Principal curvatures and parallel surfaces of wave fronts
Abstract.
We give criteria for which a principal curvature becomes a bounded function at nondegenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance squared functions of wave fronts. Moreover, we relate these singularities to some geometric invariants of fronts.
Key words and phrases:
principal curvature, singularity, wave front, parallel surface, extended distance squared function2010 Mathematics Subject Classification:
57R45, 53A05, 58K051. Introduction
Wave fronts in the Euclidean space are surfaces which may have certain singularities. Since wave fronts have a welldefined unit normal vector even at singularities, they might be considered as generalizations of immersed surfaces in . Recently, there have been several studies of wave fronts from differential geometric viewpoints (see [7, 9, 13, 16, 17, 19, 24, 25, 26, 28], for example). In particular, the behavior of Gaussian and mean curvature of wave fronts are well investigated, and relations between boundedness of Gaussian curvature near nondegenerate singular points and geometric invariants of wave fronts are known (cf. [17, 25]). For principal curvatures, Murata and Umehara [18] showed that at least one principal curvature is unbounded near a singular point. However, another principal curvature may be a bounded function. Hence it is natural to ask which properties of wave fronts determine boundedness of principal curvatures at singular points.
In this paper, we give an explicit criterion for which a principal curvature becomes a bouded function near nondegenerate singular points of wave fronts in terms of geometric invariants (Theorem 3.1). (This kind of criteria for the case of cuspidal edges is given in [28, Proposition 2.2].) For a bounded principal curvature, we can define a principal vector with respect to it. On the other hand, the image of the set of nondegenerate singular points (singular locus) is a curve on a wave front. Thus we can extend the notion of a line of curvature to a singular locus by using the principal vector and give a condition for the singular locus to be a line of curvature on wave fronts (Proposition 3.3).
As an application, we consider singularities of parallel surfaces on wave fronts. We studied parallel surfaces of cuspidal edges and gave a characterization for swallowtails appearing on parallel surfaces of cuspidal edges in terms of geometric properties of cuspidal edges in [28]. However, we have not characterized other singularities which appear on parallel surfaces of cuspidal edges or wave fronts, in their differential geometric contexts. Thus we show relations between types of singularities of parallel surfaces on wave fronts and geometric properties of initial wave fronts (Theorem 4.2). To characterize singularities, the notion of ridge points for wave fronts will play important roles. (Ridge points for regular surfaces are introduced by Porteous [21].) In addition, we consider constant principal curvature (CPC) lines near cuspidal edges. It is known that CPC lines correspond to the set of singular points of parallel surfaces ([5, 6]). Using parallel surfaces, we define special points (landmark in the sense of [22]) on cuspidal edge as cusps of CPC lines, which seems not to have appeared in the literature (Subsection 4.2).
Finally, we study the extended distance squared function on wave fronts. For the case of generic regular surfaces, singularities of extended distance squared functions correspond to types of singularities of parallel surfaces (cf. [5, Theorem 3.4]). However, for wave fronts, the same statement does not hold, in fact, different kinds of singularities (type) will appear (Theorem 5.3).
All maps and functions considered here are of class unless otherwise stated.
2. Preliminaries
2.1. Wave fronts
A map is called a wave front (or a front) if there exists a unit normal vector to such that the pair gives an immersion, where is a domain and denotes the unit sphere in (cf. [1, 13, 25]). A map is called a frontal if just a unit normal vector to exists. A point is said to be a singular point of if is not an immersion at . We denote by the set of singular points of .
For a frontal , the function as
is called the signed area density function (cf. [13, 25]). By the definition of , holds. We call nondegenerate if . Let be nondegenerate. Then there exist a neighbourhood of and a regular curve with such that is locally parametrized by . Moreover, there exists a vector field such that along . We call and the singular curve and the null vector field, respectively. Moreover, we call the image of the singular curve the singular locus.
A nondegenerate singular point is said to be of the first kind if is transverse to , that is, . Otherwise, it is said to be of the second kind ([17]). Moreover, we call a nondegenerate singular point of the second kind admissible if the singular curve consists of points of the first kind except at . Otherwise, we call nonadmissible.
Definition 2.1.
Let be a mapgerm around . Then at is a cuspidal edge if the mapgerm is equavalent to the mapgerm at , at is a swallowtail if the mapgerm is equivalent to the mapgerm at , at is a cuspidal butterfly if the mapgerm is equivalent to the mapgerm at , at is a cuspidal lips if the mapgerm is equivalent to the mapgerm at , at is a cuspidal beaks if the mapgerm is equivalent to the mapgerm at and at is a singularity (resp. singularity) if the mapgerm is equivalent to (resp. ) at , where two mapgerms are equivalent if there exist diffeomorphismgerms on the source and on the target such that holds.
We note that generic singularities of fronts are cuspidal edges and swallowtails and generic singularities of oneparameter bifurcation of fronts are cuspidal lips/beaks, cuspidal butterflies and singularities in addition to above two (see [1, 10]).
Remark 2.2.
Cuspidal edges are nondegenerate singular points of the first kind. On the other hand, swallowtails and cuspidal butterflies are of the admissible second kind (cf. [17]). Thus generic singularities of fronts are admissible.
Fact 2.3 ([11, 12, 13, 24]).
Let be a front germ, a unit normal to and a corank one singular point, namely, .

Suppose that is a nondegenerate singular point.

at is equivalent to a cuspidal edge if and only if .

at is equivalent to a swallowtail if and only if and .

at is equivalent to a cuspidal butterfly if and only if and .


Suppose that is a degenerate singular point.

at is equivalent to a cuspidal lips if and only if .

at is equivalent to a cuspidal beaks if and only if and .

Here is the signed area density function, the null vector field and the Hessian matrix of .
We note that there is a criterion for a cuspidal cross cap which appears on a frontal surface defined as a mapgerm equivalent to at ([4, Theorem 1.4]).
We recall behavior of curvatures of fronts near nondegenerate singular points . Let be a front and a unit normal vector. Let and denote the Gaussian and the mean curvature of a front . It is known that is unbounded near ([25, Corollary 3.5]). On the other hand, for the Gaussian curvature , it is known that is bounded near if and only if the second fundamental form vanishes along the singular curve ([25, Theorem 3.1]).
Next we recall behavior of principal curvature maps of a front at singular points. Let us assume that there are no umbilic points on . Then there exists a local coordinate system centered at such that and resp. and are linearly dependent on . In particular, the pair (resp. ) does not vanish at the same time ([18, Lemma 1.3]). Such a coordinate system is called a principal curvature line coordinate introduced in [18]. For this local coordinate system , we define the maps which are called the principal curvature maps ([18]) as the proportional ratio of the real projective line by
(2.1) 
Fact 2.4 ([18, Lemma 1.7]).
Let be a front and be the principal curvature maps. Then is a singular point if and only if either or holds.
By Fact 2.4, one principal curvature function of a wave front is bounded and the other is unbounded near a singular point.
2.2. Invariants of a cuspidal edge
Let be a frontal, a nondegenerate singular point and a unit normal vector. Then we can take the following local coordinate system around .
Definition 2.5 ([13, 17, 25]).
A local coordinate system centered at a singular point of the first kind
(resp. of the second kind) is called
adapted if it is compatible with the orientation of and satisfies the following conditions:
(1) the axis is the singular curve,
(2) (resp. with )
gives the null vector field on the axis,
(3) there are no singular points other than the axis.
Let be a cuspidal edge and an adapted coordinate system centered at . Since along the axis, there exists a map such that . We note that holds along the axis. Since on the axis by Fact 2.3, the pair gives a frame (cf. [17, 28]).
Lemma 2.6 ([28, Lemma 2.1]).
It holds that
where , , , , and .
For cuspidal edges, several geometric invariants are studied (for example, see [16, 17, 19, 25, 26, 27]). By using an adapted coordinate system and the frame , we set the following invariants along the axis:
, , and are called the singular curvature, the limiting normal curvature, the cuspidal curvature and the cuspdirectional torsion, respectively. See [7, 16, 17, 25, 27] for details of their geometric meanings. We note that these invariants can be defined on frontals with singular points of the first kind, and for , we can define it at singular points of the second kind (cf. [17, (1.2)]).
Lemma 2.7.
Under the above settings, , and can be expressed as
(2.2) 
along the axis, where depends on the orientation of the frame .
Proof.
One can check that can be expressed as above by defintions of functions. We show and can be written as the above formulas. Since is perpendicular to both and , can be written as .
First, we show that can be written as above. We note that holds on the axis. Since , on the axis is expressed as
on the axis.
Next, we consider . Since and on the axis, we see that
∎
It is known that does not vanish if is a cuspidal edge (cf. [17, Lemma 2.11]). In particular, never vanishes on the axis by Lemma 2.7. Take an adapted coordinate system with . Then holds on the axis (see Lemma 2.7). If , holds.
We define the following functions on as
(2.3) 
where . (The reason why can be defined as these forms is found in [28, page 55].) These functions are welldefined on . We remark that (resp. ) becomes (resp. ) if we change to . Let and be the Gaussian and the mean curvature of defined on . Then and hold. Thus we may treat and as principal curvatures of defined on . Here and can be expressed as
on the set of regular points. We note that hold on the set of regular points. If we take a principal curvature line coordinate ([18]), then fractional expressions of principal curvature maps as in (2.1) coincide with principal curvatures .
2.3. Invariants of a singular point of the second kind
Let be a frontal, a nondegenerate singular point of the second kind and a unit normal vector to . We fix an adapted coordinate system in the following (see Definition 2.5). Taking a null vector field , there exists a function on the axis with so that (see [17]). (We note that if is nonadmissible, holds on the axis, namely, .) Thus it follows that holds along the axis. On the other hand, since the axis gives the singular curve, there exists a function such that . Hence we have . We remark that and are linearly independent since holds on the axis.
Lemma 2.8.
Under the adapted coordinate system , and on can be written as
where , , , , and .
We now define two functions on by
(2.4) 
where
Since the Gaussian curvature and the mean curvature of satisfy and , we may regard as principal curvatures of on , where and are written as
on . We remark that hold on the set of regular points.
We put . This is a function on . It follows that
(2.5) 
holds along the axis (cf. [17]). We note that holds. It is known that does not vanish on the axis if and only if is a front ([17, Proposition 3.2]). We set
This is a geometric invariant called the normalized cuspidal curvature defined in [17]. By (2.5) and the definition of , we see that and hold if is a front.
Lemma 2.9.
Under the above conditions, the limiting normal curvature can be written as at if is of the admissible second kind.
Proof.
By [17, Proposition 1.9], , and , we get the conclusion. ∎
3. Principal curvatures, principal vectors and ridge points
3.1. Boundedness of a principal curvature
In this subsection, we consider boundedness of principal curvatures of fronts by using the above arguments.
Theorem 3.1.
Let be a front and a nondegenerate singular point.

Let be a cuspidal edge. If , then the principal curvature is a bounded function at . Moreover, .

Let be of the second kind. If , then the principal curvature is a bounded function at . Moreover, if is an admissible.
Converses are also true. Moreover, if one of is bounded at , then the another is unbounded.
Proof.
We prove the first asserion. Let be a front and a cuspidal edge. Take an adapted coordinate system centered at . We show the case of . In this case, holds along the axis. For the case of , one can show similarly.
We now assume that . Then by (2.2). Since and (2.3), we see that is a bounded function on and holds at . Conversely, we assume that the principal curvature is a bounded function near . In this case, it follows that is positive along the axis. This implies that is positive along the axis by (2.2). Unboundedness of near follows from the fact that the mean curvature is unbounded near .
Next, we prove the second assertion. Take an adapted coordinate system centered at a nondegenerate singular point of the second kind . Suppose that . It follows that holds near from (2.5). Since , and along the axis, it follows that and hold on the axis. Hence by (2.4), we have along the axis, and is a bounded function. By Lemma 2.9, we see that at if is admissible. The converse and unboundedness can be shown by using similar arguments to the first assertion. ∎
Remark 3.2.
We assume that is bounded near nondegenerate singular point . Although is unbounded near , is bounded near . In fact, can be rewritten as
on (cf. [28]). Thus is written as
In particular, is proportional to when is a cuspidal edge, and is proportional to when is of the second kind. Thus does not vanish.
3.2. Principal vectors and ridge points
By Theorem 3.1, one of of fronts can be defined as a bounded function near nondegenerate singular points. This implies there is a principal vector with respect to such a principal curvature at the singular point. Hence we consider explicit representation of the principal vector under an adapted coordinate system.
Let be a front, a singular point of the second kind and a unit normal vector to . Then we take an adapted coordinate system around . Assume that , namely, is a bounded function near in the following. We investigate the principal vector relative to .
Let and denote the first and the second fundamental matrices given by
The principal vector with respect to is a never vanishing vector satisfying . We can write this equation as
(3.1) 
We note that does not vanish at . Thus we can take the principal vector as
(3.2) 
by factoring out from (3.1). For the case of cuspidal edges, the principal vector with respect to is given as follows [28]:
(3.3) 
We can extend the notion of a line of curvature as follows. The singular locus is a line of curvature if the principal vector is tangent to .
Proposition 3.3.
Let be a front, a nondegenerate singular point and the singular curve passing through . Then the following assertions hold

Suppose that is a cuspidal edge. Then is a line of curvature of if and only if vanishes identically along .

Suppose that is of the second kind. Then can not be a line of curvature.
Proof.
First, we show assertion (1). Take an adapted coordinate system centered at a cuspidal edge satisfying . Assume that is bounded on . Then the principal vector relative to is given by (3.3). Since on the axis, can be written as
along the axis by Lemma 2.7. Thus vanishes on the axis if and only if vanishes along the axis, and we get the conclusion.
Next, we show (2). Take an adapted coordinate system around and assume that holds. In this case, is bounded on and the principal vector of is given as (3.2). The second component is written as
along the axis. Thus we have at . This implies that the axis can not be the line of curvature. ∎
Using the principal curvature and the principal vector relative to , we define ridge points for . Ridge points play important role to study parallel surfaces.
Definition 3.4.
Under the above settings, a point is called a ridge point if holds, where denotes the directional derivative of with respect to . Moreover, a point is called a kth order ridge point if and hold, where means the th directional derivative of with respect to .
4. Parallel surfaces of wave fronts
For the case of regular surfaces, principal curvatures relate to singularities of parallel surfaces. In this section, we consider singularities of parallel surfaces of fronts and give criteria in terms of principal curvatures and other geometric properties. Swallowtails on parallel surfaces of cuspidal edges are studied in [28]. Here we give criteria for other singularities on parallel surfaces of fronts.
4.1. Singularities of parallel surfaces of wave fronts
In this subsection, we shall deal with fronts which have singular points of the second kind (swallowtails, for example). Needless to say, the following arguments can be applied to the case of cuspidal edges.
Let be a front, a unit normal to and a nondegenerate singular point of the second kind. Then the paralle surface of is defined by , where is constant. We note that is also a front since is a unit normal to .
Lemma 4.1.
Let be a front, its unit normal vector and a nondegenerate singular point of . Suppose that is a bounded function near and . Then is a singular point of if and only if . Moreover, is nondegenerate singular point of if and only if is not a critical point of .
Proof.
We show the case that is of the second kind. Let be an adapted coordinate system centered at with the null vector field . Then the signed area density function of can be written as
by Lemma 2.8, where . Since does not vanish at , is a singular point of if and only if holds. Thus we may treat as the signed area density function of . Nondegeneracy follows from . ∎
Theorem 4.2.
Let be a front and be a nondegenerate singular point. Suppose that the principal curvature is a bounded function near and . Then for the parallel surface with , the following conditions hold.

Assume . Then the following hold

The mapgerm at is equivalent to a cuspidal edge if and only if is not a ridge point of .

The mapgerm at is equivalent to a swallowtail if and only if is a first order ridge point of .

The mapgerm at is equivalent to a cuspidal butterfly if and only if is a second order ridge point of .


Assume . Then the following hold

The mapgerm at is equivalent to a cuspidal lips if and only if and hold.

The mapgerm at is equivalent to a cuspidal beaks if and only if is a first order ridge point of , and hold.
Here is the Hessian matrix of at .

Proof.
Let be a front, a nondegenerate singular point of the second kind and a unit normal vector. Then we take an adapted coordinate system around . By Lemma 4.1, we can take the signed area density function of the parallel surface with as .
First, we prove the assertion (1). In this case, is a smooth curve near and there exists a null vector field of . We set , where are functions on . By Lemma 2.8, is written as