Abstract
We study a generalization of constant Gauss curvature \(-1\) surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyse the singularities of these surfaces, dividing them into those of characteristic and non-characteristic type. We give methods for constructing all non-degenerate singularities of both types, as well as many degenerate singularities. We also give a method for solving the singular geometric Cauchy problem: construct a pseudospherical frontal containing a given regular space curve as a non-degenerate singular curve. The solution is unique for most curves, but for some curves there are infinitely many solutions, and this is encoded in the curvature and torsion of the curve.
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1 Introduction
It is a well-known theorem of Hilbert that there do not exist complete isometric immersions in \({{\mathbb {R}}}^3\) of surfaces with constant negative Gauss curvature \(K=-1\). These surfaces have nevertheless been much studied since classical times. The integrability condition is the sine-Gordon equation \(\phi _{xy}= \sin \phi \), where x and y are unit speed asymptotic coordinates and \(\phi \) is the angle between the asymptotic directions. Most of the literature on these surfaces deals with them via the solutions of this equation, naturally leading to singularities along the curves \(\phi =n \pi \) for integers n. A more general approach for pseudospherical surfaces is the formulation in terms of Lorentz harmonic maps. The Gauss map N of a pseudospherical surface is harmonic with respect to the Lorentzian metric induced by the second fundamental form. Conversely, if we restrict to weakly regular harmonic maps, i.e. those where the derivatives \(N_x\) and \(N_y\) with respect to a null coordinate system never vanish, then these maps correspond to solutions of the sine-Gordon equation. The associated surfaces are called weakly regular pseudospherical surfaces, and this has been the standard class of pseudospherical surfaces investigated in the literature.
In this article, we aim to study the natural singularities of pseudospherical surfaces. We will drop the weak regularity assumption, as it serves only to make a connection with the sine-Gordon equation. This connection is not needed in the harmonic map approach. Given a harmonic map \(N: S \rightarrow {{\mathbb {S}}}^2\), from a simply connected Lorentz surface, there is a canonically associated map \(f: S \rightarrow {{\mathbb {R}}}^3\), unique up to a translation, that is pseudospherical wherever it is immersed, and such that \(\text {d}f\) is orthogonal to N (see Sect. 2.2). We take such maps f as the definition of a generalized pseudospherical surface.
Abandoning the identification with solutions of the sine-Gordon equation is advantageous for two reasons. In the first place, in order to solve the Cauchy problem along an arbitrary non-characteristic curve, it is necessary to choose asymptotic coordinates (x, y) such that the curve is given by \(y=\pm x\). This can always be achieved, at least locally, but not if we require that the coordinate lines are constant speed, the choice for which the angle between the coordinate curves is a solution of sine-Gordon. The second advantage is that we are interested in the natural singularities of these surfaces, and for many of these (for example the bifurcating cusp lines in Fig. 1, or the rank zero singularities in Fig. 10), there is no corresponding local solution of the sine-Gordon equation.
We will use a variant of the generalized d’Alembert method given by Toda [19] to study the surfaces. In brief, a loop group lift \(\hat{F}\) of a harmonic map is obtained, via integration and a loop group decomposition, from a potential pair \((\hat{\chi }, \hat{\psi })\) of loop algebra valued 1-forms along a pair of transverse null coordinate lines. Essentially, the solution is thus given by more or less arbitrary functions of one variable along two characteristic lines, in analogue with the d’Alembert solution of the wave equation. The challenge is to find the potentials that correspond to particular geometric properties, as the geometry is difficult to see in the potentials. To address this problem, in joint work with Svensson [8], we defined special potentials that allow one to solve a geometric Cauchy problem: find a surface that contains a given curve with prescribed surface normal. Here, we generalize these potentials to the case where the curve is required to be a singular curve, in place of prescribing the surface normal.
1.1 Main results
A frontal is a differentiable map f from a surface M into \({{\mathbb {R}}}^3\) that locally has a well-defined unit normal, that is a map N into \({{\mathbb {S}}}^2 \subset {{\mathbb {R}}}^3\) such that \(\text {d}f\) is orthogonal to N. Generalized pseudospherical surfaces, as defined here, are frontals, and we may thus call them pseudospherical frontals. If the map \((f,N): M\rightarrow {{\mathbb {R}}}^3 \times {{\mathbb {S}}}^2\) is everywhere regular, then f is called a (wave) front. A pseudospherical frontal is a wave front if and only if it is weakly regular. That is, wave front solutions are exactly those that correspond to solutions of the sine-Gordon equation.
A point p on a frontal f is called a singular point if the derivative \(\text {d}f\) has rank less than 2 at p, and the local singular locus is called a singular curve. The singular point p is non-degenerate if the singular curve is locally a regular curve in M. The image in \({{\mathbb {R}}}^3\) of a non-degenerate singular curve need not be a regular curve, demonstrated by the case of a swallowtail singularity or a cone singularity (Fig. 3). Below we will divide non-degenerate singular curves into two types, characteristic singular curves that are always tangent to a null coordinate direction and non-characteristic those that are never tangent to a null direction.
Theorem 4.2 gives the potentials for constructing all non-degenerate non-characteristic singular curves, together with the conditions on the data for cuspidal edges, swallowtails and cone singularities. We then use this to prove Theorem 4.3, which states that given an arbitrary space curve with non-vanishing curvature \(\kappa \), and torsion \(\tau \ne \pm 1\), there is a unique pseudospherical wave front that contains this curve as a cuspidal edge. Moreover, the potentials are given by a very simple formula in terms of \(\kappa \) and \(\tau \). We use this formula to compute several examples. In fact, the potentials in Theorem 4.2 generate a pseudospherical frontal from an arbitrary pair of functions \(\kappa \) and \(\tau \). At a point where \(\kappa \) vanishes, the singular curve is degenerate. At a point where \(|\tau |=1\), the surface is a frontal but not a wave front, and the singular curve is also degenerate. Examples are shown in Figs. 1, 7 and 11.
In Sect. 5, we analyse the problem for characteristic singular curves. These singularities are non-generic, but nevertheless of some interest. For example, a weakly regular pseudospherical surface (i.e. a wave front) contains a non-degenerate characteristic singular curve if and only if this curve is a straight line segment. For a general frontal, the singular curve, if it is not a straight line, must instead have non-vanishing curvature and constant torsion \(\tau = \pm 1\), incidentally the same conditions that are satisfied by asymptotic curves on a regular pseudospherical surface. In the characteristic case, the solution is not unique, and there are infinitely many pseudospherical frontals containing a prescribed curve of the allowed type. We give the precise statement and the potentials for all solutions in Theorem 5.1.
We have computed many examples of solutions using a numerical implementation of the generalized d’Alembert method.Footnote 1 We have tried to include some representative images throughout the article, as well as further examples illustrating degenerate singularities and surfaces generated from curves with unbounded curvature in Sect. 6. The surfaces are coloured here according to mean curvature, which generally blows up near singularities, showing the singular curves more clearly (Fig. 2).
1.2 Concluding remarks
This work is part of a series investigating how to analyse the singularities arising naturally in integrable systems formulations of geometric problems [4, 5, 7, 9]. The singularities in each case studied arise in a different way. For spacelike and timelike constant mean curvature surfaces in Minkowski 3-space [4, 7, 9], singularities are caused by the break down of the Iwasawa and Birkhoff loop group decompositions for non-compact groups. Approaching such points, the direction of the surface normal becomes null, and so the (harmonic) unit normal is not defined. For constant Gauss curvature surfaces in \({{\mathbb {R}}}^3\), the loop group decompositions are globally defined, and hence, the unit normal is well defined everywhere, but this does not guarantee that the surface is regular. This is because the unit normal is harmonic with respect to the metric induced by the second fundamental form, so the existence of conformal coordinates with respect to this metric does not imply surface regularity. Positive curvature surfaces, studied in [5], differ substantially in treatment from negative curvature surfaces, because the former constitute an elliptic problem corresponding to Riemannian harmonic maps, and the latter case, treated here, is hyperbolic and corresponds to Lorentzian harmonic maps.
We generally consider maps to be in the smooth category. The methods we use involve only integration and loop group decompositions, which preserve smoothness: if real analytic data are given, then the solutions are also real analytic. Our solutions, as frontals, are defined globally, because the Birkhoff decomposition used is shown in [3] to be global. We work with a simply connected (which implies contractible) Lorentz surface S. For non-trivial topologies, this amounts to working on the universal cover. Note, however, that by Kulkarni’s theorem [12], there are infinitely many Lorentzian conformal structures on the plane, and not all of these can be realized as conformal submanifolds of the Lorentz plane \({{\mathbb {R}}}^{1,1}\). This raises interesting questions for the global theory of pseudospherical frontals.
Andrey Popov [16] proved the existence and uniqueness part of our Theorem 4.3, by using the sine-Gordon equation. The potentials given in Theorem 4.3 improve this result by including solutions for curves where \(\kappa \) vanishes or \(|\tau |\) takes the value 1 and by providing a means of easily computing the solutions. Popov concluded that a pseudospherical surface is uniquely determined by a cuspidal edge on its boundary, but this is not strictly accurate: even if we restrict to the class of pseudospherical wave fronts (as he did), there exist cuspidal edges (necessarily straight lines) that are characteristic curves. For such a curve, there are infinitely many different pseudospherical wave fronts that contain it as a cuspidal edge.
An important motivation for studying the singularities of pseudospherical surfaces is to characterize the natural boundaries of the regular surfaces, given that there are no complete immersions. See, e.g. [1, 21]. Generalizations that include the singular curves as a part of the surface have previously been studied within the framework of weakly regular surfaces. In this article, we construct real analytic pseudospherical frontals (examples 5.4 and 5.5 ) that are immersed on open dense sets, but have non-degenerate singular curves where the surface is not weakly regular. This demonstrates that the weakly regular framework is not sufficiently general for the task of including even regular boundary curves of immersed pseudospherical surfaces. Given this, and the direct relationship between arbitrary Lorentz harmonic maps and globally defined pseudospherical frontals, we conclude that frontals are a more natural candidate for a global theory of pseudospherical surfaces.
2 Generalized pseudospherical surfaces
We first summarize necessary background material on pseudospherical surfaces and the loop group representation. For more references, see, for example [2, 13, 15].
2.1 Lorentz surfaces and box charts
Any pseudospherical immersion has a natural Lorentz structure induced by the second fundamental form. We therefore outline a little background on Lorentz surfaces from Weinstein [20].
A Lorentz surface (S, [h]) is an oriented \({{\mathbb {C}}}^\infty \) surface S equipped with a conformal equivalence class of indefinite metrics [h]. There is naturally associated an ordered pair of nowhere parallel null direction fields \({\mathscr {X}}\) and \({\mathscr {Y}}\). A local proper null coordinate system with respect to [h] is a local coordinate chart (x, y) such that \(\partial _x\) and \(\partial _y\) are parallel to \({\mathscr {X}}\) and \({\mathscr {Y}}\), respectively, and \(h=2 B \text {d}x \text {d}y\) for some positive function B.
The Lorentzian analogue to a holomorphic chart of a Riemann surface is a box chart. A pair of charts \(\phi = (x,y)\), \(\hat{\phi } = (\hat{x}, \hat{y})\), on a surface S are \(C^\square \hbox {-}related\) if the orientation and the directions \(\partial _x\) and \(\partial _y\) are preserved by the transition function, that is \(\hat{\phi } \circ \phi ^{-1}(x,y) = (f(x),g(y))\) with \(f^\prime g^\prime >0\). A \(C^\square \hbox {-}atlas\,{\mathscr {A}}^\square \) is a subatlas of the atlas of S in which all charts are \(C^\square \)-related. A box surface is an ordered pair \((S, {\mathscr {A}}^\square )\), consisting of a surface and a maximal \(C^\square \)-atlas, and any element of \({\mathscr {A}}^\square \) is called a box chart.
By Theorem 1 of [20], box surfaces are in one-one correspondence with Lorentz surfaces (S, [h]), where [h] is a conformal equivalence class of Lorentz metrics. In particular, given a Lorentz surface (S, h), the set of all proper null coordinate charts is a maximal \(C^\square \) atlas on S.
A grid box in \({{\mathbb {R}}}^2\) is a product of intervals \(B=(a,b)\times (c,d)\) where \(-\infty \le a < b\le \infty \) and \(-\infty \le c < d\le \infty \). Since the property of being a grid box is preserved by the transition functions of \(C^\square \)-related charts, the concept of a grid box is well defined on a Lorentz surface. We call \(\phi ^{-1}(B)\) a grid box on S if B is a grid box and \(\phi \) is a box chart.
2.2 Lorentz harmonic maps and the associated pseudospherical frontal
Let (S, h) be a simply connected Lorentz surface. Suppose \(N: S \rightarrow {{\mathbb {S}}}^2\) to be a smooth map. Then, N is harmonic if and only if the mixed partial derivative \(N_{xy}\) is proportional to N otherwise stated as \(N \times N_{xy}=0\), where (x, y) are any null coordinate system (box chart).
Consider now the system
for a map \(f: S \rightarrow {{\mathbb {R}}}^3\). The compatibility of the system (2.1), i.e. \(\partial _y ( N \times N_x ) = \partial _x (- N \times N_y)\), is equivalent to the equation \(N \times N_{xy}=0\), i.e. to the harmonicity of N.
Definition 2.1
The smooth map \(f: S \rightarrow {{\mathbb {R}}}^3\), unique up to a translation, obtained by integrating the system (2.1) is called the pseudospherical frontal associated with N. The map \(L=(f,N): S \rightarrow {{\mathbb {R}}}^3 \times {{\mathbb {S}}}^2\) is called the Legendrian lift of f.
Clearly \(\text {d}f\) is orthogonal to N, and so f is a frontal. At points where f is an immersion, the Gauss curvature is \(-1\), and the null coordinates are asymptotic coordinates for f (see below). Hence the name “pseudospherical frontal”.
Conversely, if \(\tilde{f}: S \rightarrow {{\mathbb {R}}}^3\) is a regular constant Gauss curvature \(-1\) surface, where S is simply connected, it is well known that one can find a global asymptotic coordinate system for \(\tilde{f}\), and that the unit normal is a harmonic map with respect to the Lorentz structure defined by the second fundamental form. Hence, all standard pseudospherical surfaces are obtained in the above manner from their Gauss maps.
2.3 The extended frame
Let K denote the diagonal subgroup of SU(2) and represent \({{\mathbb {S}}}^2\) as the symmetric space SU(2) / K, with projection \(\pi : SU(2) \rightarrow {{\mathbb {S}}}^2\) given by \(\pi (g) = {{\mathrm{Ad}}}_{g}e_3\), where
are an orthonormal basis for \(\mathfrak {su}(2)\), with respect to the inner product \(\langle X, Y \rangle = -2{{\mathrm{trace}}}(XY)\). We have the commutators \([e_1,e_2]=e_3\), \([e_2,e_3]=e_1\) and \([e_3,e_1]=e_2\), so that the cross-product in \({{\mathbb {R}}}^3 =\mathfrak {su}(2)\) is
Let \(N: S \rightarrow {{\mathbb {S}}}^2 = SU(2)/K\) be a harmonic map, as above, and \(F: S \rightarrow SU(2)\) any lift of N, i.e. a map such that \(N=\pi (F) = {{\mathrm{Ad}}}_F e_3\). We can express the Maurer–Cartan form of F as
where the \(\mathfrak {k}\) and \(\mathfrak {p}\) components are with respect to the Lie algebra decomposition \(\mathfrak {k} = \text {span}\{e_3\}\), \(\mathfrak {p} = \text {span}\{e_1,e_2\}\).
Equation (2.1) for the associated pseudospherical frontal can be written
and f is immersed precisely at the points where \(U_\mathfrak {p}\) and \(V_\mathfrak {p}\) are linearly independent. At such a point, the first and second fundamental forms are
where \(\phi \) is the angle from \(U_\mathfrak {p}\) to \(-V_\mathfrak {p}\), and \(|\cdot |\) is the standard norm in \({{\mathbb {R}}}^3 \equiv \mathfrak {su}(2)\). Thus, x and y are asymptotic coordinates for f, and the Gauss curvature is \(-1\).
To characterize the harmonicity of N in terms of F, we differentiate \(N={{\mathrm{Ad}}}_F e_3\) to obtain
Hence, \(N_{xy}\) is proportional to \({{\mathrm{Ad}}}_F e_3\) if and only if the \(\mathfrak {p}\) part of the right hand side vanishes, i.e. if and only if \(\left[ \partial _x V_\mathfrak {p}+ [U_\mathfrak {k}, V_\mathfrak {p}] , e_3 \right] = 0\), and this holds if and only if
If \(\alpha \) is the Maurer–Cartan form of a frame F for an arbitrary smooth map \(N: S \rightarrow {{\mathbb {S}}}^2\), we can define
where the parameter \(\lambda \) takes values in \({{\mathbb {C}}}^* :={{\mathbb {C}}}\setminus \{0\}\). The basis of the loop group set-up is that the Maurer–Cartan equation
is satisfied for all \(\lambda \) if and only if Eq. (2.2) holds, if and only if N is harmonic.
Fix some point \(p \in S\) with \(F(p)=F_0\). We want to retain the twisted structure that \(\alpha _\lambda \) already has, namely that diagonal and off-diagonal matrix components are, respectively, even and odd functions of \(\lambda \). We therefore set
Give that N is harmonic, the Maurer–Cartan equation (2.3) means that, for any value of \(\lambda \), we can solve the equations
to obtain a family of maps \(F^\lambda : S \rightarrow SL(2,{{\mathbb {C}}})\), which take values in SU(2) for real values of \(\lambda \), and we have an associated family \(N^\lambda : S \rightarrow {{\mathbb {S}}}^2\) of harmonic maps given by
Given a fixed basepoint p, the family \(N^\lambda \) is independent of the choice of lift F of N. Any other lift is of the form \(\tilde{F} = F D\) where D is a diagonal matrix-valued function, and the extended frame works out to be \(\tilde{F}^\lambda = F^\lambda D\), leaving \(N^\lambda = {{\mathrm{Ad}}}_{F^\lambda }e_3\) unchanged. Let us call the family \(N^\lambda \) the extended harmonic map, or the extended unit normal, and \(F^\lambda \) an extended frame. There is a convenient way to obtain the associated pseudospherical frontal f from \(F^\lambda \). The Sym formula is defined as:
This formula is independent of the choice of extended frame \(F^\lambda \) (given a fixed basepoint) and hence well defined on \(N^\lambda \). By computing the derivatives, one verifies:
Lemma 2.2
For each \(\lambda \in {{\mathbb {R}}}^*\), the map \(f^\lambda : S \rightarrow {{\mathbb {R}}}^3 = \mathfrak {su}(2)\), given by the Sym formula: \(f^\lambda = {\mathscr {S}}_\lambda (N^\lambda )\), is (up to a translation) the unique pseudospherical frontal associated with the harmonic map \(N^\lambda \).
The Sym formula was given by A. Sym [18]. A geometric explanation of this formula can be found in [6].
Finally, we remark that the choice of basepoint in the construction of the extended harmonic map \(N^\lambda \) has no geometric significance. Choosing a different basepoint will result in a translation of the surface obtained from the formula \(f= {\mathscr {S}}_1(N^\lambda )\), and this is the same freedom we have in the definition of the associated pseudospherical frontal.
3 Singularities of pseudospherical frontals
For notational convenience, we now use \(\hat{X}\) instead of \(X^\lambda \) to denote a family of objects parametrized by \(\lambda \). For such an object, we also write X for \(\hat{X} |_{\lambda =1}\).
Analysis of singularities is local, and so, in this section, we are generally discussing a harmonic map \(N:R \rightarrow {{\mathbb {S}}}^2\), where R is a grid box \(I_x \times I_y \subset {{\mathbb {R}}}^2\), a product of open intervals. A harmonic map N is called weakly regular if the kernel of \(\text {d}N\) is everywhere of dimension at most 1 and never contains a nonzero null vector.
Definition 3.1
An admissible connection is an integrable family of 1-forms
on \(R:=I_x \times I_y\), where \(U_{\mathfrak {k}}\), \(V_{\mathfrak {k}}\) and \(U_\mathfrak {p}\), \(V_\mathfrak {p}\) take values, respectively, in \(\mathfrak {k}\) and \(\mathfrak {p}\) in \(\mathfrak {su}(2)\). The connection is weakly regular at \(p \in R\), if both \(U_\mathfrak {p}\) and \(V_\mathfrak {p}\) are nonzero at p and regular if \(U_\mathfrak {p}\) are \(V_\mathfrak {p}\) are linearly independent at p. The connection is weakly regular or regular if these conditions hold on the whole of R. An admissible frame is a family of maps \(\hat{F}: R \rightarrow SU(2)\) such that \(\hat{F}^{-1} \text {d}\hat{F}\) is an admissible connection.
The problem of constructing harmonic maps \(R \rightarrow {{\mathbb {S}}}^2\) is essentially equivalent to that of finding admissible connections. The only freedom in the choice of admissible frame \(\hat{F}\) is a gauge \(\hat{F} \mapsto \hat{F} D\), where D takes values in the diagonal subgroup \(K \subset SU(2)\). Equivalently, \(\hat{\alpha } \mapsto D^{-1} \hat{\alpha } D + D^{-1} \text {d}D\). The harmonic map \(N = {{\mathrm{Ad}}}_F e_3\) is (weakly) regular if and only if the admissible connection is.
Lemma 3.2
Let \(\hat{F}\) be an admissible frame, with associated harmonic map \(N= {{\mathrm{Ad}}}_F e_3\) and \(f={\mathscr {S}}_1(\hat{F})\). The connection \(\hat{\alpha } := \hat{F}^{-1} \text {d}\hat{F}\) is weakly regular if and only if f is a wave front.
Proof
We have
If \(\hat{\alpha }\) is not weakly regular, then at least one of \(U_{\mathfrak {p}}\) and \(V_{\mathfrak {p}} \) is zero at some point. Since the derivatives \(\text {d}N\) and \(\text {d}f\) are computed in terms of these, the rank of \(\text {d}L=(\text {d}f, \text {d}N)\) is at most 1 at this point and f is not a wave front.
Now suppose that \(\hat{\alpha }\) is weakly regular. We need to show that \(\text {d}L = (\text {d}f, \text {d}N)\) has rank 2. Define \(W: R \rightarrow {{\mathbb {S}}}^1 \subset \mathfrak {p}\) by \(W= U_\mathfrak {p}/|U_\mathfrak {p}|\). We can write
where A and B are smooth positive real-valued functions, \(\phi \) is smooth and real valued and \(R_\phi \) denotes the rotation of angle \(\phi \) in the \(e_1 e_2\) plane. The connection is regular when \(\phi \) is not an integer multiple of \(\pi \). Writing \(W=R_\gamma e_1\), let us multiply the extended frame \(\hat{F}\) on the right by \(D=\text {diag}(e^{i\gamma /2}, e^{-i \gamma /2})\). This has no effect on the harmonic map \(N={{\mathrm{Ad}}}_F e_3\) or the map \(f={\mathscr {S}}_1(\hat{F})\). Thus, it is equivalent to consider the admissible connection \(D^{-1} \hat{\alpha } D + D^{-1} \text {d}D\), which we now denote by \(\hat{\alpha }\). The conclusion is that we can assume that
Now
and
Since A and B are non-vanishing, it follows that \(\text {d}L = (\text {d}f, \text {d}N)\) has rank 2 and f is a wave front. \(\square \)
3.1 The singular curve for pseudospherical wave fronts
Assume that \(\hat{\alpha }\), N and f are as above, and \(\hat{\alpha }\) is weakly regular. Using the same choices as in the previous lemma, we have
Thus,
Since A and B are assumed non-vanishing, the singular set is the set of points \(\sin \phi = 0\), i.e. \(\phi = k \pi \), for \(k \in {\mathbb Z}\). A singular point q on a frontal is non-degenerate if and only if one can write \(f_x \times f_y = \mu N\), where \(\mu (q)=0\) and \(\text {d}\mu |_q \ne 0\). Here, we have \(\mu = A B \sin \phi \) and \(\text {d}\mu = \pm A B\text {d}\phi \). Thus, the non-degeneracy condition in our case is
In a neighbourhood of a non-degenerate singular point, the singular set is a regular curve in the coordinate domain, and there is a well-defined 1-dimensional direction field \(\eta \) along the curve called the null direction (not to be confused with null coordinate directions!) such that
The generic singularities of pseudospherical surfaces were studied by Ishikawa and Machida [10] and shown to be cuspidal edges and swallowtails. For general wave fronts, these singularities can be identified by the following characterization:
Proposition 3.3
([11]) Let f be a wave front and q a non-degenerate singular point. Let \(\sigma (t)\) be a local parametrization for the singular curve around q, with \(\sigma (0)=q\). Then, the image of f in a neighbourhood of q is diffeomorphic to:
-
(1)
A cuspidal edge if and only if \(\eta (0)\) is not proportional to \(\sigma ^\prime (0)\);
-
(2)
A swallowtail if and only if \(\eta (0)\) is proportional to \(\sigma ^\prime (0)\), and
$$\begin{aligned} \frac{\text {d}}{\text {d}t}(\det (\sigma ^\prime (t), \eta (t))\big |_{t=0} \ne 0. \end{aligned}$$
In our situation, assuming, for concreteness’ sake that the singular curve is given locally by \(\phi (x,y)=0\), we have \(\text {d}f = (A \text {d}x + B \text {d}y){{\mathrm{Ad}}}_F e_1\), and so the null direction is given on this curve by
Assume first that the singular curve is not tangent to either \(\partial _x\) or \(\partial _y\). In that case, we can, after a change of box coordinates (see, e.g. [8]), assume that our singular curve is locally given by \(y=\varepsilon x\), where \(\varepsilon =\pm 1\). Note that this special choice of coordinates means that we cannot assume that A and B are constant. Now we have, in the basis \(\partial _x\), \(\partial _y\),
Let us add here that the special case that \(A(t)+\varepsilon B(t)\equiv 0\) corresponds to a cone singularity, i.e. a non-degenerate singular curve that maps to a single point. This follows from the formula \(\text {d}f(\sigma ^\prime (t)) = (A(t)+\varepsilon B(t)){{\mathrm{Ad}}}_F e_1\). Constructing pseudospherical wave fronts with cone singularities is discussed by Pinkall [15].
Now consider the case that the singular curve is tangent, at a point p, to one of the coordinate directions \(\partial _x\) or \(\partial _y\). Then, it is not proportional to \(\eta \), because both B and A are nonzero. In this case, by the proposition above, the surface is a cuspidal edge at p. We summarize this as:
Theorem 3.4
Let f be a pseudospherical wave front. Suppose that q is a non-degenerate singular point. If the singular curve is tangent at q to a null coordinate direction, then the surface is locally diffeomorphic to a cuspidal edge at q. Otherwise, there exist box coordinates (x, y) such that, in a neighbourhood of \(q=(0,0)\), the singular set is parametrized by \((x(t),y(t))= (t, \varepsilon t)\), and the image of f is diffeomorphic to:
-
(1)
A cuspidal edge if \(A(0) + \varepsilon B(0) \ne 0\);
-
(2)
A swallowtail if \(A(0) + \varepsilon B(0) =0\) and \(A^\prime (0)+\varepsilon B^\prime (0) \ne 0\).
-
(3)
A cone singularity if \(A(t) + \varepsilon B(t) \equiv 0\),
where \(A(t)=|f_x(t, \varepsilon t)|\) and \(B(t)=|f_y(t, \varepsilon t)|\).
3.2 Singular curves that are not wave fronts
Let us now consider the case that \(\hat{\alpha }\) is semi-regular – meaning that the derivative of the associated harmonic map N has rank at least 1 – but not weakly regular. This means that at least one of \(U_\mathfrak {p}\) and \(V_\mathfrak {p}\) is nonzero, but the other may vanish. We assume then that \(U_\mathfrak {p} \ne 0\), the other case being analogous. We can, as before, assume that \(U_\mathfrak {p}=A e_1\). After a change of box coordinates, we can take \(A=1\). The angle \(\phi \) is not well defined at points where \(V_\mathfrak {p}\) vanishes, so we now have:
where \(u_0\), \(v_0\), a and b are real-valued functions. The integrability condition \(\text {d}\hat{\alpha } + \hat{\alpha } \wedge \hat{\alpha }=0\) is equivalent to the following set of equations
Now we have
Thus, the frontal f has a singular point precisely when b vanishes, i.e. the singular set is given by
and the non-degeneracy condition is \(\text {d}b \ne 0\). If a is non-vanishing, then we are at a weakly regular point, already discussed. We therefore consider now a point q at which
We relabel coordinates so that \(q=(0,0)\). The integrability conditions above for a and b give, along the line \(y=0\), the system:
which has the unique local solution
Hence, assuming the non-degeneracy condition, which is now
the singular curve is locally given by
The other integrability condition becomes \(\partial _y u_0=0\) along \(y=0\). The null direction is \(\eta = \partial _y\), which is transverse to the singular curve, but the singularity is not a standard cuspidal edge because the surface is not a wave front along this curve. We call such a singularity a higher-order cuspidal edge, because it is non-degenerate and the image of the singular curve is a regular curve in \({{\mathbb {R}}}^3\). A fold singularity is of this type.
We have shown that if a pseudospherical surface has a non-degenerate singularity at a point where the surface is not a wave front, then the singular curve at that point is a characteristic curve, or null coordinate curve. However, we saw in the previous section that it is also possible for a weakly regular singular curve to be tangent to a characteristic direction.
4 Prescribed non-characteristic singular curves
4.1 The generalized d’Alembert method
A well-known method for producing essentially all admissible frames is the generalized d’Alembert representation given by Toda in [19]. Here, is a summary, using definitions and notation as in [8]: let \({\mathscr {G}}:= \Lambda SL(2,{{\mathbb {C}}})_{\sigma \rho }\) denote the group of smooth maps \(\gamma : {{\mathbb {S}}}^1 \rightarrow SL(2,{{\mathbb {C}}})\) that are fixed by the involutions \(\sigma \) and \(\rho \) given by
where \(P=\hbox {diag}(-1,1)\), and \(\lambda \) is the \({{\mathbb {S}}}^1\) parameter. All loops considered here extend holomorphically to \({{\mathbb {C}}}\setminus \{0\}\), and the reality condition given by \(\rho \) means that they take values in SU(2) for real values of the loop parameter \(\lambda \). We also consider the subgroups \({\mathscr {G}}^\pm \) consisting of loops the Fourier expansions of which are power series in \(\lambda ^{\pm 1}\). We denote the corresponding Lie algebras by \(\hbox {Lie}({\mathscr {G}})\), \(\hbox {Lie}({\mathscr {G}}^\pm )\).
Definition 4.1
Let \(I_x\) and \(I_y\) be two real intervals, with coordinates x and y, respectively. A potential pair \((\hat{\chi }, \hat{\psi })\) is a pair of smooth \(\hbox {Lie}({\mathscr {G}})\)-valued 1-forms on \(I_x\) and \(I_y\), respectively, with Fourier expansions in \(\lambda \) as follows:
We will call the potential pair semi-regular at a point p if at least one of the “leading coefficients” \(\chi _1\) and \(\psi _{-1}\) is nonzero at p and regular if both are nonzero, and the potential pair is called (semi-)regular if the condition holds at every point.
An admissible frame \(\hat{F}\) is then obtained by solving \(\hat{X}^{-1} \text {d}\hat{X} = \hat{\chi }\), and \(\hat{Y}^{-1} \text {d}\hat{Y} = \hat{\psi }\) for \(\hat{X}(x)\) and \(\hat{Y}(y)\), each with initial condition the identity matrix, thereafter performing, at each (x, y), a Birkhoff decomposition (see [3, 17]):
and finally defining \(\hat{F}\) by:
The admissible frame is semi-regular if and only if the potential pair is semi-regular and weakly regular if and only if the potential pair is regular.
Conversely, any admissible frame \(\hat{F}\) is associated with a potential pair \(\left( \!\hat{X}_+^{-1} \text {d}\hat{X}_+,\! \hat{Y}_-^{-1} \text {d}\hat{Y}_- \!\right) \), where \(\hat{X}_+\) and \(\hat{Y}_-\) are obtained by the pair of pointwise normalized Birkhoff factorizations
Note that the special form of an admissible connection automatically implies that \(\hat{X}_+\) and \(\hat{Y}_-\) depend only on x and y, respectively. Because of the normalization, these potentials are uniquely determined by \(\hat{F}\) and have particularly simple forms:
and are called normalized potentials.
4.2 Potentials for non-characteristic singularities
Given the d’Alembert representation just described, a generalized pseudospherical surface is locally determined by an arbitrary pair of (real)-differentiable complex-valued functions \(\zeta (x)\) and \(\xi (y)\). A generic function \({{\mathbb {R}}}\rightarrow {{\mathbb {C}}}\) is non-vanishing, and so a generic normalized potential pair is regular, and the corresponding pseudospherical surface is a wave front.
Our aim here is to give potentials that produce prescribed singular curves. We will consider separately two cases: that the singular set is or is not a characteristic curve, starting with the non-characteristic case. For this, rather than normalized potentials, a better choice is a form of the boundary potential pairs, introduced in [8] for the purpose of giving prescribed values of \(\hat{F}\) along a non-characteristic curve. We assume that the singular curve is non-degenerate and never parallel to a null curve. Then, we can always find local box coordinates (x, y) such that the curve is given by
Suppose given the value for \(\hat{F}(x,y)\), along the curve \(y=\varepsilon x\). In the coordinates
the curve is given by \(v=0\), and the value of \(\hat{F}\) along the curve is given by
Since \(\hat{F}\) is assumed to be an admissible frame we have, from Definition 3.1,
Since the highest and lowest powers of \(\lambda \) appearing are 1 and \(-1\), respectively, this 1-form is valid as either \(\hat{\chi }\) or \(\hat{\psi }\) or both in a potential pair. Hence, setting
gives a valid potential pair \((\hat{X}^{-1} \text {d}\hat{X}, \hat{Y}^{-1} \text {d}\hat{Y})\), called the boundary potential pair relative to the curve \(v=0\). For this potential pair, the Birkhoff decomposition (4.1) is trivial along the curve \(v=0\), since \(\hat{X}(v=0) = \hat{Y}(v=0)\), and so the admissible frame \(\widetilde{F}\) obtained by (4.2) agrees with \(\hat{F}\) along this curve. A uniqueness argument using normalized potentials (see [8]) then shows that \(\widetilde{F}\) and \(\hat{F}\) determine the same harmonic map.
We now want to construct \(\hat{F}_0(u)\) along a curve \(v=0\) from geometric data of a pseudospherical frontal f prescribed along the curve. Since the curve is non-characteristic, and assumed non-degenerate, f is necessarily a wave front (see Sect. 3.2). From Sect. 2.3, we can assume that we are given box coordinates (x, y) that are asymptotic coordinates for f, the angle \(\phi \) is the oriented angle between \(f_x\) and \(f_y\), and the first and second fundamental forms are:
where \(A=|f_x|\) and \(B=|f_y|\). Using the same frame F as in the proof of Lemma 3.2, defined by (3.1), we have:
In the coordinates (u, v) we have \(\phi _x=\frac{1}{2}(\phi _u+\phi _v)\). If \(v=0\) is a singular curve, we have \(\phi =k \pi \) constant along the curve, so \(\phi _u(u,0)=0\). Without loss of generality, we take \(k=0\), i.e. \(\phi (u,0)=0\). The basic data that determine the boundary potential are thus
where \(A(u)=|f_x(u,0)|\) and \(B(u)=|f_y(u,0)|\). Substituting into (4.3) and applying Theorem 3.4, we conclude that all non-degenerate non-characteristic singular curves on pseudospherical frontals are obtained from the following theorem:
Theorem 4.2
Let J be an open interval, \(A, \,\, B: J \rightarrow (0,\infty )\) and \(\beta : J \rightarrow {{\mathbb {R}}}\) three differentiable functions. Let \(\varepsilon = \pm 1\) and set
Consider the potential pair \((\hat{\eta }, \hat{\eta })\) defined on the intervals \(I_x=J\) and \(I_y=\varepsilon J\). Let \(f:I_x \times I_y \rightarrow {{\mathbb {R}}}^3\) be the generalized pseudospherical surface obtained from \((\hat{\eta }, \hat{\eta })\) via the generalized d’Alembert method. Then,
-
(1)
The set \(C:=\{y=\varepsilon x\}\) is a singular set for f.
-
(2)
C is non-degenerate at a point \((x_0,\varepsilon x_0)\) if and only if \(\beta (x_0) \ne 0\). In this case,
-
(a)
C is diffeomorphic to a cuspidal edge in a neighbourhood of \((x_0,\varepsilon x_0)\) if and only if \(A(x_0)+\varepsilon B(x_0) \ne 0\).
-
(b)
C is diffeomorphic to a swallowtail in a neighbourhood of \((x_0,\varepsilon x_0)\) if and only if \(A(x_0)+\varepsilon B(x_0) = 0\) and \(A^\prime (x_0) + \varepsilon B^\prime (x_0) \ne 0\).
-
(c)
C is diffeomorphic to a cone singularity if and only if \(A(x)+\varepsilon B(x) \equiv 0\).
-
(a)
Three non-degenerate examples are computed in Fig. 3, all with \(\beta (t)=2\). Some degenerate examples are shown in Fig. 7.
4.3 Prescribed non-characteristic cuspidal edges
Theorem 4.2 gives the boundary potential pair for the generic non-characteristic singularities of pseudospherical surfaces, as well as cones. We now adapt this to produce pseudospherical surfaces with a given curve in \({{\mathbb {R}}}^3\) as a singular curve. We treat the case that the curve is regular in \({{\mathbb {R}}}^3\), which means that the singular curve, where non-degenerate, must be a cuspidal edge.
The geometric Cauchy problem for regular pseudospherical surfaces was studied in [8]. For a non-characteristic curve, there is a unique immersed solution containing a given curve \(\gamma \) and with the surface normal N prescribed along the curve, with a regularity condition \(\langle \gamma ^\prime (t), N^\prime (t) \rangle \ne 0\). For the non-characteristic singular geometric Cauchy problem, we replace the regularity condition with a singularity condition, \(\langle \gamma ^\prime (t), N^\prime (t) \rangle = 0\):
Non-characteristic singular geometric Cauchy data along an open interval J:
-
(1)
A regular curve \(\gamma : J \rightarrow {{\mathbb {R}}}^3\);
-
(2)
A unit vector field \(Z: J \rightarrow {{\mathbb {S}}}^2 \subset {{\mathbb {R}}}^3\), satisfying
$$\begin{aligned} \langle Z(t), \gamma ^\prime (t) \rangle = 0, \quad \quad \langle Z^\prime (t), \gamma ^\prime (t) \rangle =0. \end{aligned}$$ -
(3)
Weak regularity condition:
$$\begin{aligned} |\gamma ^\prime (t) | \ne |Z^\prime (t)|. \end{aligned}$$
The above conditions are necessarily satisfied along a non-characteristic singular curve on a pseudospherical frontal. We also find that the singular curve is non-degenerate at a point if and only if the curvature \(\kappa \) of the curve \(\gamma \) is nonzero at that point. Adding this assumption then simplifies the above description of the geometric Cauchy data. Suppose that \(\gamma (s)\) is parameterized by arc length. Let \(\mathbf{t}\), \(\mathbf{n}\) and \(\mathbf{b}\) be the Frenet–Serret frame along the curve. The vector field Z must satisfy: \(\langle Z, \mathbf{t} \rangle =0\) and \(\langle Z^\prime , \mathbf{t} \rangle =0\). Differentiating the first equation gives
Hence, the assumptions \(\langle Z, \mathbf{t} \rangle =0\) and \(\kappa \ne 0\) imply that \(\langle Z, \mathbf{n}\rangle =0\). It follows that \(Z=\pm \mathbf{b}\), where \(\mathbf{b}\) is the unit binormal to the curve. Since \(\mathbf{b}^\prime = -\tau \mathbf{n}\), where \(\tau \) is the torsion, the weak regularity condition \(|\gamma ^\prime | \ne |N^\prime |\) becomes \(\tau \ne \pm 1\). To simplify matters, we will also take \(\tau >-1\). Hence, for non-degenerate singular curves, the geometric Cauchy data are the curve given in the following result:
Theorem 4.3
Let \(\gamma : J \rightarrow {{\mathbb {R}}}^3\) be a regular arc-length parameterized curve, with curvature \(\kappa \) and torsion \(\tau \) satisfying
along J. Let \(\varepsilon := \hbox {sign}(\tau -1)\). Then,
-
(1)
There exists, unique up to a Euclidean motion, a pseudospherical wave front f(u, v), with box coordinates (x, y) and \(u=(x+\varepsilon y)/2\), \(v=(x-\varepsilon y)/2\), containing \(\gamma \) as a non-characteristic singular curve in the form \(f(u,0)=\gamma (u)\). The singular curve is non-degenerate.
-
(2)
The surface f is given by the d’Alembert method, with potential pair \((\hat{\eta }, \hat{\eta })\) on \(J \times \varepsilon J\), with
$$\begin{aligned} \hat{\eta } = \left( \frac{\tau -1}{2}e_1 \lambda ^{-1} + \kappa e_3 + \frac{\tau + 1}{2} e_1 \lambda \right) \text {d}s. \end{aligned}$$ -
(3)
All non-degenerate non-characteristic singular curves of pseudospherical frontals that have a regular image in \({{\mathbb {R}}}^3\) are obtained this way.
Proof
By Theorem 4.2, there is a generalized pseudospherical surface generated by any triple of functions A, B and \(\beta \). The surface is a wave front if and only if both A and B are non-vanishing, which in this case means \(\tau \ne \pm 1\). The non-degeneracy condition is \(\beta = -2\kappa (t) \ne 0\).
Now suppose the existence of a pseudospherical wave front \(f:J \times \varepsilon J \rightarrow {{\mathbb {R}}}\) with \(f(u,0)=\gamma (u)\) a non-degenerate non-characteristic singular curve. As described above, it follows that the surface normal satisfies \(N(u,0)= \pm \mathbf{b}(u)\). Since we are only looking for the potential up to a Euclidean motion, we can take
Along the singular curve, the vectors \(f_u\), \(f_v\), \(f_x\) and \(f_y\) are all parallel. As previously, let F be the frame defined at (3.1), so that, on \(v=0\),
which is to say that
We have already shown in Sect. 4.2 that along \(v=0\)
where \(A(u)=|f_x(u,0)|\) and \(B(u)=|f_y(u,0)|\). Differentiating \(\mathbf{b} = {{\mathrm{Ad}}}_F (e_3)\), we have
so that
We also have \(\gamma ^\prime (u)=f_u = f_x + \varepsilon f_y\), from which
There are, in general, two solutions for positive A and B, but the surfaces obtained from the corresponding potentials are congruent after interchanging x and y. Hence, we can take the solution:
To find \(\beta \), we use
so \(\beta =-2 \kappa \). Substituting the expressions for A, B, \(\varepsilon \) and \(\beta \) into the potential \(\hat{\eta }\) of Theorem 4.2 gives the potential in the theorem statement. Since the above data were obtained from an arbitrary solution of the geometric Cauchy problem, this also proves uniqueness, and so items (1) and (2) are proved. Item (3) follows from the fact, already explained, that, for a non-degenerate non-characteristic singular curve the curvature is non-vanishing and the torsion satisfies \(|\tau |\ne 1\) (Fig. 4). \(\square \)
Example 4.4
Circles: take \(\gamma (t)=R(\cos t, \sin t, 0)\), where \(R>0\). The arc-length parameter, curvature and torsion are \(s=Rt\), \(\kappa =1/R\) and \(\tau =0\). The potential is thus:
and this gives the well-known pseudospherical surfaces of revolution. The case \(R=1\) is the pseudosphere.
Example 4.5
Helices: taking \(\kappa \) and \(\tau \) both constant, with \(\tau \ne 0\), gives a surface containing a circular helix as a cuspidal edge (Fig. 5). Helical, as well as rotational, constant curvature surfaces, were studied by Minding in [14]. These surfaces are generally periodic in the v direction, which can be seen by considering that the curve is invariant under a 1-parameter family of rigid motions (a screw motion). The surface must also have this symmetry by uniqueness of the solution to the geometric Cauchy problem. Hence, the next singular curve encountered when moving in the v direction is also a circular helix. By the symmetry of the initial data, it follows that every second singular curve is congruent.
As with the case of the circle, there are essentially three types:
-
(1)
Case \(\kappa ^2+\tau ^2>1\): here, there are two sets of helices with the same axis but different radius. The initial curve is on the outer cylinder when \(|\tau |>1\), and the inner when \(|\tau |<1\).
-
(2)
Case \(\kappa ^2+\tau ^2=1\): the special case where the inner helices degenerate to a straight line. These are Dini’s surfaces, which can be parametrized as
$$\begin{aligned} f(\zeta ,\xi )=(a \cos \zeta \sin \xi , a \sin \zeta \sin \xi , a(\cos \xi + \ln (\tan (\xi /2))) + b\xi ), \end{aligned}$$where, for the case of constant curvature \(K=-1\), we must have \(a^2+b^2=1\). The surface has singularities at \(\cos (\xi )=0\), so we can take the helix
$$\begin{aligned} \gamma (t) = f(t,\pi /2)=(a \cos t, a \sin t, bt) \end{aligned}$$as the initial curve. We then have \(\kappa =|a|\) and \(\tau =b\). Hence, Dini’s surfaces are given by constant \(\kappa \) and \(\tau \), with \(\kappa ^2 + \tau ^2=1\).
-
(3)
Case \(\kappa ^2+\tau ^2<1\): here, the inner helix disappears completely, so that all singular curves are congruent.
Example 4.6
The closed curve \(\gamma (t)=(\cos (3t), \sin (3t), -\sin (t))\) lies on a round cylinder and has two self-intersections. Computing \(\kappa (t)=3(8\cos ^2(t)+82)^{1/2}(\cos ^2(t)+9)^{3/2}\), \(\tau =-12\cos (t)/(4\cos ^2(t)+41)\) and \(\text {d}s = \sqrt{\cos ^2(t)+9} \text {d}t\), we see that \(\kappa \) is non-vanishing and \(|\tau |<1\). The surface that contains this curve as a cuspidal edge is shown in Fig. 6.
Example 4.7
Examples with inflections and with \(\tau \) taking the value 1: Theorem 4.3 is stated for curves with \(\kappa \) non-vanishing and \(\tau \ne \pm 1\). However, we can use any functions \(\kappa \) and \(\tau \) and still obtain a valid potential pair and therefore a pseudospherical frontal. If we take \(\kappa \equiv 0\), the solution degenerates to a straight line. If we take \(\tau \equiv \pm 1\), the solution degenerates to a helix curve.
If \(\kappa \) vanishes at just one point, we will get a singular curve that is degenerate at this point, but non-degenerate elsewhere, provided \(|\tau |\ne 1\). The most basic example is \(\kappa (t)=t\), \(\tau (t)=1/2\), shown in Fig. 7. At the point (0, 0), there are two cuspidal edges crossing each other. For the example \(\kappa (t)=t\), \(\tau (t)=0\), the surface appears to have a degenerate cone point. The case \(\kappa (t)=t^2\), \(\tau (t)=1/2\) is also computed and shown in Fig. 11. In this case, the singular set is a single curve through the point (0, 0).
If we take \(\tau =\pm 1\) at just one point, the surface is not a wave front at this point. This is because the potential pair is \((\hat{\chi }, \hat{\psi })\), where
so exactly one of \(\chi _1\) and \(\psi _{-1}\) vanishes. The potential pair is semi-regular but not regular. Moreover, the singular curve must be degenerate at this point, because we showed in Sect. 3.2 that if the singular curve is non-degenerate at a point where the surface is not a wave front, then the curve is a characteristic curve on a neighbourhood of this point, which is not the case here. The surface shown at Fig. 1 is generated by the Viviani figure-8 space curve \(\gamma (t)=0.3(1+ \cos (t), \sin (t), 2 \sin (t/2)\). The torsion takes the values \(\pm 1\) twice each, and at each such point another singular curve branches off from the figure eight (Fig. 1, right).
5 Prescribed characteristic singular curves
Now we want to give potentials for non-degenerate characteristic singular curves. As expected for a Cauchy problem along a characteristic, we will find that data along a curve do not specify a unique solution: further data must be provided along another, transverse, characteristic curve. Moreover, with our solution, the non-degeneracy is only guaranteed in a neighbourhood of the intersection of these two curves.
As explained in Sect. 3.2, given that the map is semi-regular, we can assume that box coordinates are chosen such that the singular curve is locally given as \(\{y=0\}\), and can choose a local frame satisfying
where
The surface is a wave front at points where \(a(x,0) \ne 0\). The curve \(\gamma (x)=f(x,0)\) is already arc-length parameterized. Hence, differentiating the expression for \(f_x\), we have:
along \(y=0\). Thus, up to a change of orientation, \(u_0(x,0)= \kappa (x)\), the curvature of \(\gamma \). Note that if \(\kappa (x) \ne 0\) for all x, then the curve has a well-defined normal \(\mathbf{n}={{\mathrm{Ad}}}_F e_2\), and hence, the binormal is \(\mathbf{b}= {{\mathrm{Ad}}}_F e_3=N\). We then have
from which we conclude that \(\tau (x)=1\) along the whole curve. Although the curve is singular, this is the same property that asymptotic curves (of non-vanishing curvature) have on a regular pseudospherical surface, namely that \(\tau =\pm 1\).
Now differentiating the expression \(f_x \times f_y = b {{\mathrm{Ad}}}_Fe_3\), using \(b(x,0)=b_x(x,0)=0\), we also have
Hence, if the surface is a wave front we must have \(\kappa (x)=0\) for all x. In other words, the only possible non-degenerate characteristic singular curve on a pseudospherical wave front is a straight line.
Theorem 5.1
Let \(I_x\) be an open interval containing 0, and \(\gamma : I_x \rightarrow {{\mathbb {R}}}^3\) a regular space curve, parameterized by arc length, with either non-vanishing curvature function \(\kappa \), and constant torsion \(\tau =\pm 1\), or with curvature everywhere zero on \(I_x\). Let \(I_y\) be an open interval containing 0. For every choice of differentiable 1-form of type
with
and
there corresponds a unique pseudospherical frontal \(f:I_x \times I_y \rightarrow {{\mathbb {R}}}^3\), such that
-
(1)
f is semi-regular on an open set containing \(I_x \times \{0\}\), and
-
(2)
\(f(x,0)=\gamma (x)\) is a characteristic singular curve in the surface, non-degenerate on a neighbourhood of (0, 0).
Up to a Euclidean motion, the surface f is given by the d’Alembert method with potential pair \((\hat{\chi }, \hat{\psi })\) on \(I_x \times I_y\), where
and all such surfaces f satisfying (a) and (b) are obtained this way.
Proof
The 1-forms defined satisfy the requirements for a potential pair, and therefore integrating \(\hat{X}^{-1} \text {d}\hat{X} = \hat{\chi }\), and \(\hat{Y}^{-1} \text {d}\hat{Y} = \hat{\psi }\), with initial conditions \(\hat{X}(0)=I\) and \(\hat{Y}(0)=I\), performing a Birkhoff decomposition
give us an admissible frame \(\hat{F}= \hat{X} \hat{H}_- = \hat{Y} \hat{H}_+^{-1}\). We write \(O_\pm (\lambda ^{\pm k})\) for any convergent Fourier series of the form \(\sum _{j=k}^\infty a_k \lambda ^{\pm j}\). The normalization of \(\hat{H}_-\) means that its Fourier expansion is \(\hat{H}_- = I + O_-(\lambda ^{-1}) \), so
Since the coefficient of \(\lambda \) is \(e_1 \text {d}x\), we can apply the analysis of Sect. 3.2 to conclude that
Along the curve \(y=0\), we have \(\hat{Y} = I\), and so the unique factor \(\hat{H}_-\) in the Birkhoff decomposition above satisfies \(\hat{H}_-(x,0)=I\). Thus, \(\hat{F}(x,0)=\hat{X}(x)\), and along \(y=0\) we have
Hence,
To check the non-degeneracy condition on \(\partial _y b(x,0)\), we will use the expression \(\hat{F}= \hat{Y} \hat{H}_+^{-1}\). Since \(\hat{H}_+\) is \({\mathscr {G}}^+\)-valued, we can write
We have \(\hat{H}_+^{-1}(x,0)=\hat{X}(x)\), and so, along \(y=0\),
whilst along \(x=0\), we also have \(\hat{H}_+(0,y)=I\). Hence,
From \(\hat{F}= \hat{Y} \hat{H}_+^{-1}\), we obtain
which gives
Differentiating this, using \(\beta (0)=0\):
For the case \(\kappa (x)\equiv 0\), we have \(\theta _x(x,0)=0\), so \(\theta \) is constant along \(x=0\), and \(\cos (\theta (x,0)) = 1\), \(\sin (\theta (x,0)=0\) by the initial condition at (0, 0). Thus,
Since \(\theta _y(0,0)=0\) and \(\beta ^\prime (0) \ne 0\), it follows that the non-degeneracy condition \(b_y(x,0) \ne 0 \) is satisfied on an open set containing (0, 0). On the other hand, for the case \(\kappa \ne 0\), where we take \(\alpha (0)=0\), we have
In this case, we use \(\cos (\theta (0,0)) = 1\), \(\sin (\theta (0,0)=0\) to again conclude that \(b_y(x,0) \ne 0 \) is satisfied on an open set containing (0, 0).
To see that the singular curve f(x, 0), of the solution f, coincides with \(\gamma \), the discussion preceding the statement of this theorem shows f(x, 0) has curvature \(\kappa \) and if \(\kappa \) is non-vanishing, constant torsion \(\tau =1\). Since a curve is determined by its curvature and torsion, we must have, up to a Euclidean motion, \(f(x,0)=\gamma (x)\). If \(\kappa \) is everywhere zero, then the curve is just a straight line segment of the same length as \(I_x\), again identical with \(\gamma (x)\) up to a Euclidean motion.
For uniqueness given the potential \(\hat{\psi }\), it is enough to observe that \(\hat{\psi }\) is a normalized potential, with normalization point (0, 0), which is uniquely determined by the surface \(f: I_x \times I_y \rightarrow {{\mathbb {R}}}^3\) and the choice of normalization point. Thus, given any surface \(\tilde{f}\) satisfying \(\tilde{f}(x,0)=\gamma (x)\), we obtain \(\hat{\chi }\) from the knowledge of \(\kappa \) and the frame \(\tilde{\hat{F}}(x,0)\), and we recover \(\hat{\psi }\) from a normalized Birkhoff decomposition of \(\tilde{\hat{F}}(x,y)\) as described at the end of Sect. 4.1. Hence, \(\tilde{f} = f\). Since \(\hat{\psi }\) is the most general normalized potential satisfying the regularity conditions, all possible solutions are obtained this way. \(\square \)
Remark 5.2
-
(a)
Because \(\beta (0)=0\) and \(\beta ^\prime (0) \ne 0\), we can, on a neighbourhood of \(y=0\), change y-coordinates to \(\tilde{y}(y)\) so that \(\beta (y) \text {d}y = \tilde{y} \text {d}\tilde{y}\). In these coordinates, the potential \(\hat{\psi }\) is of the form
$$\begin{aligned} \hat{\psi } = (\tilde{\alpha }(\tilde{y}) e_1 + \tilde{y} e_2) \lambda ^{-1} \text {d}\tilde{y}. \end{aligned}$$Thus, given \(\kappa \), the unique solution is determined, on an open set containing the curve, by a single function \(\tilde{\alpha }(\tilde{y})\) that is arbitrary if \(\kappa \equiv 0\) but in the general case must satisfy \(\tilde{\alpha }(0)=0\)
-
(2)
For the case that \(\kappa (x) \equiv 0\), adding the assumption \(\alpha (0)=0\) guarantees that the entire singular curve is non-degenerate.
-
(3)
Suppose coordinates are chosen such that \(\beta (y)=y\), as just described. Then, if \(\alpha \) is an odd function of y, the surface has a fold singularity along \(y=0\), i.e f satisfies \(f(x,y)=f(x,-y)\). This can be seen from the symmetry \(\hat{\psi }(-y)= \hat{\psi }(y)\). Such a singularity, at least if \(\alpha \) is analytic, can be “removed” in the sense that one half of the folded surface is part of a regular pseudospherical surface which contains the same curve: writing \(\alpha (y)=y(a_1 + a_3 y^2 + \cdots )\), and setting \(2 \tilde{y}= y^2\), we have, for \(y>0\), the expressions \(\tilde{\alpha }(\tilde{y}) \text {d}\tilde{y}= \alpha (y) \text {d}y = (a_1 + a_3 2 \tilde{y} + a_5 (2\tilde{y})^2 + \cdots ) \text {d}\tilde{y}\) and \(y \text {d}y = \text {d}\tilde{y}\). Hence, the surface corresponding to the pair \(\hat{\psi } = (\tilde{\alpha }(\tilde{y}) e_1 + e_2) \lambda ^{-1} \text {d}\tilde{y}\) and \(\hat{\chi } = (\kappa (x) e_3 + \lambda e_1) \text {d}x\) is regular on an open set containing the x-axis and agrees with the folded surface on the set \(y>0\).
Of course, the Lorentz structure corresponding to the two surfaces is different here at the line \(y=0\). For a given global Lorentz structure, there is no way to remove this singularity because the vanishing of a 1-form \(g(y) \text {d}y\) is well defined with respect to changes of box charts. An example of a folded Amsler surface is shown in Fig. 8.
Example 5.3
Weakly regular characteristic singularities: these are all given by data of the form \(\kappa \equiv 0\), \(\beta (y)=y\) and an arbitrary choice of \(\alpha \) with \(\alpha (0) \ne 0\). The singular curve is guaranteed to be non-degenerate in a neighbourhood of (0, 0). An example is shown in Fig. 2.
Example 5.4
Straight lines that are not weakly regular: these are given by \(\kappa \equiv 0\), \(\beta (y)=y\) and any choice of \(\alpha \) with \(\alpha (0)=0\). The entire line is a non-degenerate singularity. These are all higher-order cuspidal edges. See Fig. 8. If \(\alpha \) is an odd function, we have a fold. If \(\alpha \) is not an odd function, then we cannot “remove” the singular curve as can be done with the fold. For example, for the case \(\alpha (y)=y^2\) and \(\beta (y)=y\), let \(S_+\) denote the surface generated by \((\hat{\chi }, \hat{\psi })\), for \(y > 0\). Then, \(S_+\) does not extend to a pseudospherical wave front over the curve \(y=0\). If it did, because asymptotic directions are well defined on a pseudospherical surface, the surface would be generated by a potential pair \((\hat{\chi }, \tilde{\hat{\psi }})\), where \(\hat{\chi }\) is unchanged and the one-form \(\tilde{\hat{\psi }}\) agrees with \(\hat{\psi }\) on the set \(y \ge 0\), but where \(\tilde{\hat{\psi }}\) is regular at \(y=0\). In other words, we are looking for a change of coordinates \(\tilde{y}(y)\) valid on \(y>0\) such that the 1-form \((y^2 , y) \text {d}y= \left( y^2 \frac{\text {d}y}{\text {d}\tilde{y}} , y\frac{\text {d}y}{\text {d}\tilde{y}}\right) \text {d}\tilde{y}\) extends to a regular 1-form at \(y=0\). By definition, this means that both components are smooth and at least one non-vanishing at \(y=0\). If \(y \frac{\text {d}y}{\text {d}\tilde{y}}\) is non-vanishing, we can assume that \(\tilde{y}\) is chosen so that \(y \frac{\text {d}y}{\text {d}\tilde{y}} = 1\), that is \(\tilde{y} = y^2/2\), and hence, \((y^2, y) \text {d}y = \left( \sqrt{2 \tilde{y}}, 1\right) \text {d}\tilde{y}\), which is not differentiable at \(\tilde{y}=0\). A similar argument shows that coordinates cannot be found such that the first component \(y^2 \text {d}y\) is nonzero.
Example 5.5
Figure 8 (right) shows a pseudospherical frontal that contains a helix curve. The surface is not a wave front because the singular curve is characteristic and not a straight line. The singularity is non-degenerate in a neighbourhood of (0, 0), but degenerates at some points, which can be seen where it is intersected by other singular curves.
Example 5.6
Weakly regular characteristic singularities: these are all given by data of the form \(\kappa \equiv 0\), \(\beta (y)=y\) and an arbitrary choice of \(\alpha \) with \(\alpha (0) \ne 0\). The singular curve is guaranteed to be non-degenerate in a neighbourhood of (0, 0). An example is shown in Fig. 2.
6 Examples and numerics
In this section, we use numerics to give a picture of some degenerate singularities, as well as to show the global appearance of solutions generated by certain types of singular curve.
6.1 Degenerate singularities
In Example 4.7, we saw some degenerate singularities where \(\kappa \) vanishes or \(|\tau |\) takes the value 1 on a singular curve generated by Theorem 4.3. Theorem 4.2 is slightly more general, and the condition for a degenerate singularity for the potential \(\hat{\eta } = (-B(t)e_1 \lambda ^{-1} - \beta (t)/2 e_3 + A(t) e_1 \lambda ) \text {d}t\) is that \(\beta \) vanishes. Two examples are shown in Fig. 9. Both are degenerate cone points.
6.2 Singularities where the derivative vanishes
Any potential pair \((\hat{\chi }, \hat{\psi })\) with \(\chi _1(x_0)= \psi _{-1}(y_0) =0\) produces a pseudospherical frontal the derivative of which has rank 0 at \((x_0,y_0)\). Examples computed with \(\hat{\chi } = \hat{\psi } = (-B(t)e_1 \lambda ^{-1} - \beta (t)/2 e_3 + A(t) e_1 \lambda ) \text {d}t\), \(\beta (t)=1\) and A and B vanishing are shown in Fig. 10.
6.3 Global properties of solutions
If we consider a surface generated by singular curve data \((\kappa , \tau )\), where \(|\kappa (t)| \rightarrow \infty \) as \(t \rightarrow \pm \infty \) and \(\tau \) is bounded, then the solution becomes concentrated spatially for large (u, v), with a spiral in the u direction and many singularities in the v direction. This means that computing a finite subdomain gives a realistic sense of what the surface looks like, as in Fig. 7. More examples are shown in Fig. 11.
Notes
Currently available at http://davidbrander.org/software.html.
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Brander, D. Pseudospherical surfaces with singularities. Annali di Matematica 196, 905–928 (2017). https://doi.org/10.1007/s10231-016-0601-8
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DOI: https://doi.org/10.1007/s10231-016-0601-8
Keywords
- Differential geometry
- Integrable systems
- Loop groups
- Pseudospherical surfaces
- Constant Gauss curvature
- Singularities