Singularities of parallels to tangent developable surfaces

Given a surface in $\mathbf{R}^{3}$, a “parallel surface” or a ”parallel” to the surface is simply defined as a surface which have the common family of normal affine lines with the original surface. In fact, given a surface $(u,v)\mapsto f(u,v)$ with a unit normal $\nu(u,v)$, its parallels are given by the surfaces $f(u,v)+r\nu(u,v)$ with the parameter $r\in\mathbf{R}$.

In general a line congruence, i.e., a two-dimensional family of affine lines in $\mathbf{R}^{3}$ is called a system of rays if it forms a (possibly singular) Lagrangian surface in the space of affine lines in $\mathbf{R}^{3}$ [1, 2]. The condition is equivalent to that the family is expressed in a parametric form as

for some “frontal” surface $(u,v)\mapsto f(u,v)\in\mathbf{R}^{3}$ and its unit normal field $(u,v)\mapsto\nu(u,v)\in S^{2}$ so that $(u,v)\mapsto(f(u,v),\nu(u,v))\in\mathbf{R}^{3}\times S^{2}$ is Legendrian, i.e. it is an integral map for the standard contact structure on the unit tangent bundle $\mathbf{R}^{3}\times S^{2}$ of the Euclidean space $\mathbf{R}^{3}$.

Parallel deformations are regarded as an important and interesting transformations of surfaces in differential geometry. For instance, constant mean curvature surfaces and positive constant Gaussian curvature surfaces are related as parallels to each other. Parallels are studied deeply from geometric point of view of differential geometric surface theory (see [21] p.185, [18] p.225 for instance). It is characteristic also that, in the process of taking parallels, the surfaces may have singularities with a geometric origin. Singularites which appear in parallels are studied deeply from singularity theory viewpoint [5, 6, 17].

A surface in Euclidean 3-space is called developable if it is isometric to the plane and the class of developable surfaces. The class is characterized also as the surfaces with zero Gaussian curvature. Developable surfaces are roughly classified into cylinders, cones and tangent developable surfaces. A tangent developable surface is defined as a ruled surface by tangent lines to a space curve and has singularities at least along the space curve, called the directrix or the edge of regression ([11]). Note that the class of developable surfaces is preserved under the parallel transformation. Moreover any conical surface can be regarded as a tangent surface of the “vertex” i.e. a constant curve endowed with a tangential framing along the constant curve, while the class of cylindrical surfaces are preserved under the parallel transformations.

Then we are led naturally to ask which singularities appear on parallels to tangent developable surfaces of curves and the possibility to classify such generic singularities.

The notion of tangent developable surfaces is generalized to frontal curves. A possibly singular curve $t\mapsto f(t)$ is called a frontal curve if there exists a family of unit vector field $\tau(t)$ such that $f^{\prime}(t)$ is a scalar multiple of $\tau(t)$ for any $t$. A possibly singular surface $(u,v)\mapsto F(u,v)$ is called a frontal surface if there exists a family of unit vector field $\nu(u,v)$ such that $\nu(u,v)$ is orthogonal to both $F_{u}(u,v)$ and $F_{v}(u,v)$ for any $(u,v)$ (see §2).

Under the context mentioned above, we are going to study in this paper parallel frontals for the case when a system of rays in $\mathbf{R}^{3}$ is “degenerate”, i.e. when the corresponding Lagrangian surface in the space of affine lines in $\mathbf{R}^{3}$ is expressed as

where $f(t)$ is a frontal curve in $\mathbf{R}^{3}$, $\tau(t)$ is a unit tangential frame along $f$ and $\nu(t)$ is a unit normal to the surface $(t,s)\mapsto f(t)+s\tau(t)$ of $f$, the tangent developable surface, or simply, the tangent surface of $f$. Here the degeneracy means that the direction vector field $\nu(t)$ for the normal lines is independent of $s$.

The same idea is applied to frontal hypersurfaces or systems of rays in $\mathbf{R}^{n+1}$ and also even to $n$-dimensional frontal submanifolds or systems of “multi-dimensional rays” in $\mathbf{R}^{n+p}$ when the frontal submanifolds are assumed to be “normally flat” (see §2 and [16]).

Then, in fact, we clarify that any parallels to the tangent surface of a curve turns to be again a tangent surface of some curve, under a mild condition, and thus the class of tangent developable surfaces of curves is in fact preserved by the surface transformation by taking parallels. Furthermore we show that the observation remains true for frontal curves not only in $\mathbf{R}^{3}$ but also in $\mathbf{R}^{1+p},p\geq 2$ in general (see §2 and [16]). Actually we obtain the explicit formula for directrixes which produce parallels by taking tangent lines in terms of differential geometric data on the original curve (Theorem 2.2). In this paper we proceed to classify singularities which appear in parallel surfaces to the tangent surfaces of generic frontal curves in $\mathbf{R}^{3}$ and $\mathbf{R}^{4}$ (Theorem 3.6 and Theorem 3.8).

In the next section §2 we make clear the framework of this paper, introducing Theorem 2.2, and in §3 we formulate our classification results in this paper (Theorems 3.5, 3.6 and 3.8). In §4 we prove Theorem 2.2 briefly and study singularities of parallels to tangent surfaces to curves in $\mathbf{R}^{n+p}$ in general to show Theorems 3.5 3.6 and 3.8. In §5 and in §6 we study the classification problem of some singularities which appear in the list of Theorem 3.8.

All manifolds and mappings in this paper are assumed of class $C^{\infty}$ unless otherwise stated.

Let $M$ be an $n$-dimensional manifold and $\mathbf{R}^{n+p}$ the $(n+p)$-dimensional Euclidean space.

A map-germ $f:(M,a)\to\mathbf{R}^{n+p}$ at a point $a\in M$ is called a frontal-germ if there exists an orthonormal frame $\nu_{1},\nu_{2},\dots,\nu_{p}:(M,a)\to T\mathbf{R}^{n+p}$ along $f$ such that

for any $u\in M$ nearby $a$ and for any $i=1,2,\dots p$ ([14]). Hear $\cdot$ means Euclidean inner product and $f_{*}:TM\to T\mathbf{R}^{n+p}$ the differential of $f$. Then we consider the subbundle $N_{f}$ generated by $\nu_{1},\nu_{2},\dots,\nu_{p}$ as above in the pull-back bundle $f^{*}(T\mathbf{R}^{n+p})$. We call $N_{f}$ the normal bundle of the frontal-germ $f$.

A mapping $f$ is called a frontal if the germ of $f$ at any point $a\in M$ is a frontal-germ.

We remark that, in general, $N_{f}$ is not uniquely determined if $f$ is very degenerate. We call a mapping $f$ defined on $M$ proper if the singular locus $\Sigma(f)$, i.e. the non-immersive locus of $f$, is nowhere dense in $M$.

For a proper frontal $f:M\to\mathbf{R}^{n+p}$, the normal bundle $N_{f}$ is uniquely determined on $M$, because it is determined as the orthogonal complement to $f_{*}(TM)$ which is regarded as a subbundle of $f^{*}(T\mathbf{R}^{n+p})$ over the regular locus $M\setminus\Sigma(f)$ of $f$ ([14] Proposition 6.2). Then, in turn, the tangent bundle of the frontal $f$ is defined as the orthogonal complement to $N_{f}$ in $f^{*}(T\mathbf{R}^{n+p})$ over the whole $M$. Thus the pull-back bundle $f^{*}(T\mathbf{R}^{n+p})$ is decomposed into the sum $T_{f}\oplus N_{f}$ by the tangent bundle $T_{f}$ of rank $n$ and the normal bundle $N_{f}$ of rank $p$.

A frontal $f$ is called normally flat (resp. tangentially flat) if the induced connection on $N_{f}$ (resp. the induced connection on $T_{f}$) from the Euclidean connection on $f^{*}(T\mathbf{R}^{n+p})$ is flat ([19, 22]). If $f$ is normally flat, then there exists an orthonormal frame $\{\nu_{1},\dots,\nu_{p}\}$ along $f$ such that the covariant derivative of each $\nu_{i}$ by any vector field over $M$ belongs to $T_{f}$. We call such a frame a normally parallel orthonormal frame or Bishop frame of the normally flat frontal (see [4]). Normally parallel sections of $N_{f}$ form a $p$-dimensional vector space and any normally parallel section $\nu$ of $N_{f}$ is written as a linear combination $\nu=\sum_{i=1}^{p}r_{i}\nu_{i},(r_{1},\dots,r_{p}\in\mathbf{R})$ (See [16] Lemma 3.1). A normally parallel vector field $\nu$ is said to be generic if $r_{1},\dots,r_{p}$ are taken to be generic.

Frontal curves and hypersurfaces are normally flat. In [16], we define, for a given normally flat frontal in a Euclidean space, its parallel frontals using normally flat frame (Bishop frame). Moreover it is shown that the tangent surface of a frontal curve which is generated by tangent lines to the curve is normally flat, provided it is a frontal, and every parallel to the tangent surface turns to be right equivalent to the tangent surface of a frontal curve, called the directrix or the edge of regression.

Let $f:M\to\mathbf{R}^{n+p}$ be a normally flat frontal, and $\{\nu_{1},\dots,\nu_{p}\}$ a Bishop frame of $N_{f}$ defined over $M$. Then we define the parallels to $f$ by

Then the parallels are normally flat and have the same Bishop frame with $f$.

Parallels are basic and interesting objects to be studied in the cases of both hypersurfaces and curves [6, 8].

Let $f:M\to\mathbf{R}^{n+p}$ be a frontal. If $\{\tau_{1},\dots,\tau_{n}\}$ is a frame of the tangent bundle $T_{f}$ over $M$, then the tangent map ${\mbox{\rm Tan}}(f):M\times\mathbf{R}^{n}\to\mathbf{R}^{n+p}$ is defined by

The right equivalence class of ${\mbox{\rm Tan}}(f)$ is independent of the choice of $\{\tau_{1},\dots,\tau_{n}\}$. When $n=1$, then the tangent map is called a tangent surface.

Then we have that tangent surfaces of frontal curves are normally and tangentially flat:

Let $f:I\to\mathbf{R}^{1+p}$ be a frontal curve. A point $t\in I$ is called an inflection point of $f$ if there exists a unit frame $\tau:I\to S^{p}(\subset\mathbf{R}^{1+p})$ of $T_{f}$ such that $t$ is a singular point of $\tau$, regarded as a curve in the sphere $S^{p}$.

Suppose $F={\mbox{\rm Tan}}(f):I\times\mathbf{R}\to\mathbf{R}^{1+p}$ is a proper frontal and $T_{F}=T_{{\mbox{\rm Tan}}(f)}$ is the tangent bundle of ${\mbox{\rm Tan}}(f)$. Let $\tau$ be a unit section of $T_{f}$ regarded as a unit section of $T_{{\mbox{\rm Tan}}(f)}$ along $I\times\{0\}$. Note that ${\mbox{\rm Tan}}(f)(t,0)=f(t)$. Take an orthonormal frame $\{\tau,\mu\}$ of $T_{{\mbox{\rm Tan}}(f)}$ along $I\times\{0\}$. Then there exists unique function $\kappa:I\to\mathbf{R}$ with $\tau^{\prime}(t)=\kappa(t)\mu(t)$, which is called the curvature function of the frontal curve $f$ with respect the frame $\{\tau,\mu\}$. Remark that $t_{0}\in I$ is an inflection point of $f$ if and only if $\kappa(t_{0})=0$. Take a normally parallel orthonormal frame $\{\nu_{1},\dots,\nu_{p-1}\}$ of $N_{{\mbox{\rm Tan}}(f)}$ along $I\times\{0\}$. Then there exist functions $\ell_{1},\dots,\ell_{p-1}:I\to\mathbf{R}$ uniquely, which is called torsion functions of $f$, such that $\mu^{\prime}(t)=-\kappa(t)\tau(t)+\sum_{i=1}^{p-1}\ell_{i}(t)\nu_{i}(t)$. The frame $\{\nu_{1},\dots,\nu_{p-1}\}$ extends to a normally parallel orthonormal frame of $N_{{\mbox{\rm Tan}}(f)}$ over $I\times\mathbf{R}$ as $\nu_{i}(t,s)=\nu_{i}(t)$, by the constancy of $N_{{\mbox{\rm Tan}}(f)}$ along $\{t\}\times\mathbf{R}$.

Now let us consider parallels ${\mbox{\rm P}}_{r}({\mbox{\rm Tan}}(f))$ of ${\mbox{\rm Tan}}(f)$ generated by $\{\nu_{1},\dots,\nu_{p-1}\}$ of $N_{{\mbox{\rm Tan}}(f)}$. Then we have that parallels to tangent surfaces of frontal curves are again tangent surfaces of frontal curves as follows:

To give the classification results in this paper for singularities of parallels to tangent surfaces of frontal curves, we recall several necessary notions.

We define, for a curve $f:I\to\mathbf{R}^{1+p}$, the $(n+p)\times k$-Wronskian matrix of $f$ by

for any positive integer $k$. Note that the rank of $W_{k}(f)(t)$ is independent of the choices of local coordinate on $I$ and affine coordinates of $\mathbf{R}^{n+p}$.

By Theorem 2.2, the singularity of a parallel of ${\mbox{\rm Tan}}(f)$ are described by the tangent surface ${\mbox{\rm Tan}}(g)$ of a curve $g$, provided $f$ has no inflection points.

For the basic notions of singularity theory of mappings, see [7, 3] for instance.

Note that generic plane frontal curves $I\to\mathbf{R}^{2}$, $p=1$, may have inflection points, which are of type $(1,3)$.

Now we give the classification of types of singularities appearing in directrixes of parallels to tangent surfaces of generic frontal curves generated by its generic normally parallel normal fields.

It is known that, in several cases, singularities of tangent surfaces ${\mbox{\rm Tan}}(g)$ are described by types of $g$ ([11, 13]).

We give in this paper the exact generic classification of parallels to tangent developables of frontal curves in $\mathbf{R}^{3}$ and in $\mathbf{R}^{4}$.

Let $p=2$. Then the possible types are $(1,2,3),(1,2,4),(2,3,4),(2,3,5)$ and $(3,4,5)$ by Theorem 3.5. Therefore we have

The first column indicates the possible types of the directrix. The second column shows the names of right-left diffeomorphism class of singularities of parallels. The third column gives codimension of the locus of the type on the $(t,s;r)$-space $I\times\mathbf{R}\times\mathbf{R}$.

It is known that the diffeomorphism type of tangent developables to curves in $\mathbf{R}^{3}$ of types in Theorem 3.6 is uniquely determined except for the case $(2,3,5)$ ([10]). The tangent surfaces of curves of type $(2,3,5)$ fall into two diffeomorphism classes, the generic folded pleat and the non-generic folded pleat ([12]). We call a map-germ the generic folded pleat if it is diffeomorphic (right-left equivalent) to a map-germ $(\mathbf{R}^{2},0)\to(\mathbf{R}^{3},0)$ defined by

Note that the above normal form is diffeomorphic to that given in [12].

Let $p=3$. Then the possible types are

by Theorem 3.5. Thus we have

It is known the diffeomorphism type of tangent surfaces to curves of types $(1,2,3,4),(1,2,3,5)$ and $(2,3,4,5)$ in $\mathbf{R}^{4}$ is uniquely determined (see [13]). We show that it is also the case for $(3,4,5,6)$ (Proposition 5.1 of §5).

The singularities of tangent surfaces to curves of type $(2,3,4,6)$ turn to fall into several diffeomorphism classes (see §6). The generic normal form is given by

which is called the unfurled swallowtail.

The following bifurcation occurs by parallel deformations of tangent surfaces of curves of type $(2,3,5,6)$ (resp. $(3,4,5,6)$):

We recall here a sufficient condition on a curve to be a frontal.

(2) We write $\tau^{\prime}(t)=a(t)\tau(t)+\kappa(t)\mu(t)$ for some functions $a,\kappa$. Since $\tau^{\prime}(t)\cdot\tau(t)=0$ and $\mu(t)\cdot\tau(t)=0$, we see $a(t)=0$. The tangent map ${\mbox{\rm Tan}}(f)$ is given by $(t,s)\mapsto f(t)+s\tau(t)$ up to diffeomorphisms preserving $(\mathbf{R},0)\times 0$. The Jacobi matrix of ${\mbox{\rm Tan}}(f)$ is given by $(f^{\prime}(t)+s\tau^{\prime}(t),\tau(t))$. The ideal generated by $2\times 2$-minors of $(f^{\prime}(t)+s\tau^{\prime}(t),\tau(t))$ is equal to that for $s\kappa(t)(\mu(t),\tau(t))$ and it is generated by the function $s\kappa(t)$. By the argument in (1), we see that the ideals generated by $st^{a_{2}-a_{1}-1}$ and by $s\kappa(t)$. Therefore in particular we have that ${\mbox{\rm ord}}_{0}\,\kappa=a_{2}-a_{1}-1$. Moreover $\kappa(0)=0$ if and only if $a_{2}-a_{1}\geq 2$ as required. $\Box$

The types appeared in Theorem 3.5 satisfy $a_{2}-a_{1}=1$. Therefore we see that generically a frontal curve $f$ has no inflection points and that ${\mbox{\rm Tan}}(f)$ is a proper frontal by Lemma 4.1.