪·, ·⟫ denotes the Minkowski inner product on R n+2 1 . A regular curve c in S n = P(L n+1 ) is an immersion of some interval (a, b) into P(L n+1 ), where the notation c indicates that such a map may also be described by pointwise taking the linear span of a light cone lift c : (a, b) → L n+1 of c . For simplicity, we always assume c to be smooth. A polarized curve ( c , Q) in S n is a regular curve c in S n together with a nowhere zero quadratic differential Q on (a, b). To a polarized curve ( c , Q), we associate the po R n+2 1 -valued 1-form ω given by where c is any light cone lift of c and the function Q is related to Q via that lift c by The 1-form ω is independent of the choice of lift c of c . Let G be a Lie group with Lie algebra g and ψ a g-valued 1-form on (a, b). Then, for any p ∈ (a, b), denote by From the defining properties (3) of Γ p (ψ) and its uniqueness, it follows readily that Therefore, the map Γ p (ψ) : (a, b) t → Γ p t (ψ) ∈ G is the composition of Γ p (ψ) with taking the inverse in G and thus satisfies Under a gauge transformation ψ → g ψ := g −1 ψ g + g −1 d g using a smooth map g : (a, b) → G, the primitives Γ p (ψ) transform as Definition 1 Let ( c , Q) be a polarized curve in S n with associated 1-form ω given by (2). For λ ∈ R\{0}, p ∈ (a, b) and ĉ p ∈ S n , the curve For λ ∈ R and p ∈ (a, b), the curve c λ, p := Γ p (λω) c is called the λ-Calapso transform of ( c , Q) normalized at p.
Any λ-Calapso transform of a polarized curve ( c , Q) is an immersion. Similarly, any λ-Darboux transform with initial point ĉ p not lying on the image of c λ, p immerses. 1 One may turn the immersed transforms of ( c , Q) into polarized curves themselves by equipping them with some polarization, for example again Q. This is useful in the study of repeated transforms of polarized curves, as it is done in [6]. However, in this work we do not study repeated transforms and thus, for simplicity, we consider the transforms merely as curves and not as polarized curves. We remark that due to (4), two λ-Calapso transforms normalized at p and q, respectively, differ by a Möbius transformation. Moreover, the Darboux and Calapso transformations of a polarized curve ( c , Q) in S n are invariant under Möbius transformations of S n and reparametrizations of c .
This paper is devoted to the study of the limiting behaviour of the Darboux and Calapso transforms at a in the case that c can be extended regularly to some (a − , b), but the polarization has a pole of first or second order at a in the sense of If c has a smooth extension to (a − , b), then the polarization Q has a pole of first or second order at a if and only if the 1-form ω associated with ( c , Q) has a pole of first or second order at a.
The above definition of a pole is invariant under those diffeomorphisms of (a, b) to (ã,b) which extend to diffeomorphisms of some larger interval Together with the reparametrization invariance of the Darboux and Calapso transformation, we can thus make the convenient choice a = 0.
Our definitions of the transformations of a polarized curve are formulated entirely in the projective model of Möbius geometry, where PO R n+2 Thus, the maps Γ p (Ψ ) are exactly the primitives of ψ. More generally, for any real, nowhere vanishing function h on (0, b), we have In particular, this identity allows to compute the primitives of a po R n+2 1 -valued 1-form via arbitrary nowhere zero rescalings of the primitives of its orthogonal lift.
Relation (6) also enables us to show that our Definition 1 agrees with the corresponding ones in [6]. Namely, let ( c , Q) be a polarized curve with associated 1-form ω and orthogonal lift Ω. Using (6), it follows that withĉ p ∈ ĉ p the liftĉ = Γ p (λΩ)ĉ p of a λ-Darboux transform ĉ of ( c , Q) as defined in Definition 1 is a parallel section of the connection and hence yields a − λ 2 -Darboux transform 2 of ( c , Q) in the sense of [6, Definition 2.5]. Conversely, any parallel sectionĉ of D λ d t yields a λ-Darboux transform ĉ according to our Definition 1.
Not surprisingly, the main advantage of the linearization of the projective formalism is its linear structure. For example, integration of (3) shows that the primitives of an o R n+2 Its disadvantage, on the other hand, is its reliance on the choice of lifts. As we will see in Sect. 3, when a po R n+2 1 -valued 1-form ψ and thus also its orthogonal lift Ψ on (0, b) have a pole, it may happen that the O + R n+2 1 -valued primitives Γ p (Ψ ) tend towards infinity as the pole is approached, while the corresponding PO R n+2 If this is the case for the 1-form ω associated with a polarized curve and its orthogonal lift Ω, then the liftsĉ = Γ p (λΩ)ĉ p and c λ, p = Γ p (λΩ)c diverge as the singularity is approached, while the corresponding maps ĉ and c λ, p converge. In such a situation, a lift-independent formalism is clearly beneficial.

Main tools
In this section, we study the limiting behaviour at 0 of the primitives Γ p (ψ) of a certain class of po R n+2 1 -valued 1-forms ψ on (0, b) with a pole of first order at 0. In Sect. 3.1, we restrict to 1-forms of a particularly simple form, which we call pure pole forms. For those, the primitives and their limiting behaviour at 0 can be computed explicitly. In Sect. 3.2, we then relate the limiting behaviour of primitives of the more general pole forms to that of the pure pole forms. In view of our definition of the transformations of a polarized curve via the primitives of multiples λω of the associated 1-form ω (Definition 1), these tools will provide the means to investigate the limiting behaviour of the transforms of a singular polarized curve in Sects. 4 and 5.

Primitives of pure pole forms
We call a po(R n+2 Pure pole forms come in three types corresponding to the three signatures which the linear span v, w ⊂ R n+2 1 can have. Accordingly, we speak of Minkowski, degenerate and spacelike pure pole forms. In addition to the quadratic form · 2 induced by the indefinite Minkowski inner product ⟪·, ·⟫, we introduce the standard, Euclidean, positive-definite norm |·| on R n+2 and R (n+2) 2 .
On End R n+2 1 R (n+2) 2 , the norm |·| is submultiplicative and satisfies |A * | = |A| for Proposition 1 Let ξ be a pure pole form on (0, b) and Ξ its orthogonal lift. Then, there is a constant B ∈ R such that where but Γ p (ξ ) does not converge as 0 is approached as it is a rotation in the plane v, w with speed increasing towards infinity; Then, where Proof First we note that Γ t p (Ξ ) is given by the exponential 1. For for all s ∈ R and k ∈ N, the exponential computes to Since ζ > 0, taking the limit t → 0 yields (9). Since |t/ p| ≤ 1 for all p ∈ (0, b) and t ∈ (0, p], there is a constant B ∈ R such that (8) is bounded for all p, t ∈ (0, b) such that (10) holds for some B ∈ R. To see that Γ p (ξ ) does not have a limit at 0, denote by iζ a nonzero eigenvalue of v ∧ w. Under a Lie group isomorphism O(R 2 ) → R mod 2π, the primitive Γ t p (Ξ ) gets mapped to ± ln t p |ζ | mod 2π which does not have a limit for any p ∈ (0, b) as t tends to 0. Hence, also Γ t p (Ξ ) does not have a limit for any p ∈ (0, b) as t tends to 0.
The one-parameter families (Γ t p (ξ )) t∈(0,b) have simple Euclidean interpretations: For Minkowski ξ , stereographically project S n \{ v + } to R n . Then, (Γ t p (ξ )) t∈(0,b) acts as a family of similarities with centre v − of R n and a t-dependent scale factor that tends towards infinity as t → 0. The limiting map v + v * − maps all points to the point at infinity, v + , of Euclidean rotations. As t approaches 0, the rotation speed increases towards infinity.
For degenerate ξ and v 0 ∈ v, w ∩L n+1 , stereographically project S n \{ v 0 } to R n . Then, but not compact. The limits (9) and (12) can thus not be computed in O + R n+2 1 , which demonstrates the advantage of the projective model.

Primitives of pole forms
In this section, we define pole forms and relate the limiting behaviour of their primitives at 0 to that of pure pole forms. For a pole form ψ, the pure pole form ξ for which ψ − ξ is bounded is clearly unique. Therefore, we may define Definition 4 Let ψ be a pole form on (0, b) with ψ − ξ bounded for the pure pole form ξ . Then, ψ is called Minkowski, spacelike or degenerate according to whether ξ has that property. For Minkowski ψ, denote by ζ the positive eigenvalue of v ∧ w. If ψ is Minkowski with ζ < 1, spacelike or degenerate, we say that ψ is of the first kind. Otherwise, it is of the second kind.
We remark that these properties of a pole form are invariant under those diffeomorphisms of (0, b) to (ã,b) which extend to diffeomorphisms of (− , b) to (ã −˜ ,b) for some ,˜ > 0 (c.f. [13]). Different techniques are necessary to investigate the limiting behaviour of primitives of pole forms of the first and of the second kind. We start with those of the first kind. (0, b), and let χ be a bounded, continuous map from (0, b) to po R n+2 1 . Then, the family Γ p (ξ )χΓ p (ξ ) p∈(0,b) of maps

Lemma 1 Let ξ be a pure pole form of the first kind on
where (B p ) p∈(0,b) is a family of integrable functions on (0, b) that satisfies Moreover, if ξ is degenerate or spacelike, then Proof Let Ξ be the orthogonal lift of ξ . If ξ is degenerate, it follows from the submultiplicativity of |·|, Proposition 1 and the assumed boundedness of χ that there is a C ∈ R such that where ( p, t) is as in (13). Now set B p (t) = C ( p, t) 2 . For all i ∈ N, by l'Hôpital's rule (ln x) i x converges to zero as x → 0 such that (17) holds. Moreover, for any i ∈ N, the integral of (ln x) i is a linear combination of terms of the form x(ln x) k with 0 ≤ k ≤ i. Again, since all terms x(ln x) k converge to zero as x → 0, it follows that for all p ∈ (0, b) the function B p is integrable and that (18) holds. If ξ is spacelike, again by Proposition 1, Γ p (Ξ ) is bounded on (0, b). Therefore, Γ p (ξ )χΓ p (ξ ) is bounded on (0, b). Thus, we can choose B p to be the same constant for all p ∈ (0, b). Clearly, a constant B p is integrable and satisfies (17) and (18).
Let finally ξ be Minkowski with positive eigenvalue ζ < 1. Denote by v ± the eigenvectors of v ∧ w with eigenvalues ±ζ . According to the decomposition Thus, there is a constant C such that By assumption ζ < 1. Therefore, for all p ∈ (0, b), the function B p is integrable over (0, b) and the product t B p (t) converges to zero as t tends towards b.
We remark that the bounding functions B p that we defined in (19) for the Minkowski case do not satisfy (18). This lemma can now be used to prove that a primitive of a pole form of the first kind factorizes into a continuous map from [0, b) to PO R n+2 1 and the primitive of a pure pole form, whose limiting behaviour at 0 we know from Proposition 1. ∈ (0, b), define the gauge-transformed 1-form

Corollary 1 Let ψ be a pole form of the first kind so that ψ − ξ is bounded for the pure pole form ξ . For all p
Then, the primitives Γ p (ξ p ψ) have limits in PO R n+2 1 at 0 and Moreover, if ξ is spacelike or degenerate, Proof Since ξ takes values in a fixed one-dimensional subalgebra, the orthogonal lift Ξ p Ψ of ξ p ψ can be written as  (6), Γ p (ξ p ψ) has a limit in PO R n+2 1 at 0. The relation (21) follows from (5) because ξ p ψ is a gauge transform of ψ by the map Γ p (ξ ). The limit (22) follows from where the first inequality can be derived from the integral Eq. (7) and in the last step we used (18).
From the factorization (21) and Proposition 1, we conclude that the primitives of spacelike pole forms do not have limits at the singularity, while those of degenerate and Minkowski pole forms of the first kind do have limits. The following corollary deals with these limits for the degenerate case. In Lemma 2 and Proposition 2, we then derive an analogous result for Minkowski pole forms of the first and second kind.
and is such that such that a = A . The adjoint A * with respect to the Minkowski inner product yields a well-defined element a * := A * ∈ PEnd R n+2 1 , which is independent of the chosen A ∈ a. If a ∈ PO R n+2 1 , then a * = a −1 .
Lemma 1 and Corollary 1 do not hold for pole forms of the second kind. In particular, the gauge transforms (20) are in general not integrable and although the factorization (21) also exists in that case, it seems to be of little use because we do not know whether the first factor, Γ p (ξ p ψ), has a limit at 0. Thus, we pursue a different strategy. The following lemma and proposition are based on ideas from [17].  ∈ (0, b) such that Proof The equality follows from |A −1 | = |A * | = |A| for A ∈ O R n+2 1 . We now prove the inequality. Let Ξ and Ψ be the orthogonal lifts of ξ and ψ. For p, t ∈ (0, b), one verifies by differentiation 3 (28) By assumption, ψ − ξ and hence Ψ − Ξ are bounded by some D ∈ R w.r.t. |·|. By (8), We now prove by contradiction that there is ab ∈ (0, b) such that Namely, chooseb ∈ (0, b) sufficiently small such that Suppose there existed a p ∈ (0,b) and a t ∈ (0, p] such that t p ζ Γ t p (Ψ ) = 2B. Denote byt the smallest of those t. Then, evaluating (29) att and using (31) yield 2B < 1 2 2B + B, a contradiction. Hence, indeed (30) holds.
We can now prove the analogue of Corollary 2 for Minkowski pole forms. 3 Or by using the integral Eq. (7) for the 1-form ξ p Ψ and the identity Γ p (ξ p Ψ ) = Γ p (Ψ )Γ p (Ξ ). ψ on (0, b) with ψ − ξ bounded for the pure pole form ξ . Denote by v ± eigenvectors of v ∧ w with eigenvalues ±ζ , ζ > 0. Then, there is a continuous map k :

Proposition 2 Let Ψ be the orthogonal lift of a Minkowski pole form
and is such that In particular, Proof First, we note that (34) follows from (33) and one limit in (33) follows from the other because (Γ t p (Ψ )) * = Γ p t (Ψ ) and taking the adjoint is continuous. We prove the first limit in (33).
Let Ψ and Ξ be the orthogonal lifts of ψ and ξ , respectively. Consider again the identity (28). For t ∈ (0, p), it can be written in the form where In order to prove (33), we wish to take the limit t → 0 of (35). To that end, we convince ourselves that we may evaluate the limit t → 0 under the integral in (35) by applying the dominated convergence theorem. Let firstb be as in Lemma 2 and fix p ∈ (0,b). Let (t i ) i∈N be any sequence in (0, p] converging to 0 and consider the sequence (τ → M(t i , τ, p)) i∈N of functions from (0, p] to R (n+2) 2 . Due to (9), this sequence of functions converges pointwise. Moreover, due to Lemma 2, the assumed boundedness of Ψ − Ξ and (8), the sequence of functions is uniformly bounded by some constant. Thus, the conditions of the dominated convergence theorem are satisfied and we conclude where we used (9). Again due to Lemma 2 and the assumed boundedness of Ψ − Ξ , the integrand in k is bounded uniformly in p and so the integral in k exists for all p ∈ (0,b) and lim p→0 k( p) = v + /⟪v + , v − ⟫. The first identity in (32) also follows from (4) and (36).
The map k has to take values in the light cone because for all p This proves the proposition for all p ∈ (0,b). But using (4), it can easily be seen to hold for all p ∈ (0, b).

Pole of first order
In this section, we investigate the limiting behaviour at 0 of the Darboux and Calapso transforms of a polarized curve ( c , Q) on (0, b), where c has a smooth extension to (− , b) for some > 0 and Q has a pole of first order at 0. For convenience, we further assume that c and Q have smooth extensions to some (0, b +˜ ),˜ > 0. We assume that Q t = Q(t) d t 2 satisfies Q(t) < 0. This is convenient and no restriction of our results because replacing Q by −Q has the same effect as replacing λ by −λ and there is no restriction on the range or sign of λ. With this assumption, we can choose a smooth lift c of c such that Q(t) = Q(t) c (t) 2 = − 1 t and the 1-form associated with ( c , Q) reads Now define the degenerate pure pole form Then, λω − λξ has a finite limit at 0 because and c is assumed to be smoothly extendible to (− , b). Thus, λω − λξ is bounded on (0, b) and λω is a degenerate pole form. In particular, it is a pole form of the first kind. With the help of Corollary 1, we can now show that every Calapso transform of ( c , Q) converges. (0, b) such that c has a regular extension to (− , b) and Q has a pole of first order at 0. Then, for all p ∈ (0, b) and λ ∈ R, the λ-Calapso transform c λ, p normalized at p has a limit at 0.

Theorem 1 Let ( c , Q) be a polarized curve on
Proof If λ = 0, then c λ, p = c and the statement is trivial. So let λ = 0. Factorize Γ p (λω) as in (21) to get Since λω is of the first kind, from Corollary 1 we know that Γ p (λξ p λω) has a limit in PO R n+2 1 at 0. To see that Γ p (λξ ) c has a limit at 0, write Proof From Corollary 2, we know that for every λ = 0, there is a continuous map k λ : and k λ satisfies (24). Therefore, if ĉ p = k λ ( p) , then If on the other hand ĉ p = k λ ( p) , then we use the properties (24) of k λ to find

Pole of second order
We now come to the more intricate and diverse case of a polarized curve ( c , Q) on (0, b), where c is smoothly extendible to (− , b) for some > 0 and Q has a pole of second order at 0. In this case, also λω has a pole of second order at 0 and we cannot apply the results of Sect. 3 directly. Instead, we use a gauge transformation λω → g λω = g −1 λω g + g −1 d g as in (5) to write As we will see in Sect. 5.1, it is possible to choose g such that g λω is a pole form and, in particular, has only a pole of first order. This is achieved by a gauge transformation which is singular in the sense that the transforming map g : (0, b) → PO(R n+2 1 ) converges to a map of the kind vw * ∈ PEnd R n+2 1 with v, w ∈ L n+1 as 0 is approached. In Sects. 5.2 and 5.3, we will then apply the results of Sect. 3 to the pole form g λω to investigate the behaviour of the Darboux and Calapso transforms of ( c , Q) at the singular point 0.
Again, for convenience, we assume that c and Q have smooth extensions to some (0, b +˜ ),˜ > 0.

The singular gauge transformation
We assume that the polarization Q is of the Again, this is no restriction of our results: We can assume Q(t) to be positive because replacing Q by −Q has the same effect as replacing λ by −λ, and we make no restriction on the range of λ. Furthermore, the Darboux and Calapso transformations are invariant under reparametrizations of c . Since Q has a pole of second order (see Definition 2), there certainly is a parameter t for c such that Q t = d t 2 t 2 and c remains smoothly extendible to (− , b). For the gauge transformation, we use a product g = F R of a frame F : (0, b) → O(R n+2 1 ), smoothly extendible to (− , b), and a particularly simple singular factor R : . We first construct the frame F. To this end, let c be the flat lift of c , for which c 2 = 1. Then, with the above assumptions, Q(t) = 1 t 2 . Let N 1 , . . . , N n−1 be parallel, orthonormal unit normal fields of c that satisfy 4 N 1 (t), . . . , N n−1 (t)}. The Maurer-Cartan form of F is then of the form Next, define the singular factor R : Then, R has Maurer-Cartan form The product g := F R is a singular frame for the singular lifts t −1 c(t) and tc(t) of c(t) and c(t) , respectively. It satisfies Using g for a gauge transformation (5) of λω, the gauge-transformed 1-form g λω reads Indeed, g λω has only a pole of first order and the pure pole form ξ λ given by (0, b). Thus, g λω is a pole form. The next Lemma, which summarizes the algebraic properties of ξ λ , can be verified by direct computation.
such that ξ λ can be written as If 1−2λ > 0, then v, w is Minkowski and v ± are eigenvectors of v∧w with real eigenvalues ± √ 1 − 2λ. If 1 − 2λ = 0, then v, w is degenerate and v + = −v − ∈ v, w is null. If 1 − 2λ < 0, then v, w is spacelike and v ± are complex conjugate eigenvectors of v ∧ w with imaginary eigenvalues ± √ 1 − 2λ ∈ iR. In particular, for λ > 0, the pole form λω is of the first kind. For λ ≤ 0, it is of the second kind.
By Proposition 2, Corollary 1 and Proposition 1, for 1 − 2λ ≥ 0, the primitives Γ t p (g λω) converge as t → 0 while for 1 − 2λ < 0 they do not have a limit at 0. We treat these cases separately in the next two sections.
By (39), g(t) has a limit at 0. Since 1 − 2λ ≥ 0, by Lemma 3 the pole form g λω is Minkowski or degenerate such that its primitives also have limits at 0. Very little work is required to prove convergence of the Darboux and Calapso transforms at 0 in the following two theorems using the results of Sect. 3.

The behaviour at the singularity for 1 − 2 < 0
In this case, ξ λ takes values in an algebra of infinitesimal Euclidean rotations. Thus, by Proposition 1, the primitives of ξ λ do not have a limit at 0 and hence by Corollary 2 neither do the primitives of g λω converge. From this, it follows that the λ-Calapso transforms of c do not converge. the result that also c(0) is a limit point of ĉ , this implies that ϕ converges to plus or minus infinity such that indeed the frequency of the rotation of ĉ on the curvature circle tends towards infinity (Fig. 3).