A_{k} singlarities of wave fronts.

# Ak singlarities of wave fronts.

Kentaro Saji Department of Mathematics, Faculty of Educaton, Gifu University, Yanagido 1-1, Gifu 151-1193, Japan Masaaki Umehara Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan  and  Kotaro Yamada Faculty of Mathematics, Kyushu University, Higashi-ku, Fukuoka 812-8581, Japan 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

 ⟨X,Y⟩\!\tinyK=n∑j=1xjyj(X=(x1,…,xn), Y=(y1,…,yn)∈Kn).

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

 PT∗Kn+1=Kn+1×Pn(K),

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

 L:=(f,[ν]):U⟶Kn+1×Pn(K)

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

 ⟨df(u),ν⟩\!\tinyK=0(u∈TU).

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) X⟼((k+1)tk+2+k∑j=2(j−1)tjxj,−(k+2)tk+1−k∑j=2jtj−1xj,X1)

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

 (1.2) F(t,u0,…,un):=tk+2+uktk+⋯+u1t+u0.

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 Ak+1-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) λ:=det(fx1,…,fxn,ν),

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

 S1:=S(f)={q∈U;λ(q)=0}

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

 f1:=f|S1:S1⟶Kn+1

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

 ~η=(det(f1x2f1x3f2x2f2x3),−det(f1x1f1x3f2x1f2x3),det(f1x1f1x2f2x1f2x2))

is an extended null vector field.

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

 (2.2) S1:={q∈U;f′(q)=0},

where . The following assertion holds:

###### Lemma 2.3.

The singular set of has the following expressions:

 S2 ={q∈S1;ηq∈TqS1}={q∈S1;λ′(q)=0}={q∈U;λ(q)=λ′(q)=0} ={q∈S1;f′′(q)=0}={q∈U;f′(q)=f′′(q)=0}.
###### 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

 λ′(q)={φdet(fz1,…,fzn−1,f′,ν)}′(q)=φ(q)det(fz1,…,fzn−1,f′′,ν)(q)=0,

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:

 λ(0) :=λ, λ(1) :=λ′, λ(l) :=dλ(l−1)(~η), f(0) :=f, f(1) :=f′, f(l) :=df(l−1)(~η),

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

 Sj−1:=S(fj−2)

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

 fj−1:=f|Sj−1:Sj−1⟶Kn+1.

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

 λ(j−1)(q) ={φdet(fz1,…,fzn−1,f′,ν)}(j−1)(q) =φ(q)det(fz1,…,fzn−1,f(j),ν)(q),

satisfies

 (2.3) Sj ={q∈Sj−1;ηq∈TqSj−1} ={q∈Sj−1;λ(j−1)(q)=0}={q∈U;λ(q)=⋯=λ(j−1)(q)=0} ={q∈Sj−1;f(j)(q)=0}={q∈U;f′(q)=⋯=f(j)(q)=0}.

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

 (2.4) TqSj−1⊄ker(dλ(j−1))q.

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) λ=λ′=⋯=λ(k−1)=0,λ(k)≠0

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) μ:=det(v1,…,vn−1,η),

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

 μ′=dμ(η),μ′′=dμ′(η),⋯,μ(k−1)=dμ(k−2)(η).

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) μ=μ′=⋯=μ(k−2)=0,μ(k−1)≠0at p

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

 μ(p)=μ′(p)=0,μ′′(p)≠0,

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

 n⊥:={v∈Tf(p)Kn+1;⟨v,n⟩\!% \tinyK=0}

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 k-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 k-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

 ^f(z1,…,zn)=(^f1,…,^fn+1)=Θ∘f(z1,…,zn)

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

 TpSi=Span\!\tinyK{∂zi+1,…,∂zn},

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

 ⎧⎨⎩Span\!\tinyK{ξk,…,ξn−1}=TpSkif k

and

 Span\!\tinyK{ξi+1,…,ξn−1,~ηp}=TpSi(1≤i≤k−1andk≥2).

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

 T1:=Span\!\tinyK{∂w1,∂w2,…,∂wn−1},∂wn∈Span\!\tinyK{~η}(=T2).

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) f(p)=0,ν(p)=e1,∂f∂wi(p)=ei+2(i=1,…,n−1),

where denotes the -th canonical vector

 (3.2) ej:=(0,…,0,1,0,…,0)(j=1,2,…,n+1)

of . We set . Then it holds that

 (3.3) ∂fj+2∂wl(p)=δjl(j,l=1,…,n−1),

where is Kronecker’s delta. Define new functions by

 gj(z1,…,zn−1;w1,…,wn):=fj+2(w1,…,wn)−zj(j=1,…,n−1).

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

 gj(z1,…,zn−1,φ1(z1,…,zn−1,wn),…,φn−1(z1,…,zn−1,wn),wn)=0.

If we set and

 wj:=φj(z1,…,zn−1,wn)(j=1,…,n−1),

then gives a new local coordinate system of around such that

 (3.4) zj =fj+2(φ1(z1,…,zn−1,zn),…,φn−1(z1,…,zn−1,zn),zn), (3.5) ∂wj∂zl(p) =δjl(j=1,…,n−1, l=1,…,n).

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

 0=n−1∑l=1∂fj+2∂wl∂φl∂zn+∂fj+2∂wn=n−1∑l=1∂fj+2∂wl∂φl∂znon S1.

Since the matrix is regular near by (3.3),

 ∂wl∂zn=∂φl∂zn=0(on S1 near p)

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

Thus if we set

 ^f(z1,…,zn):=f(φ1(z1,…,zn),…φn−1(z1,…,zn),zn),

then

 ∂^f∂zn=n−1∑j=1∂f∂wj∂φj∂zn+∂f∂wn=0on S1 near p,

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

 ∂^f∂zi(p)=(n−1∑j=1∂f∂wj(p)∂wj∂zi(p))+∂f∂wn(p)=∂f∂wi(p)=ei+2,

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

 (λ′z2(p),…,λ′zn−1(p),λ′′(p))≠0.
2. If , then and

 (3.6) λ(m−2)zm(p)=⋯=λ(m−2)zn−1(p)=λ(m−1)(p)=0

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

 (λ(k)zk+1(p),…,λ(k)zn−1(p),λ(k+1)(p))≠0.
###### 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