A_{k} singlarities of wave fronts.

singlarities of wave fronts.

Kentaro Saji Department of Mathematics, Faculty of Educaton, Gifu University, Yanagido 1-1, Gifu 151-1193, Japan ksaji@gifu-u.ac.jp Masaaki Umehara Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan umehara@math.sci.osaka-u.ac.jp  and  Kotaro Yamada Faculty of Mathematics, Kyushu University, Higashi-ku, Fukuoka 812-8581, Japan kotaro@math.kyushu-u.ac.jp Dedicated to Professor Yoshiaki Maeda on the occasion of his sixtieth birthday.
October 25, 2008
Abstract.

In this paper, we discuss the recognition problem for -type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role in generalizing the authors’ previous work “the geometry of fronts” for surfaces. The crucial point to prove our criteria for -singularities is to introduce a suitable parametrization of the singularities called the “-th KRSUY-coordinates” (see Section 3). Using them, we can directly construct a versal unfolding for a given singularity. As an application, we prove that a given nondegenerate singular point on a real (resp. complex) hypersurface (as a wave front) in (resp. ) is differentiably (resp. holomorphically) right-left equivalent to the -type singular point if and only if the linear projection of the singular set around into a generic hyperplane (resp. ) is right-left equivalent to the -type singular point in (resp. ). Moreover, we show that the restriction of a -map to its Morin singular set gives a wave front consisting of only -type singularities. Furthermore, we shall give a relationship between the normal curvature map and the zig-zag numbers (the Maslov indices) of wave fronts.

2000 Mathematics Subject Classification:
Primary 57R45; Secondary 57R35, 53D12.
Kentaro Saji was supported by the Grant-in-Aid for Young Scientists (Start-up) No. 19840001, from the Japan Society for the Promotion of Science. Masaaki Umehara and Kotaro Yamada were supported by the Grant-in-Aid for Scientific Research (A) No. 19204005 and Scientific Research (B) No. 14340024, respectively, from the Japan Society for the Promotion of Science.

1. Introduction

Throughout this paper, we denote by the real number field or the complex number field . Let and be positive integers. A map is called -differentiable if it is a -map when , and is a holomorphic map when . Let be the -inner product given by

Let be the -projective space and the canonical projection. By using the above -inner product, the -projective cotangent bundle has the following identification

which has the canonical -contact structure. Let be a domain and

a Legendrian immersion, where is a locally defined -differentiable map into such that

In this situation, is called a wave front or front, and is called the -normal vector field of . If , might vanish. A point is called a singular point if the front is not an immersion at .

In this paper, we shall discuss the recognition problem for -type singularities on wave fronts. These are fundamental singularities on wave fronts (see [2]). We give a simple and computable necessary and sufficient condition that a given singular point on a hypersurface (as a wave front) in is -right-left equivalent to the -type singular point (; Theorem 2.4, Corollaries 2.5, 2.6 and 2.8 in Section 2), where two -differentiable map germs are -right-left equivalent if there exist -diffeomorphism germs and such that holds. Here, the -type singularity (or -front singularity) is a map germ defined by

(1.1)

at the origin, where , . The image of it coincides with the discriminant set of the versal unfolding

(1.2)

By definition, -front singularities are regular points. A -cusp in a plane is an -front singularity and a swallowtail in is an -front singularity.

When , useful criteria for cuspidal edges and swallowtails are given in [7]. We shall give a generalization of the criteria here. The crucial point to prove our criteria for -singularities is to introduce the “-th KRSUY-coordinates” as a generalization of the coordinates for cuspidal edges and swallowtails in [7]. Using them, we can directly construct a versal unfolding whose discriminant set coincides with the given singularity.

As an application, when , we show that the restriction of a -map into its Morin singular set (see the appendix) gives a wave front consisting of only -type singularities . Moreover, in the final section, we shall give a relationship between the normal curvature map and the zig-zag numbers (the Maslov indices) of wave fronts.

2. Criteria for -front singularities

Let be a domain and a front. Since we will work on a local theory, we may assume that the -normal vector field of is defined on . We define a -differentiable function on as the determinant

(2.1)

where (). A point is -singular (or singular) if , that is, is a singular point. A point is -nondegenerate (or non-degenerate) if is -singular and the exterior derivative does not vanish at . The following assertion is obvious:

Lemma 2.1.

The definition of -nondegeneracy is independent of the choice of local coordinate system and the choice of a -normal vector field . Moreover, if is -nondegenerate, the -linear map has a kernel of -dimension exactly one.

If is 1-nondegenerate, the singular set of

is an embedded -differentiable hypersurface of near . We denote by the -differentiable tangent bundle of -differentiable manifold . Then

is a -differentiable map. Since , we can take a sufficiently small neighborhood () of and non-zero -differentiable vector field on which belongs to the kernel of , that is for . We call a null vector field of . Moreover, we can construct a -differentiable vector field on (called an extended null vector field) whose restriction on the singular set gives a null vector field.

Remark 2.2.

Since on , we can write the components of explicitly using determinants of -submatrices of the Jacobian matrix of . Then this explicit expression of gives a -differentiable vector field on a sufficiently small neighborhood of a singular point. Thus, we get an explicit procedure to construct . For example, let be a front such that . Then

is an extended null vector field.

Let and be the derivatives , and , respectively. Then by definition, we have

(2.2)

where . The following assertion holds:

Lemma 2.3.

The singular set of has the following expressions:

Proof.

A point belongs to if and only if there exists such that . Since , the vector must be proportional to . Thus we get . Since is a level set of , if and only if , which proves the first equation. By (2.2) and by on , the derivative of with respect to vanishes on , that is . Next we suppose (). Since on , by taking a new coordinate system on such that , we have

where is the Jacobian. Consequently holds, that is . ∎

A -nondegenerate singular point is called -singular if . Moreover, a -singular point is -nondegenerate if on . In this case, is an embedded hypersurface of around if it is -nondegenerate.

Now, we define the -singularity and the -nondegeneracy inductively (): Firstly, we give the following notations:

where . We fix a -nondegenerate singular point of . Then the -st singular set

is an embedded hypersurface of around . (Here we replace by a sufficiently small neighborhood of at each induction step if necessary. In particular, we may assume .) We set

As in the proof of Lemma 2.3, by an inductive use of the identity

satisfies

(2.3)

Each point of is called a -singular point. A point is called -nondegenerate if on , that is

(2.4)

These definitions do not depend on the choice of a coordinate system of . Under these notations, a criterion for -front singularities is stated as follows:

Theorem 2.4.

Let be a front, where is a domain in . Then at is -right-left equivalent to the -front singularity if and only if is -nondegenerate but is not -singular .

Though -singular points are defined only for , we define any points are not -singular. This theorem is proved in Sections 3 and 4.

Corollary 2.5.

Let be a front. Then at is -right-left equivalent to the -front singularity if and only if

(2.5)

at and the Jacobian matrix of -differentiable map is of rank at .

Corollary 2.6.

Let be a domain in , a front, and a nondegenerate singular point. Take a local tangent frame field of the singular set (a smooth hypersurface in ) around and set

(2.6)

as the determinant function on where is a null vector field. Then is -singular if and only if . In particular, is -right-left equivalent to the -singularity if and only if . Moreover, a -singular point is -nondegenerate if and only if . Furthermore, is -right-left equivalent to the -singularity if and only if and hold. (If , then the null vector at is tangent to and is well-defined.)

Remark 2.7.

These assertions for have been already proved in [7]. The criterion for -singularities for general has been shown in [10].

Let a front, and a -nondegenerate singular point. Then is a submanifold of codimension in and Lemma 2.3 yields that the null vector field is a tangent vector field of , and we can set

Then these functions are -functions on .

Corollary 2.8.

Let be a domain in , a front, and a -nondegenerate singular point. Then is -right-left equivalent to the -singularity if and only if

(2.7)

hold, and the Jacobian matrix of the map is of rank at .

Remark 2.9.

In Corollary 2.8, the criterion for -singularities reduces to -nondegeneracy and

since the condition implies that the map is of rank at .

Proofs of these assertions are given in Section 4.

Let be a front and a -nondegenerate singular point. Then we can consider the restriction of into . Let us denote the limiting tangent hyperplane by

The following assertion can be proved straightforwardly.

Corollary 2.10.

Let be a front and a -nondegenerate singular point. For any vector , the following are equivalent:

  1. at is an -front singularity .

  2. The projection at is an -front singularity .

Here, is the normal projection with respect to to the hyperplane

and an -front singularity means a regular point.

The following assertion follows immediately from Theorem A.1 in the appendix.

Corollary 2.11.

Let be a domain in and a -differentiable map. Suppose that is a -nondegenerate singular point, namely, the exterior derivative of the Jacobian of does not vanish at . Then the following are equivalent:

  1. is an -Morin singular point of .

  2. is a front, and is an -front singularity of .

Here, the -Morin singularities are defined in the appendix, and the -Morin singularity means a regular point.

3. Adopted coordinates and -nondegeneracy

In [7], a certain kind of special coordinate system was introduced to treat the recognition problem of cuspidal edges and swallowtails in . In this section, we give a generalization of such a coordinate system around a singular point, which will play a crucial role.

Lemma 3.1 (Existence of the -th KRSUY-coordinate system).

Let be a front and a -nondegenerate singular point . Then there exists a -differentiable coordinate system around and a non-degenerate -affine transformation such that

satisfies the following properties:

  1. The point corresponds to .

  2. gives an extended null vector field on .

  3. If , then the -vector space is spanned by , that is

    where , .

  4. Suppose . If is -singular, then . On the other hand, if is not -singular, .

  5. and .

  6. for and for .

  7. If , holds for .

We call a coordinate system on satisfying the properties (), (), () and () a “-adopted coordinate system”, and we call by the “-th KRSUY-coordinate system” a pair of coordinate systems on and satisfying all above conditions. (In [7], the existence of the coordinates for , and was shown.)

Proof of Lemma 3.1.

By replacing to be a smaller neighborhood if necessary, we can take an extended null vector field on (see Remark 2.2). Now we assume that is -nondegenerate. Then is a submanifold of of codimension . So we can take a basis of such that

and

We now take a local coordinate system around such that (), and consider the following two involutive distributions and of rank and respectively

where is a sufficiently small neighborhood of . Since two distributions and span , the lemma in [6, page 182] yields the existence of local coordinate system around such that

Moreover, by a suitable affine transformation of , we may assume that () and (). Replacing to be sufficiently smaller, we may assume that this new coordinate system is defined on . Moreover, we may reset . Thus satisfies (), (), () and (). By a suitable affine transformation of , we may assume that

(3.1)

where denotes the -th canonical vector

(3.2)

of . We set . Then it holds that

(3.3)

where is Kronecker’s delta. Define new functions by

Since for (), the implicit function theorem yields that there exist functions () such that

If we set and

then gives a new local coordinate system of around such that

(3.4)
(3.5)

Since is the null direction, we have . Differentiating (3.4) with respect to , we have

Since the matrix is regular near by (3.3),

hold for . In particular, (3.5) holds for .

Thus if we set

then

holds which implies (). By (3.5), for . Hence we have () and (). Moreover,

which implies (). Now () and () follow immediately from (3.1) and (3.4) respectively. ∎

Now we use the notation under the -adopted coordinate system.

Lemma 3.2.

Let be a -nondegenerate singular point of . Under a -th adopted coordinate system , the following hold:

  1. If , then is -singular if and only if . Moreover is -nondegenerate if and only if

  2. If , then and

    (3.6)

    hold for . Moreover is -singular if and only if . Furthermore is -nondegenerate if and only if

Proof.

When , is -singular if and only if by (2.3). Suppose now is -singular (). By definition, is -nondegenerate if and only if . Since by (), this is equivalent to , which proves (1) and the latter part of (2). Next, we assume and prove the first part of (2): By (2.3), we have . By the assumption (), we have . In particular, is constant along these directions, which implies (3.6). On the other hand, since , is -nondegenerate. By (2.4), we have