Boundedness of geometric invariants near a singularity which is a suspension of a singular curve

Luciana F. Martins , Kentaro Saji , Samuel P. dos Santos and Keisuke Teramoto Universidade Estadual Paulista (UNESP), Instituto de Biociências Letras e Ciências Exatas, R. Cristóvão Colombo, 2265, Jd Nazareth, 15054-000 São José do Rio Preto, São Paulo, Brazil luciana.martins@unesp.br samuel.paulino@unesp.br Department of Mathematics, Graduate School of Science, Kobe University, Rokkodai 1-1, Nada, Kobe, 657-8501, Japan saji@math.kobe-u.ac.jp Graduate School of Sciences and Technology for Innovation, Yamaguchi University, Yamaguchi, 753-8512, Japan kteramoto@yamaguchi-u.ac.jp

Abstract.

Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of diverge, in particular the boundedness about these invariants represent geometry of the surface and the curve. In this paper, we study boundedness and orders of several geometric invariants near a singular point of a surface which is a suspension of a singular curve in the plane and those of curves passing through the singular point. We evaluates the orders of Gaussian and mean curvatures and them of geodesic, normal curvatures and geodesic torsion for the curve.

Key words and phrases:

cuspidal edge; geodesic curvature; normal curvature; geodesic torsion

2020 Mathematics Subject Classification:

Primary 57R45; Secondary 58K05

Partly supported by the Japan Society for the Promotion of Science KAKENHI Grants numbered 18K03301, 19K14533, 22K03312, 22K13914, the Japan-Brazil bilateral project JPJSBP1 20190103 and the São Paulo Research Foundation grant 2018/17712-7.

1. Introduction

In this paper, we study boundedness of several geometric invariants near a singular point of a surface which is a suspension of a singular curve in the plane. More precisely, let $\sigma$ be an $\mathcal{A}$ -equivalence class of singular plane curve-germs. A $\sigma$ -edge is a map-germ $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ such that it is $\mathcal{A}$ -equivalent to $(u,v)\mapsto(u,c_{1}(v),c_{2}(v))$ , where $c=(c_{1},c_{2})$ is a representative of $\sigma$ , namely, a one-dimensional suspension of $\sigma$ . Here, two map-germs $h_{1},h_{2}:(\boldsymbol{R}^{m},0)\to(\boldsymbol{R}^{n},0)$ are $\mathcal{A}$ -equivalent if there exist diffeomorphisms $\Phi_{s}:(\boldsymbol{R}^{m},0)\to(\boldsymbol{R}^{m},0)$ and $\Phi_{t}:(\boldsymbol{R}^{n},0)\to(\boldsymbol{R}^{n},0)$ such that $h_{2}=\Phi_{t}\circ h_{1}\circ\Phi_{s}^{-1}$ . A cuspidal edge ( $\mathcal{A}$ -equivalent to the germ $(u,v)\mapsto(u,v^{2},v^{3})$ at the origin) and a $5/2$ -cuspidal edge ( $\mathcal{A}$ -equivalent to the germ $(u,v)\mapsto(u,v^{2},v^{5})$ ) are examples of $\sigma$ -edges, and $\sigma$ are a $3/2$ -cusp and $5/2$ -cusp respectively. If $\sigma$ is of finite multiplicity, then the $\sigma$ -edge is a frontal. A frontal is a class of surfaces with singular points, and it is well known that surfaces with constant curvature are frequently in this class. In these decades, there are several studies of frontals from the viewpoint of differential geometry and various geometric invariants at singular points are introduced (for instance [3, 5, 6, 7, 11, 12, 14]). If a surface is invariant under a group action on $\boldsymbol{R}^{3}$ , then $\sigma$ -edges will appear naturally. Singularities appearing on surfaces of revolution and a helicoidal surface are examples of such surfaces [13, 16]. Moreover, such singularities appear on the dual surface at cone like singular points of a constant mean curvature one surface in the de Sitter $3$ -space (see [8]).

In this paper, we study geometry of $\sigma$ -edges. For this, we consider two classes of singular map-germs, that we shall call $m$ -type and $(m,n)$ -type edges, the first including $(m,n)$ -type edges and also $\sigma$ -edges when $\sigma$ has finite multiplicity (see Section 2). One observes that $m$ -type edges are frontals. In order to proceed our study we find a normal form for each one of these map-germs preserving the geometry of the initial map, since we only use isometries in the target (Proposition 2.9). In [11, 14] the authors define singular, normal and cuspidal curvatures, as well as cuspidal torsion for frontals. In an analogous way, we define similar geometric invariants for $m$ -type edges, getting same names, except for the cuspidal curvature, which we called $(m,m+i)$ -cuspidal curvature. These invariants are related with the coefficients of the normal form given in Proposition 2.9. It is worth mention that these cuspidal curvatures are similar. In fact, we know that a frontal-germ is a front if and only if the cuspidal curvature is not zero. We conclude from Proposition 2.11 that a $m$ -type edge is a front if and only if the $(m,m+1)$ -cuspidal curvature is non zero at 0. In particular, we study orders of geometric invariants and geometric invariants of curves passing through the singular point. We evaluates the orders of Gaussian and mean curvatures (Theorem 2.17) and the minimum orders of geodesic, normal curvatures and geodesic torsion for a singular curve passing through the singular point (Theorem 3.5). These minimum orders are written in terms of singular, cuspidal and normal curvatures and the cuspidal torsion. As a corollary, we give the boundedness of these curvatures under certain generic conditions (Corollary 3.6).

2. Geometry of $\sigma$ -edges

We give several classes similar to $\sigma$ -edges. It includes $\sigma$ -edges, and these classes will be useful to treat. We recall that a map-germ $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ is a frontal if there exists a unit vector field $\nu$ along $f$ such that $\left\langle{df_{p}(X_{p})},{\nu(p)}\right\rangle=0$ holds at any $p\in(\boldsymbol{R}^{2},0)$ and any $X_{p}\in T_{p}\boldsymbol{R}^{2}$ , where $\left\langle{\cdot},{\cdot}\right\rangle$ is the canonical inner product of $\boldsymbol{R}^{3}$ . The vector field $\nu$ is called a unit normal vector field of $f$ . A map-germ $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ is an $m$ -type edge if it is $\mathcal{A}$ -equivalent to $(u,v^{m},v^{m+1}a(u,v))$ for a function $a(u,v)$ . A map-germ $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ is a $(m,n)$ -type edge ( $m<n$ ) if it is $\mathcal{A}$ -equivalent to $(u,v^{m},v^{n}h(u,v))$ , where $h(0,0)=1$ . This is equivalent to being $\mathcal{A}^{n}$ -equivalent to $(u,v^{m},v^{n})$ . Two map-germs are $\mathcal{A}^{n}$ -equivalent if their $n$ -jets at the origin are $\mathcal{A}$ -equivalent. We show:

Lemma 2.1.

Let $f$ be a $\sigma$ -edge $($ respectively, an $m$ -type edge, an $(m,n)$ -type edge $)$ . Then an intersection curve of $f$ with a surface $T$ which is transversal to $f(S(f))$ passing through $p\in S(f)$ near $0$ is $\mathcal{A}$ -equivalent to $\sigma$ $($ respectively, $\mathcal{A}^{m}$ -equivalent to $(t^{m},0)$ , $\mathcal{A}^{n}$ -equivalent to $(t^{m},t^{n}))$ .

Proof.

Since the assumption and the assertion do not depend on the choice of the coordinate systems, we can assume $f$ is given by $(u,c_{1}(v),c_{2}(v))$ , where $c=(c_{1},c_{2})$ is $\mathcal{A}$ -equivalent to $\sigma$ . Then $T$ can be represented by the graph $\{(x,y,z)\,|\,x=h(y,z)\}$ in $(\boldsymbol{R}^{3},0)$ as the $xyz$ -space, and the intersection curve is $(h(c_{1}(v),c_{2}(v)),c_{1}(v),c_{2}(v))$ . Since $T$ is transverse to the $x$ -axis, the orthogonal projection of $T$ to the $yz$ -plane is a diffeomorphism, we see the assertion. One can show the other claims by the similar way. ∎

2.1. A sufficient condition

We give a sufficient condition of a frontal-germ being an $m$ or $(m,n)$ -type edge under the assumption $n<2m$ . We assume $n<2m$ throughout this subsection. Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be a frontal-germ satisfying $\operatorname{rank}df_{0}=1$ . Then there exists a vector field $\eta$ such that $\eta_{p}$ generates $\ker df_{p}$ if $p\in S(f)$ . We call $\eta|_{S(f)}$ a null vector field, and $\eta$ an extended null vector field. An extended null vector field is also called a null vector field if it does not induce a confusion. We assume that the set of singular points $S(f)$ is a regular curve, and the tangent direction of $S(f)$ is not in $\ker df_{0}$ . Let $\xi$ be a vector field such that $\xi_{p}$ is a non-zero tangent vector of $S(f)$ for $p\in S(f)$ . We consider the following conditions for $(\xi,\eta)$ :

[2.1]

$\eta^{i}f=0$ $(1\leq i\leq m-1)$ on $S(f)$ ,
[2.2]

$\operatorname{rank}(\xi f,\eta^{m}f)=2$ on $S(f)$ ,
[2.3]

$\operatorname{rank}(\xi f,\eta^{m}f,\eta^{i}f)=2$ $(m<i<n)$ on $S(f)$ ,
[2.4]

$\operatorname{rank}(\xi{f},\eta^{m}f,\eta^{n}f)=3$ at $p$ .

Here, for a vector field $\zeta$ and a map $f$ , the symbol $\zeta^{i}f$ stands for the $i$ -times directional derivative of $f$ by $\zeta$ . Moreover, for a coordinate system $(u,v)$ and a map $f$ , the symbol $f_{v^{i}}$ stands for $\partial^{i}f/\partial v^{i}$ .

Proposition 2.2.

Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be a frontal-germ satisfying $\operatorname{rank}df_{0}=1$ . Assume that the set of singular points $S(f)$ is a regular curve, and the tangent direction of $S(f)$ generated by $\xi$ is not in $\ker df_{0}$ . If there exists a null vector field $\eta$ satisfying [2.1], and $(\xi,\eta)$ satisfies [2.2], then $f$ is an $m$ -type edge. Moreover, if $(\xi,\eta)$ also satisfies [2.3]–[2.4], then $f$ is an $(m,n)$ -type edge.

As we will see, the conditions [2.2]–[2.4] does not depend on the choice of null vector field $\eta$ satisfying [2.1]. To show this fact, we show several lemmas which we shall need later. Firstly we show that the conditions does not depend on the choice of the diffeomorphism on the target. In what follows in this section, $f$ is as in Proposition 2.2.

Lemma 2.3.

Let $\Phi$ be a diffeomorphism-germ on $(\boldsymbol{R}^{3},0)$ , and set $\hat{f}=\Phi(f)$ . If $f$ and $(\xi,\eta)$ satisfy the condition $C$ , then $\hat{f}$ and $(\xi,\eta)$ satisfy $C$ , where $C=\{\ref{itm:cri1}\}$ , $C=\{\ref{itm:cri1},\ref{itm:cri2}\}$ , $C=\{\ref{itm:cri1}$ – $\ref{itm:cri3}\}$ and $C=\{\ref{itm:cri1}$ – $\ref{itm:cri4}\}$ .

Proof.

Let us assume $\eta$ satisfies [2.1]. By a direct calculation, we have $\eta\hat{f}=d\Phi(f)\eta f$ , and

\eta^{i}\hat{f}=\sum_{j=0}^{i-1}c_{ij}\eta^{j}(d\Phi(f))\eta^{i-j}f\quad(c_{ij}\in\boldsymbol{R}\setminus\{0\}).

(2.1)

By [2.1], $\eta^{i}\hat{f}=0$ $(2\leq i\leq m-1)$ and $\eta^{m}\hat{f}=d\Phi(f)\eta^{m}f$ on $S(f)$ . Then we see the assertion for the cases $C=\{\ref{itm:cri1}\}$ and $C=\{\ref{itm:cri1},\ref{itm:cri2}\}$ . We assume $\eta$ satisfies [2.1]–[2.3]. By (2.1) and $c_{1n}=1(\neq 0)$ we see the assertion. ∎

It is clear that the conditions [2.2]–[2.4] do not depend on the choice of $\xi$ , i.e., non-zero functional multiple and extension other than $S(f)$ . Moreover, it does not depend on the non-zero functional multiple of $\eta$ :

Lemma 2.4.

Let $h$ be a non-zero function. If $f$ and $(\xi,\eta)$ satisfy the condition $C$ , then $f$ and $(\xi,\hat{\eta})$ satisfy $C$ , where $\hat{\eta}=h\eta$ and $C$ is the same as those in Lemma 2.3.

Proof.

Since $(h\eta)^{i}f$ is a linear combination of $\eta f,\ldots,\eta^{i}f$ , and the coefficient of $\eta^{i}f$ is $h^{i}$ , we see the assertion. ∎

A coordinate system $(u,v)$ satisfying $S(f)=\{v=0\}$ , $\eta|_{S(f)}=\partial_{v}$ is said to be adapted.

Lemma 2.5.

Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be a frontal-germ satisfying $\operatorname{rank}df_{0}=1$ . We assume that the set of singular points $S(f)$ is a regular curve. For any null vector field $\eta$ , there exists an adapted coordinate system $(u,v)$ such that $\eta=\partial_{v}$ for any $(u,v)$ .

In this lemma, we do not assume that $f$ is an $m$ -type edges.

Proof.

Since $\operatorname{rank}df_{0}=1$ , one can easily see that there exists a coordinate system $(u,v)$ such that $\eta=\partial_{v}$ for any $(u,v)$ . Since $S(f)$ is a regular curve, and the tangent direction of it is not in $\ker df_{0}$ , $S(f)$ can be parametrized as $(u,a(u))$ . Define a new coordinate system $(\tilde{u},\tilde{v})$ by $\tilde{u}=u$ and $\tilde{v}=v-a(u)$ . Then $S(f)=\{\tilde{v}=0\}$ and $\partial/\partial\tilde{v}=\partial/\partial v$ hold. This shows the assertion. ∎

Lemma 2.6.

If two null vector fields $\eta,\tilde{\eta}$ satisfy [2.1], and $(\xi,\eta)$ satisfies $C$ , then $(\xi,\tilde{\eta})$ also satisfies $C$ . Here, $C$ is the collection of the conditions: $C=\{\ref{itm:cri2}\}$ , $C=\{\ref{itm:cri2},\ref{itm:cri3}\}$ , $C=\{\ref{itm:cri2}$ – $\ref{itm:cri4}\}$ .

Proof.

Let us assume that $\eta$ and $\tilde{\eta}$ satisfy [2.1]. Since the assumption [2.1] and the assertion do not depend on the choice of the coordinate system on the source by Lemma 2.5, we take $(u,v)$ an adapted coordinate system with $\eta=\partial_{v}$ for any $(u,v)$ and $\xi=\partial_{u}$ . Since $f_{v}=\cdots=f_{v^{m-1}}=0$ on the $u$ -axis, $f_{v}$ has the form $f_{v}=v^{m-1}\psi(u,v)$ . If the pair $(\xi,\eta)$ satisfies [2.2], then $\operatorname{rank}(f_{u},\psi)=2$ on the $u$ -axis. On the other hand, any null vector field is written as $a_{1}(u,v)\partial_{u}+a_{2}(u,v)\partial_{v}$ , $(a_{1}(u,0)=0,\,a_{2}(u,v)\neq 0)$ . By Lemma 2.4, dividing this by $a_{2}$ , we may assume an extended null vector field $\tilde{\eta}$ is

\tilde{\eta}=va(u,v)\partial_{u}+\partial_{v}.

Since it holds that $\tilde{\eta}^{2}f=0$ on the $u$ -axis (when $m>2$ ) and $f_{u}(u,0)\neq 0$ , we have $a(u,0)=0$ . Continuing this argument, we may assume

\tilde{\eta}=v^{m-1}a(u,v)\partial_{u}+\partial_{v}.

(2.2)

Thus $\tilde{\eta}f=v^{m-1}(af_{u}+\psi)$ holds, and $\tilde{\eta}^{m}f=(m-1)!(af_{u}+\psi)$ holds on the $u$ -axis. Therefore $(\xi,\tilde{\eta})$ satisfies [2.2]. We assume that the pair $(\xi,\eta=\partial_{v})$ satisfies [2.1]-[2.3] and $(\xi,\tilde{\eta})$ does [2.1], [2.2]. By this assumption, $\operatorname{rank}(f_{u},\psi)=2$ , $\operatorname{rank}(f_{u},\psi,\psi_{v^{i}})=2$ $(0<i<n-m)$ . By the form of $\tilde{\eta}$ , it holds that $\tilde{\eta}^{m+1}f=(m-1)!(a_{v}f_{u}+af_{uv}+\psi_{v})$ on the $u$ -axis. Since $f_{v}=0$ on the $u$ -axis, $f_{uv}=0$ on the $u$ -axis. Thus $\operatorname{rank}(\xi f,\tilde{\eta}^{m}f,\tilde{\eta}^{m+1}f)=2$ on the $u$ -axis. Similarly, $f_{v^{m-1}}=0$ on the $u$ -axis, $f_{uv^{2}}=\cdots=f_{uv^{m-1}}=0$ on the $u$ -axis. Thus if $i\leq m-1$ , then since $n<2m$ , we have $\tilde{\eta}^{m+i}f=(m-1)!(\psi_{v^{i}}+a_{v^{i}}f_{u})$ on the $u$ -axis. Thereby we have $\operatorname{rank}(\xi f,\tilde{\eta}^{m}f,\tilde{\eta}^{m+i}f)=2$ ( $i\leq m-2$ ) on the $u$ -axis. The last assertion can be shown by the same calculation. ∎

Proof of Proposition 2.2.

We assume $f$ satisfies the condition of the proposition, and $(\xi,\eta)$ satisfies the conditions [2.1] and [2.2]. Then we take an adapted coordinate system $(u,v)$ such that $\eta=\partial_{v}$ . By the proof of Lemma 2.6, there exist $p(u)$ and $q(u,v)$ such that $f(u,v)=p(u)+v^{m}q(u,v)$ , and $(p_{1})_{u}(0,0)\neq 0$ , where $p=(p_{1},p_{2},p_{3})$ . We set $U=p_{1}(u),V=v$ . Then $f$ has the form $(U,P_{2}(U),P_{3}(U))+V^{m}Q(U,V)$ . By a coordinate change on the target, $f$ has the form $(U,0,0)+V^{m}Q(U,V)$ , where $Q(U,V)=(0,Q_{2}(U,V),Q_{3}(U,V))$ . Rewriting the notation, we may assume $f$ is written as

f(u,v)=(u,v^{m}q_{2}(u,v),v^{m}q_{3}(u,v)).

On this coordinate system, $\partial_{v}$ satisfies the condition [2.1], it satisfies [2.2] by Lemma 2.6. This implies $(q_{2}(0,0),q_{3}(0,0))\neq(0,0)$ . So we assume $q_{2}(0,0)\neq 0$ . We set $U=u$ , $V=vq_{2}(u,v)^{1/m}$ . Rewriting the notation, we may assume $f$ is written as $(u,v^{m},v^{m}q_{3}(u,v))$ . By a coordinate change on the target, we may assume $f$ is written as $(u,v^{m},v^{m+1}q_{3}(u,v))$ . This proves the first assertion. We assume that $\eta$ also satisfies [2.3] and [2.4]. We may assume $f$ is written as $(u,v^{m},v^{m+1}q_{3}(u,v))$ . By Lemma 2.6, we may assume that $\partial_{v}$ satisfies [2.3] and [2.4]. By [2.3], the function $q_{3}(u,v)$ satisfies $q_{3}=(q_{3})_{v}=\cdots(q_{3})_{v^{n-m-1}}=0$ on the $u$ -axis. Thus $f$ is written as $(u,v^{m},v^{n}q_{4}(u,v))$ . By [2.4], it holds that $q_{4}\neq 0$ , and hence the assertion is proved. ∎

By the proof of Lemma 2.6, we have the following property:

Corollary 2.7.

Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be a frontal satisfying $\operatorname{rank}df_{0}=1$ and the set of singular points $S(f)$ is a regular curve. Furthermore, assume a vector field $\eta$ satisfying the condition [2.1] with $\partial_{v}$ satisfying [2.1]. Let $(u,v)$ be an adapted coordinate system. Then there exists $\psi$ such that $\eta f(u,v)=v^{m-1}\psi(u,v)$ .

2.2. Normal form of $m$ or $(m,n)$ -type edge

Given a curve-germ $\gamma:(\boldsymbol{R},0)\to(\boldsymbol{R}^{2},0)$ , if there exists $m$ such that $\gamma^{\prime}=t^{m-1}\rho$ ( $\rho(0)\neq 0$ ), then $\gamma$ at $0$ is said to be of finite multiplicity, and such an $m$ is called the multiplicity or the order of $\gamma$ at $0$ . Moreover, if there exists $n$ ( $n>m$ and $n\neq km$ , $k=2,3,\ldots$ ) such that $\gamma$ is $\mathcal{A}^{n}$ -equivalent to $(t^{m},t^{n})$ , then $\gamma$ is called of $(m,n)$ -type. This $(m,n)$ is well-defined since if $\gamma$ is $\mathcal{A}^{r}$ -equivalent to $(t^{m},0)$ then it is not $\mathcal{A}^{r}$ -equivalent to $(t^{m},t^{i})$ for $i\leq r$ , $i\neq km$ $(k=1,2,\ldots)$ . We simplify a curve-germ of $(m,n)$ -type and an $(m,n)$ -type edge by coordinate changes on the source and by special orthonormal matrices on the target. Let $(x,y)$ be the ordinary coordinate system of $(\boldsymbol{R}^{2},0)$ . A coordinate system $(u,v)=(u(x,y),v(x,y))$ is positive if the determinant of the Jacobi matrix of $(u(x,y),v(x,y))$ is positive. We have the following results.

Lemma 2.8.

Let $\gamma:(\boldsymbol{R},0)\to(\boldsymbol{R}^{2},0)$ be a curve germ satisfying $\gamma^{(i)}(0)=0$ $(i=1,\ldots,m-1)$ , and $\gamma^{(m)}(0)\neq 0$ . Then there exist a parameter $t$ and a special orthonormal matrix $A$ on $\boldsymbol{R}^{2}$ such that

A\gamma(t)=\left(t^{m},\ t^{m+1}b(t)\right).

(2.3)

Let $\gamma:(\boldsymbol{R},0)\to(\boldsymbol{R}^{2},0)$ be a curve germ of $(m,n)$ -type. Then there exist a parameter $t$ and a special orthonormal matrix $A$ on $\boldsymbol{R}^{2}$ such that

A\gamma(t)=\left(t^{m},\ \sum_{i=2}^{\lfloor n/m\rfloor}a_{i}t^{im}+t^{n}b(t)\right)\quad(b(0)\neq 0),

(2.4)

where $\lfloor k\rfloor$ is the greatest integer less than $k$ $($ in our convention, $n/m$ is not an integer $)$ .

Proof.

One can easily see the first assertion. We assume that $\gamma$ is a curve germ of $(m,n)$ -type, then we may assume $\gamma(t)=(t^{m},t^{m+1}b(t))$ . If $t^{m+1}b(t)$ has a term $t^{i}$ $(i<n,i\neq km)$ , then $j^{n}\gamma(0)$ is not $\mathcal{A}^{n}$ -equivalent to $(t^{m},t^{n})$ . This proves the assertion. ∎

Proposition 2.9.

Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be an $m$ -type edge. Then there exist a positive coordinate system $(u,v)$ and a special orthonormal matrix $A$ on $\boldsymbol{R}^{3}$ such that

Af(u,v)=\left(u,\dfrac{u^{2}a(u)}{2}+\dfrac{v^{m}}{m!},\dfrac{u^{2}b_{0}(u)}{2}+\dfrac{v^{m}}{m!}b_{m}(u,v)\right)\quad(b_{m}(0,0)=0).

(2.5)

Moreover, if $f$ is an $(m,n)$ -type edge, then there exist a positive coordinate system $(u,v)$ and a special orthonormal matrix $A$ on $\boldsymbol{R}^{3}$ such that

Af(u,v)=\left(u,\dfrac{u^{2}a(u)}{2}+\dfrac{v^{m}}{m!},\dfrac{u^{2}b_{0}(u)}{2}+\sum_{i=2}^{\lfloor n/m\rfloor}\dfrac{v^{im}}{(im)!}b_{im}(u)+\dfrac{v^{n}b_{n}(u,v)}{n!}\right),

(2.6)

$b_{n}(0,0)\neq 0$ .

Proof.

By the proof of Proposition 2.2, we may assume

f(u,v)=(u,u^{2}a_{2}(u)+v^{m}a_{2m}(u,v),u^{2}a_{3}(u)+v^{m}a_{3m}(u,v)).

By that proof again, $(a_{2m}(0,0),a_{3m}(0,0))\neq(0,0)$ . By a rotation on $\boldsymbol{R}^{3}$ , we may assume $a_{2m}(0,0)>0$ and $a_{3m}(0,0)=0$ . By a coordinate change $v\mapsto va_{2m}(u,v)^{1/m}$ , we may assume $f(u,v)=(u,u^{2}a_{2}(u)+v^{m}/m!,u^{2}a_{3}(u)+v^{m}a_{3m}(u,v))$ , $(a_{3m}(0,0)=0)$ . This proves the first assertion. If $f$ is an $(m,n)$ -type edge, then the function $a_{3m}(u,v)$ can be expanded by

\sum_{i=0}^{n-1}v^{i}b_{i}(u)+v^{n}b_{n}(u,v).

Since $f$ is an $(m,n)$ -type edge, the curve $v\mapsto f(u,v)$ is of $(m,n)$ -type for any $u$ near $0$ . This implies that $b_{i}(u)=0$ $(i\neq km,k\geq 1)$ . By $a_{3m}(0,0)=0$ , $b_{0}(u)=0$ . This proves the assertion. ∎

Each form (2.5) and (2.6) is called the normal form of an $m$ -type edge and an $(m,n)$ -type edge, respectively. Looking the first and the second components in (2.5) and (2.6), we remark that the $m$ -jet of the coordinate system $(u,v)$ which gives the normal form is uniquely determined up to $\pm$ when $m$ is even. Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be an $m$ -type edge and $\eta$ a null vector field which satisfies the condition [2.1]. Then the subspace $V_{1}=df_{0}(T_{0}\boldsymbol{R}^{2})$ and the subspace $V_{2}$ spanned by $df_{0}(T_{0}\boldsymbol{R}^{2})$ , $\eta^{m}f(0)$ do not depend on the choice of $\eta$ . We assume that the representation $f=(f_{1},f_{2},f_{3})$ of $xyz$ -space $\boldsymbol{R}^{3}$ satisfies that $V_{1}$ is the $x$ -axis and $V_{2}$ is the $xy$ -plane. Then the coordinate system $(u,v)$ gives the normal form (2.5) if and only if $f_{1}(u,v)=u$ and $(f_{2})_{uv}$ is identically zero.

2.3. Geometric invariants

2.3.1. Cuspidal curvatures

Let $f$ be an $m$ -type edge. A pair of vector fields $(\xi,\eta)$ is said to be adapted if $\xi$ is tangent to $S(f)$ , and $\eta$ is a null vector field. We take an adapted pair of vector fields $(\xi,\eta)$ such that $\eta$ satisfies the condition [2.1], and $(\xi,\eta)$ is positively oriented. One can show the existence of such a pair by the definition of $m$ -type edge. We define

\omega_{m,m+1}(t)=\frac{|\xi{f}|^{(m+1)/m}\det(\xi{f},\eta^{m}{f},\eta^{m+1}f)}{|\xi{f}\times\eta^{m}{f}|^{(2m+1)/m}}(\mu(t))

where $\mu$ is a parametrization of $S(f)$ . We call $\omega_{m,m+1}$ the $(m,m+1)$ -cuspidal curvature. We have the following lemma:

Proposition 2.10.

The function $\omega_{m,m+1}$ does not depend on the choice of $(\xi,\eta)$ satisfying the condition [2.1].

Proof.

Since it is not appeared in the formula, $\omega_{m,m+1}$ does not depend on the choice of the coordinate system. Let $(\xi,\eta)$ be an adapted pair of vector fields satisfying the condition [2.1]. It is clear that the function $\omega_{m,m+1}$ does not depend on the choice of $\xi$ . We take an adapted coordinate system $(u,v)$ satisfying $\partial_{v}=\eta$ . Then

\omega_{m,m+1}(u)=|f_{u}|^{(m+1)/m}\det(f_{u},f_{v^{m}},f_{v^{m+1}})|f_{u}\times f_{v^{m}}|^{-(2m+1)/m}.

By Corollary 2.7, we have $f_{v}=v^{m-1}\psi$ . Let $\tilde{\eta}$ be another null vector field satisfying the condition [2.1]. We see that $\omega_{m,m+1}$ does not depend on the non-zero functional multiples of $\eta$ , we may assume $\tilde{\eta}=a(u,v)\partial_{u}+\partial_{v}$ . By the proof of Lemma 2.6, we may assume that $\tilde{\eta}$ is

\tilde{\eta}=v^{m-1}a(u,v)\partial_{u}+\partial_{v}.

(2.7)

Then by $f_{v}=v^{m-1}\psi$ ,

\tilde{\eta}f=v^{m-1}(af_{u}+\psi).

Thus

\tilde{\eta}^{m}f=(m-1)!(af_{u}+\psi)+(m-1)(m-1)!v\eta(af_{u}+\psi)+v^{2}g(u,v),

(2.8)

where $g$ is a function, and

\tilde{\eta}^{m+1}f=(m-1)!\eta(af_{u}+\psi)+(m-1)(m-1)!\eta v\eta(af_{u}+\psi)=m!(\eta af_{u}+a\eta f_{u}+\eta\psi)

hold on the $u$ -axis. Since $\psi=((m-1)!)^{-1}f_{v^{m}}$ and $\psi_{v}=(m!)^{-1}f_{v^{m+1}}$ , we have

	$\displaystyle\frac{\|\xi{f}\|^{(m+1)/m}\det(\xi{f},\eta^{m}{f},\eta^{m+1}f)}{\|\xi{f}\times\eta^{m}{f}\|^{(2m+1)/m}((m-1)!)^{1/m}}(u,0)=$	$\displaystyle\frac{\|f_{u}\|^{(m+1)/m}\det(f_{u},\psi,\psi_{v})}{\|f_{u}\times\psi\|^{(2m+1)/m}}(u,0)$
	$\displaystyle=$	$\displaystyle\frac{\|f_{u}\|^{(m+1)/m}\det(f_{u},f_{v^{m}},f_{v^{m+1}})}{\|f_{u}\times f_{v^{m}}\|^{(2m+1)/m}}(u,0).$

This shows the assertion. ∎

We have the following proposition.

Proposition 2.11.

Let $f\colon(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be an $m$ -type edge. Then $f$ at $0$ is an $(m,m+1)$ -type edge if and only if $\omega_{m,m+1}\neq 0$ at $0$ .

Proof.

Since $f$ is an $m$ -type edge, by Proposition 2.9, we may assume that $f$ is given by the right-hand side of (2.5). Since $b_{m}(0,0)=0$ , there exist $c_{1}(u)$ and $c_{2}(u,v)$ such that $b_{m}(u,v)=c_{1}(u)+vc_{2}(u,v)$ . Since we can take $\eta=\partial_{v}$ , the function $\omega_{m,m+1}$ is a non-zero functional multiple of $c_{2}(u,0)$ . Then we see the assertion. ∎

It is easy to show that $(m,m+1)$ -type edges are fronts and that an $m$ -type edge is a front if and only if $\omega_{m,m+1}\neq 0$ . In Appendix A, we define $(m,n)$ -cuspidal curvature for a curve germ of $(m,n)$ -type, denoting it by $r_{m,n}$ . An intersection curve of $(m,m+1)$ -type edge $f$ with $T$ as in Lemma 2.1, is a curve-germ of $(m,m+1)$ -type. The following holds.

Corollary 2.12.

Let $f\colon(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be a $\sigma$ -edge, where $\sigma$ is $\mathcal{A}$ -equivalent to $v\mapsto(v^{m},v^{m+1})$ . Then the $(m,m+1)$ -cuspidal curvature $\omega_{m,m+1}$ at $0$ coincides with the $(m,m+1)$ -cuspidal curvature $r_{m,m+1}$ of the intersection curve $\rho$ of $f$ with a plane $P$ which is perpendicular to the tangent line to $f$ at $0$ .

Proof.

By the assumption, we may assume that $f$ is given by the normal form (2.5). Since $f_{u}(0,0)=(1,0,0)$ and $f_{v}(0,0)=(0,0,0)$ , the plane $P$ is given by $P=\{(0,y,z)\in\boldsymbol{R}^{3}\ |\ y,z\in\boldsymbol{R}\}$ . Thus the intersection curve $\rho$ can be parametrized by

\rho(v)=f(0,v)=\left(0,\frac{v^{m}}{m!},\frac{b_{m+1}(0,v)v^{m+1}}{(m+1)!}\right).

This can be considered as a normal form of a curve which is $\mathcal{A}^{m+1}$ -equivalent to $v\to(v^{m},v^{m+1})$ . Hence we have the assertion by Example A.2. ∎

Let $f$ be an $m$ -type edge. We assume $\omega_{m,m+1}$ is identically zero on $S(f)$ . Let $\mu(t)$ be a parametrization of $S(f)$ . We define

\omega_{m,m+2}(t)=\dfrac{|\xi f|^{(m+2)/m}\det(\xi f,\eta^{m}f,\eta^{m+2}f)}{|\xi f\times\eta^{m}f|^{(2m+2)/m}}(\mu(t)).

We will see this does not depend on the choice of $(\xi,\eta)$ which satisfies the conditions [2.1] and [2.2] in Proposition 2.2 and $\det(\xi{f},\eta^{m}f,\eta^{j}f)=0$ $(j<m+2)$ . Inductively, we define $\omega_{m,m+i}$ when $\omega_{m,m+j}=0$ $(j\leq i-1)$ by

\omega_{m,m+i}=\dfrac{|\xi f|^{(m+i)/m}\det(\xi f,\eta^{m}f,\eta^{m+i}f)}{|\xi f\times\eta^{m}f|^{(2m+i)/m}}(\mu(t)).

We will also see this does not depend on the choice of $(\xi,\eta)$ satisfying the conditions [2.1] and [2.2] in Proposition 2.2 and $\det(\xi{f},\eta^{m}f,\eta^{j}f)=0$ $(j<m+i)$ . If $i=m$ , we set $\beta_{m,2m}=\omega_{m,2m}$ .

Proposition 2.13.

Under the assumption $\omega_{m,m+1}=\cdots=\omega_{m,m+i-1}=0$ , the function $\omega_{m,m+i}$ $(i=1,\ldots,m-1)$ does not depend on the choice of the pair $(\xi,\eta)$ which satisfies the conditions [2.1] and [2.2] in Proposition 2.2 and $\det(\xi{f},\eta^{m}f,\eta^{m+j}f)=0$ $(1\leq j<i)$ .

Proof.

We already showed the case $i=1$ in Proposition 2.10. Let $(\xi,\eta)$ be a pair of vector fields satisfying the assumption of lemma. We take an adapted coordinate system $(u,v)$ such that $\partial_{v}=\eta$ . By the proof of Lemma 2.6, we see $f_{v}=v^{m-1}\psi$ .

Moreover, we have:

Lemma 2.14.

There exist functions $\alpha,\beta$ , and a vector valued function $\theta$ such that

\psi_{v}=\alpha f_{u}+\beta\psi+v^{i-1}\theta.

(2.9)

Proof.

Since $f_{v^{m+1}}=(m-1)!\psi_{v}$ on the $u$ -axis, $\omega_{m,m+1}=0$ implies that there exists $\alpha_{1},\beta_{1},\theta_{1}$ such that $\psi_{v}=\alpha_{1}f_{u}+\beta_{1}\psi+v\theta_{1}$ . We assume that there exist $\alpha_{k},\beta_{k},\theta_{k}$ such that $\psi_{v}=\alpha_{k}f_{u}+\beta_{k}\psi+v^{k}\theta_{k}$ $(k=1,\ldots,i-2)$ . Differentiating this equation, we have

\psi_{v^{k+1}}=\sum_{l=0}^{k}{}{\begin{pmatrix}k\\ l\end{pmatrix}}\Big{(}(\alpha_{k})_{v^{l}}f_{uv^{k-l}}+(\beta_{k})_{v^{l}}\psi_{v^{k-l}}+(v^{k})_{v^{l}}(\theta_{k})_{v^{k-l}}\Big{)}

Thus by $f_{uv}=\cdots=f_{uv^{m-1}}=0$ holds, and $\psi_{v^{j}}\in\left\langle f_{u},\psi\right\rangle_{\boldsymbol{R}}$ $(j\leq k)$ on the $u$ -axis, we have $2=\operatorname{rank}(f_{u},\psi,\psi_{v^{k+1}})=\operatorname{rank}(f_{u},\psi,\theta_{k})$ on the $u$ -axis. Hence there exist functions $\alpha_{k+1},\beta_{k+1}$ , and a vector valued function $\theta_{k+1}$ such that $\theta_{k}=\alpha_{k+1}f_{u}+\beta_{k+1}\psi+v\theta_{k+1}$ . This shows the assertion. ∎

We continue the proof of Proposition 2.13. Since the assertion holds by multiplying the null vector field by a non-zero function, we take a null vector field $\eta$ as in the right-hand side of (2.7). By the same calculations in the proof of Proposition 2.10, we have $\eta f=v^{m-1}(af_{u}+\psi)$ . Thus

\eta^{m+i}f=\sum_{k=0}^{m+i-1}{\begin{pmatrix}m+i-1\\ k\end{pmatrix}}\eta^{k}v^{m-1}\eta^{m+i-1-k}(af_{u}+\psi).

Since $\eta^{k}v^{m-1}=0$ if $k\neq m-1$ and $\eta^{k}v^{m-1}=(m-1)!$ ,

\eta^{m+i}f={\begin{pmatrix}m+i-1\\ m-1\end{pmatrix}}(m-1)!\eta^{i}(af_{u}+\psi).

Thus $\eta^{m+i}f=vg(u,v)+(af_{u}+\psi)_{v^{i}}$ , where $g$ is a function. By $f_{uv}=\ldots=f_{uv^{m-1}}=0$ holds, and $\psi_{v^{j}}\in\left\langle f_{u},\psi\right\rangle_{\boldsymbol{R}}$ $(j\leq k)$ on the $u$ -axis by Lemma 2.14, we have

\displaystyle\dfrac{|\xi f|^{(m+i)/m}\det(\xi f,\eta^{m}f,\eta^{m+i}f)}{|\xi f\times\eta^{m}f|^{(2m+i)/m}}(u,0)=

\displaystyle\dfrac{|f_{u}|^{(m+i)/m}\det(f_{u},f_{v^{m}},f_{v^{m+i}})}{|f_{u}\times f_{v^{m}}|^{(2m+i)/m}}(u,0),

and this shows the assertion. ∎

We call $\omega_{m,m+i}$ the $(m,m+i)$ -cuspidal curvature, and $\beta_{m,2m}$ the $(m,2m)$ -bias. Note that $\beta_{m,2m}$ does not depend on the choice of $(\xi,\eta)$ satisfying [2.1], [2.2] and $\left\langle{\xi{f}},{\eta^{m}{f}}\right\rangle=0$ at $p$ by the same calculation. In this case, $a(0,0)=0$ by the additional assumption. If $f$ is an $m$ -type edge, and written as (2.5), then $\omega_{m,m+1}(0)=(m+1)(b_{m})_{v}(0,0)$ . If $f$ is an $(m,n)$ -edge $(n<2m)$ , and written as (2.6), then $\omega_{m,n}(0)=b_{n}(0,0)$ , and $\beta_{m,2m}(0,0)=b_{2m}(0)$ . See Appendix A for geometric meanings of the terms $b_{im}$ $(i=2,\ldots,{\lfloor n/m\rfloor})$ .

2.3.2. Singular, normal curvatures and cuspidal torsion

Let $f$ be an $m$ -type edge, and $\mu(t)$ be a parametrization of the singular set. Let $\nu$ be a unit normal vector field of $f$ , and we set $\lambda=\det(f_{u},f_{v},\nu)$ for an oriented coordinate system $(u,v)$ on $(\boldsymbol{R}^{2},0)$ . We set $\hat{\mu}=f\circ\mu$ . Then we define

\kappa_{s}(t)=\operatorname{sgn}\Big{(}\delta\ \eta^{m-1}\lambda(\mu(t))\Big{)}\dfrac{\det(\hat{\mu}^{\prime},\hat{\mu}^{\prime\prime},\nu(\mu))}{|\hat{\mu}^{\prime}|^{3}},\quad\kappa_{\nu}(t)=\dfrac{\left\langle{\hat{\mu}^{\prime\prime}},{\nu(\mu)}\right\rangle}{|\hat{\mu}^{\prime}|^{2}}

(2.10)

and

\kappa_{t}(t)=\dfrac{\det(\xi f,\eta^{m}f,\xi\eta^{m}f)}{|\xi f\times\eta^{m}f|^{2}}(\mu(t))-\dfrac{\det(\xi f,\eta^{m}f,\xi^{2}f)\left\langle{\xi f},{\eta^{m}f}\right\rangle}{|\xi f|^{2}|\xi f\times\eta^{m}f|^{2}}(\mu(t)),

(2.11)

where $\delta=1$ if $(\mu^{\prime},\eta)$ agrees the orientation of the coordinate system, and $\delta=-1$ if $(\mu^{\prime},\eta)$ does not agree the orientation. We call $\kappa_{s}$ , $\kappa_{\nu}$ and $\kappa_{t}$ singular curvature, normal curvature and cuspidal torsion, respectively. These definitions are direct analogies of [14, 11]. It is easy to see that the definitions (2.10) do not depend on the choice of parametrization of the singular curve. Moreover, $\kappa_{s}$ does not depend on the choice of $\nu$ , nor the choice of $\eta$ when $m$ is even. To see the well-definedness of $\kappa_{t}$ , we need:

Proposition 2.15.

The definition (2.11) does not depend on the choice of the adapted vector fields $(\xi,\eta)$ with $\eta$ which satisfies [2.1].

Proof.

One can easily to check it does not depend on the choice of functional multiplications of $\eta$ . Since the assertion does not depend on the choice of local coordinate system, one can choose an adapted coordinate system $(u,v)$ with $\partial_{v}$ satisfying [2.1]. Let $\eta$ be a null vector field which satisfies [2.1]. Then by the proof of Lemma 2.6, we may assume $\eta$ is given by (2.2). Then by (2.8), we see $\eta^{m}f=(m-1)!(af_{u}+\psi)$ on the $u$ -axis, where $\psi$ is given in the proof of Lemma 2.6. Furthermore, by (2.8), we see and $\xi\eta^{m}f=(m-1)!(a_{u}f_{u}+af_{u^{2}}+\psi_{u})$ on the $u$ -axis. Substituting these formulas into the right-hand side of (2.11), we see it is

\dfrac{\det(f_{u},\psi,\psi_{u})}{|f_{u}\times\psi|^{2}}(u,0)-\dfrac{\det(f_{u},\psi,f_{u^{2}})\left\langle{f_{u}},{\psi}\right\rangle}{|f_{u}|^{2}|f_{u}\times\psi|^{2}}(u,0),

and by $f_{v^{m}}=(m-1)!\psi$ , this shows the assertion. ∎

If an $m$ -type edge $f$ is given by the form (2.5), then $\kappa_{s}(0)=a(0)$ , $\kappa_{\nu}(0)=b(0)$ and $\kappa_{t}(0)=(b_{m})_{u}(0,0)$ .

2.4. Boundedness of Gaussian curvature and mean curvature near an $m$ -type edge

Here we study the behavior of the Gaussian and mean curvatures.

Let $g:(\boldsymbol{R}^{i},0)\to\boldsymbol{R}$ be a function-germ ( $i=1,2)$ . If there exists an integer $n$ $(n\geq 1)$ such that $g\in\mathcal{M}_{i}^{n}$ and $g\not\in\mathcal{M}_{i}^{n+1}$ , then $g$ is said to be of order $n$ , where $\mathcal{M}_{i}=\{g\colon(\boldsymbol{R}^{i},0)\to\boldsymbol{R}\ |\ g(0)=0\}$ is the unique maximal ideal of the local ring of function-germs and $\mathcal{M}_{i}^{n}$ denotes the $n$ th power of $\mathcal{M}_{i}$ (cf. [9, p. 46]). If $g\not\in\mathcal{M}_{i}$ , then the order of $g$ is $0$ . The order of $g$ is denoted by $\operatorname{ord}(g)$ . If $g$ is of order $n$ $(n\geq 0)$ , then $g$ is said to be of finite order. Let $g_{1},g_{2}:(\boldsymbol{R}^{i},0)\to\boldsymbol{R}$ be two function-germs such that $g_{i}$ is of finite order. The rational order $\operatorname{ord}(f)$ of a function $f=g_{1}/g_{2}:(\boldsymbol{R}^{i}\setminus Z,0)\to\boldsymbol{R}$ , where $Z=g_{2}^{-1}(0)$ is

\operatorname{ord}(f)=\operatorname{ord}(g_{1})-\operatorname{ord}(g_{2}).

For a function $f=g_{1}/(|g_{2}|g_{3}):(\boldsymbol{R}^{i}\setminus Z,0)\to\boldsymbol{R}$ , we define $\operatorname{ord}(f)=\operatorname{ord}(g_{1})-\operatorname{ord}(g_{2})-\operatorname{ord}(g_{3})$ , where $Z=g_{2}^{-1}(0)\cup g_{3}^{-1}(0)$ . If $g_{1}\in\mathcal{M}_{i}^{\infty}$ , then we define $\operatorname{ord}(f)=\infty$ . If $\operatorname{ord}(f)=0$ , then $f$ is called rationally bounded, and $\operatorname{ord}(f)=1$ , then $f$ is called rationally continuous ([12, Definition 3.4]). If $i=1$ , this is the usual one.

Since the property $g\in\mathcal{M}_{i}^{n}$ does not depend on the choice of coordinate system, the order and the rational order does not depend on the choice of coordinate system.

Let $f\colon(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be an $m$ -type edge, and let $(u,v)$ be an adapted coordinate system with $\partial_{v}$ satisfying [2.1]. We take $(m-1)!\psi$ in Corollary 2.7. Namely, here we set $\psi$ by $f_{v}=v^{m-1}\psi/(m-1)!$ . Since $f$ is an $m$ -type edge, $f_{u}$ and $\psi$ is linearly independent (Proposition 2.2 and the independence of the condition [2.2]). Thus the unit normal vector $\nu$ of $f$ can be taken as $\nu=\hat{\nu}/|\hat{\nu}|$ $(\hat{\nu}=f_{u}\times\psi)$ . Using $f_{u}$ , $\psi$ and $\nu$ , we define the following functions:

	$\displaystyle\widehat{E}$	$\displaystyle=\left\langle{f_{u}},{f_{u}}\right\rangle,$	$\displaystyle\widehat{F}$	$\displaystyle=\left\langle{f_{u}},{\psi}\right\rangle,$	$\displaystyle\widehat{G}$	$\displaystyle=\left\langle{\psi},{\psi}\right\rangle,$
	$\displaystyle\widehat{L}$	$\displaystyle=-\left\langle{f_{u}},{\widehat{\nu}_{u}}\right\rangle,$	$\displaystyle\widehat{M}$	$\displaystyle=-\left\langle{\psi},{\widehat{\nu}_{u}}\right\rangle,$	$\displaystyle\widehat{N}$	$\displaystyle=-\left\langle{\psi},{\widehat{\nu}_{v}}\right\rangle.$

We note that coefficients of the first and the second fundamental forms of $\sigma$ -edges being of multiplicity $m$ can be written as

	$\displaystyle E$	$\displaystyle=\widehat{E},$	$\displaystyle F$	$\displaystyle=\dfrac{v^{m-1}}{(m-1)!}\widehat{F},$	$\displaystyle G$	$\displaystyle=\left(\dfrac{v^{m-1}}{(m-1)!}\right)^{2}\widehat{G},$
	$\displaystyle L$	$\displaystyle=\dfrac{\widehat{L}}{\|\widehat{\nu}\|},$	$\displaystyle M$	$\displaystyle=\dfrac{v^{m-1}}{\|\widehat{\nu}\|(m-1)!}\widehat{M},$	$\displaystyle N$	$\displaystyle=\dfrac{v^{m-1}}{(m-1)!\|\widehat{\nu}\|}\widehat{N}.$

Lemma 2.16.

The differentials $\nu_{u}$ and $\nu_{v}$ of $\nu$ are written as

	$\displaystyle\nu_{u}$	$\displaystyle=-\dfrac{\widehat{G}\widehat{L}-\widehat{F}\widehat{M}}{(\widehat{E}\widehat{G}-\widehat{F}^{2})\|\hat{\nu}\|}f_{u}-\dfrac{\widehat{E}\widehat{M}-\widehat{F}\widehat{L}}{(\widehat{E}\widehat{G}-\widehat{F}^{2})\|\hat{\nu}\|}\psi,$
	$\displaystyle\nu_{v}$	$\displaystyle=-\dfrac{\dfrac{v^{m-1}}{(m-1)!}\widehat{G}\widehat{M}-\widehat{F}\widehat{N}}{(\widehat{E}\widehat{G}-\widehat{F}^{2})\|\hat{\nu}\|}f_{u}-\dfrac{\widehat{E}\widehat{N}-\dfrac{v^{m-1}}{(m-1)!}\widehat{F}\widehat{M}}{(\widehat{E}\widehat{G}-\widehat{F}^{2})\|\hat{\nu}\|}\psi.$

Proof.

Since $\left\langle{\nu_{u}},{\nu}\right\rangle=\left\langle{\nu_{v}},{\nu}\right\rangle=0$ , there exist functions $A,B,C,D$ on $(\boldsymbol{R}^{2},0)$ such that

\nu_{u}=Af_{u}+B\psi,\quad\nu_{v}=Cf_{u}+D\psi.

Considering $\left\langle{\nu_{u}},{f_{u}}\right\rangle,\left\langle{\nu_{u}},{\psi}\right\rangle,\left\langle{\nu_{v}},{f_{u}}\right\rangle$ and $\left\langle{\nu_{v}},{\psi}\right\rangle$ , we have

-\frac{1}{|\hat{\nu}|}\begin{pmatrix}\widehat{L}\\ \widehat{M}\end{pmatrix}=\begin{pmatrix}\widehat{E}&\widehat{F}\\ \widehat{F}&\widehat{G}\end{pmatrix}\begin{pmatrix}A\\ B\end{pmatrix},\quad-\frac{1}{|\hat{\nu}|}\begin{pmatrix}\frac{v^{m-1}}{(m-1)!}\widehat{M}\\ \widehat{N}\end{pmatrix}=\begin{pmatrix}\widehat{E}&\widehat{F}\\ \widehat{F}&\widehat{G}\end{pmatrix}\begin{pmatrix}C\\ D\end{pmatrix}.

Solving these equations, we have the assertion. ∎

By this lemma, $\nu_{v}$ can be written as

\nu_{v}=\frac{\widehat{N}}{(\widehat{E}\widehat{G}-\widehat{F}^{2})|\hat{\nu}|}(\widehat{F}f_{u}-\widehat{E}\psi)

along the $u$ -axis. Since $f_{u}$ and $\psi$ are linearly independent and $\widehat{E}\neq 0$ , the condition $\nu_{v}(0)\neq 0$ is equivalent to $\widehat{N}(0)\neq 0$ . To see this fact, we take the same setting in the proof of Proposition 2.10. Then we see

\det(f_{u},f_{v^{m}},f_{v^{m+1}})=m\det(f_{u},\psi,\psi_{v})=m\left\langle{\hat{\nu}},{\psi_{v}}\right\rangle=m\widehat{N}

(2.12)

along the $u$ -axis, where $\hat{\nu}=f_{u}\times\psi$ and $\widehat{N}=\left\langle{\hat{\nu}},{\psi_{v}}\right\rangle=-\left\langle{\hat{\nu}_{v}},{\psi}\right\rangle$ . Since $\{f_{u},\psi,\nu\}$ is a frame of $\boldsymbol{R}^{3}$ , and $\left\langle{f_{u}},{\nu_{v}}\right\rangle=\left\langle{f_{v}},{\nu_{u}}\right\rangle=0$ , $\left\langle{\nu},{\nu_{v}}\right\rangle=0$ , it holds that $\nu_{v}\neq 0$ if and only if $\left\langle{\nu_{v}},{\psi}\right\rangle\neq 0$ . Moreover, since $\left\langle{\nu},{\psi}\right\rangle=0$ , it holds that $\left\langle{\nu_{v}},{\psi}\right\rangle\neq 0$ is equivalent to $\left\langle{\hat{\nu}_{v}},{\psi}\right\rangle\neq 0$ . Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be an $(m,n)$ -type edge, and let us set

r=\min\Big{(}\{n\}\cup\{im\,|\,b_{im}(0)\neq 0\text{ in the form }\eqref{eq:edgenormal2},i=2,3,\ldots\}\Big{)}.

Theorem 2.17.

Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be an $(m,n)$ -type edge. Then the rational order of the mean curvature $H$ is $r-2m$ . If the normal curvature does not vanish at $0$ , then the rational order of the Gaussian curvature $K$ is $r-2m$ .

Proof.

We take an adapted coordinate system $(u,v)$ such that $\partial_{v}$ satisfies [2.1]. Since $\hat{L}=\left\langle{f_{uu}},{\nu}\right\rangle$ , the normal curvature does not vanish if and only if $\hat{L}(0)\neq 0$ . The Gaussian curvature $K$ and the mean curvature $H$ of $f$ are given by

K=\dfrac{(m-1)!}{v^{m-1}}\dfrac{\widehat{L}\widehat{N}-\frac{v^{m-1}}{(m-1)!}\widehat{M}^{2}}{|\widehat{\nu}|^{2}(\widehat{E}\widehat{G}-\widehat{F}^{2})},\quad H=\dfrac{(m-1)!}{v^{m-1}}\dfrac{\widehat{E}\widehat{N}-2\frac{v^{m-1}}{(m-1)!}\widehat{F}\widehat{M}+\frac{v^{m-1}}{(m-1)!}\widehat{G}\widehat{L}}{2|\widehat{\nu}|(\widehat{E}\widehat{G}-\widehat{F}^{2})}.

This and $\widehat{E}\neq 0$ , $\widehat{E}\widehat{G}-\widehat{F}^{2}\neq 0$ at $0$ , together with

\widehat{N}=\dfrac{v^{r-m-1}}{m(r-m-1)!}(b_{r}(0)+v\alpha(u,v)),

where $\alpha$ is a function, by using the form (2.6) and (2.12) give the assertion. ∎

By Theorem 2.17, the order of $K$ and $H$ coincide. Moreover since $n<2m$ , they never bounded when the normal curvature does not vanish.

3. Curves passing through $m$ -type edges

In this section, we consider geometric invariants of a curve $\gamma$ passing through an $m$ -type edge $f$ . If $\hat{\gamma}=f\circ\gamma$ is non-singular, then the usual invariants can be defined as well as the regular case. We consider the case when $\hat{\gamma}$ has a singular point, namely, $\gamma$ passing through a singular point of $f$ in the direction of a null vector.

3.1. Normalized curvatures of singular curves

Following [15, 4], we introduce normalized curvature on curves in $\boldsymbol{R}^{2}$ . Let $\hat{\gamma}:(\boldsymbol{R},0)\to(\boldsymbol{R}^{n},0)$ be a curve, and let $0$ be a singular point. We assume that there exists $k$ such that $\hat{\gamma}^{\prime}=t^{k-1}\rho$ ( $\rho(0)\neq 0$ ).

We set

s=\int|\hat{\gamma}^{\prime}|\,dt,

(3.1)

and

\tilde{s}=\operatorname{sgn}(s)|s|^{1/k},

(3.2)

we see $\tilde{s}$ is a $C^{\infty}$ function and $d\tilde{s}/dt(0)>0$ . We call this parameter an $1/k$ -arc-length.

Proposition 3.1.

The parameter $t$ is an $1/k$ -arc-length parameter of $\hat{\gamma}$ if and only if $|\hat{\gamma}^{\prime}(t)|=k|t^{k-1}|$ .

Proof.

If $|\hat{\gamma}^{\prime}(t)|=k|t^{k-1}|$ and $s(t)$ as in (3.1) it holds that

s(t)=\int_{0}^{t}k|\xi^{k-1}|d\xi=\int_{0}^{t}\varepsilon\,k\xi^{k-1}d\xi=\varepsilon\,t^{k},\ \left(\varepsilon=\begin{dcases}\operatorname{sgn}(t),\ &\text{if $k$ is even}\\ 1\ &\text{if $k$ is odd}\end{dcases}\right).

Since $\operatorname{sgn}(s)=\operatorname{sgn}(t)$ , then $|s|=|t^{k}|$ and, therefore, $t=\operatorname{sgn}(s)|s|^{1/k}$ .

Let us suppose now that $t$ is the $1/k$ -arc-length, i.e., $t=\operatorname{sgn}(s)|s|^{1/k}$ , with $s(t)$ as in (3.1). Since $\operatorname{sgn}(s)=\operatorname{sgn}(t)$ , thus $t^{k}=\operatorname{sgn}(s)^{k}|s|=\operatorname{sgn}(s)^{k+1}s$ and, consequently, $s^{\prime}(t)=\operatorname{sgn}(t)^{k+1}kt^{k-1}=k|t|^{k-1}$ . Therefore, it holds that $|\hat{\gamma}^{\prime}(t)|=k|t^{k-1}|$ . ∎

Let us set $n=2$ . Then the curvature $\kappa$ satisfies that

\tilde{\kappa}=|\tilde{s}^{k-1}|\kappa

(3.3)

is a $C^{\infty}$ function. We call $\tilde{\kappa}$ the normalized curvature. This is originally introduced in [15] and generalized in [4]. Let $f(t)$ be a given $C^{\infty}$ -function, and $k\geq 2$ be an integer. Then similarly to [15, Theorem 1.1], one can show that there exists a unique plane curve up to isometries in $\boldsymbol{R}^{2}$ with normalized curvature given by $\tilde{\kappa}(t)=f(t)$ , where $t$ is the $1/k$ -arc-length parameter.

Using the frame $\{\boldsymbol{e},\boldsymbol{n}\}$ along $\hat{\gamma}$ defined by $\boldsymbol{e}=\rho/|\rho|$ and $\boldsymbol{n}$ the $\pi/2$ -rotation of $\boldsymbol{e}$ , the normalized curvature can be interpreted as follows: We set the function $\kappa_{1}$ by the equation

{\begin{pmatrix}\boldsymbol{e}^{\prime}\\ \boldsymbol{n}^{\prime}\end{pmatrix}}={\begin{pmatrix}0&\kappa_{1}\\ -\kappa_{1}&0\end{pmatrix}}{\begin{pmatrix}\boldsymbol{e}\\ \boldsymbol{n}\end{pmatrix}},

(3.4)

where ^′ is a differentiation by the $1/k$ -arc-length. Then we have:

Proposition 3.2.

Let $\{\boldsymbol{e},\boldsymbol{n}\}$ be the above frame along $\hat{\gamma}(t)$ in the Euclidean plane $\boldsymbol{R}^{2}$ satisfying (3.4), where $t$ is the $1/k$ -arc-length parameter. Then $\kappa_{1}=k\tilde{\kappa}$ holds.

Proof.

Since $\hat{\gamma}^{\prime}(t)=t^{k-1}\rho(t)$ where $\rho(0)\neq 0$ and the $1/k$ -arc-length parameter $t$ satisfies $|\hat{\gamma}^{\prime}(t)|=k|t|^{k-1}$ , so $\hat{\gamma}^{\prime\prime}(t)=(k-1)t^{k-2}\rho(t)+t^{k-1}\rho^{\prime}(t)$ and $|\rho(t)|=k$ . Then

\kappa(t)=\dfrac{1}{k^{3}|t|^{k-1}}\det(\rho(t),\rho^{\prime}(t)).

Consequently,

\tilde{\kappa}(t)=|t|^{k-1}\kappa(t)=\dfrac{1}{k^{3}}\det(\rho(t),\rho^{\prime}(t)).

On the other hand, since $\kappa_{1}(t)=\boldsymbol{e}^{\prime}(t)\cdot\boldsymbol{n}(t)$ , where $\boldsymbol{e}(t)=\rho(t)/|\rho(t)|=\rho(t)/k$ and $\boldsymbol{n}(t)$ is the $\pi/2$ -counterclockwise rotation of $\boldsymbol{e}(t)$ , and the dot ‘ $\cdot$ ’ denotes the canonical inner product of $\boldsymbol{R}^{2}$ , it holds that:

\kappa_{1}(t)=\frac{1}{k}\rho^{\prime}(t)\cdot\boldsymbol{n}(t)=\frac{1}{k^{2}}\det(\rho(t),\rho^{\prime}(t)).

Thus we have the assertion. ∎

3.2. Normalized curvatures on frontals

According to Section 3.1, we define the normalized curvatures for curves on a frontal. Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be a frontal and $\nu$ a unit normal vector field of $f$ . Let $\gamma\colon(\boldsymbol{R},0)\to(\boldsymbol{R}^{2},0)$ be a curve. We set $\hat{\gamma}=f\circ\gamma$ . We assume there exists $k$ such that $\hat{\gamma}^{\prime}=t^{k-1}\rho$ ( $\rho(0)\neq 0$ ). The geodesic curvature $\kappa_{g}$ , the normal curvature $\kappa_{n}$ and the geodesic torsion $\tau_{g}$ are defined by

\kappa_{g}=\dfrac{\det(\hat{\gamma}^{\prime},\hat{\gamma}^{\prime\prime},\nu)}{|\hat{\gamma}^{\prime}|^{3}},\ \kappa_{n}=\dfrac{\left\langle{\hat{\gamma}^{\prime\prime}},{\nu}\right\rangle}{|\hat{\gamma}^{\prime}|^{2}},\ \tau_{g}=\dfrac{\det(\hat{\gamma}^{\prime},\nu,\nu^{\prime})}{|\hat{\gamma}^{\prime}|^{2}}

(3.5)

on regular points (see [1, page 261]). These curvatures can be unbounded near singular points. Indeed, it holds that

\kappa_{g}=\dfrac{1}{|t|^{k-1}}\dfrac{\det(\rho,\rho^{\prime},\nu)}{|\rho|^{3}},\ \kappa_{n}=\dfrac{1}{t^{k-1}}\dfrac{\left\langle{\rho^{\prime}},{\nu}\right\rangle}{|\rho|^{2}},\ \tau_{g}=\dfrac{1}{t^{k-1}}\dfrac{\det(\rho,\nu,\nu^{\prime})}{|\rho|^{2}}.

(3.6)

One can easily see that

\tilde{\kappa}_{g}=|\tilde{s}|^{k-1}\,\kappa_{g},\ \tilde{\kappa}_{n}=\tilde{s}^{k-1}\,\kappa_{n},\ \tilde{\tau}_{g}=\tilde{s}^{k-1}\,\tau_{g}

(3.7)

are $C^{\infty}$ functions, where $\tilde{s}$ is the function given by (3.2) for $\hat{\gamma}$ . We call $\tilde{\kappa}_{g},\tilde{\kappa}_{n},\tilde{\tau}_{g}$ normalized geodesic curvature, normal curvature and geodesic torsion of $\tilde{\gamma}$ , respectively. These satisfy:

Lemma 3.3.

It holds that

$\displaystyle\tilde{\kappa}_{g}$	$\displaystyle=\dfrac{1}{k^{2}\,k!^{-1/k}}\dfrac{\det\left(\hat{\gamma}^{(k)},\hat{\gamma}^{(k+1)},\nu\right)}{\|\hat{\gamma}^{(k)}\|^{2+1/k}},$	(3.8)
$\displaystyle\tilde{\kappa}_{n}$	$\displaystyle=\dfrac{1}{k^{2}\,k!^{-1/k}}\dfrac{\left\langle{\hat{\gamma}^{(k+1)}},{\nu}\right\rangle}{\|\hat{\gamma}^{(k)}\|^{1+1/k}},$	(3.9)
$\displaystyle\tilde{\tau}_{g}$	$\displaystyle=\dfrac{1}{k\,k!^{-1/k}}\dfrac{\det(\hat{\gamma}^{(k)},\nu,\nu^{\prime})}{\|\hat{\gamma}^{(k)}\|^{1+1/k}}$	(3.10)

at $t=0$ .

Proof.

Since $\hat{\gamma}^{\prime}(t)=t^{k-1}\rho(t)$ , then $\rho(0)=\frac{\hat{\gamma}^{(k)}(0)}{(k-1)!}$ , $\rho^{\prime}(0)=\frac{\hat{\gamma}^{(k+1)}(0)}{k!}$ and $\rho_{0}=|\rho(0)|=\frac{|\hat{\gamma}^{(k)}(0)|}{(k-1)!}$ . Therefore, it holds that

\tilde{s}^{k-1}=t^{k-1}\left(\dfrac{\rho_{0}^{(k-1)/k}}{k^{(k-1)/k}}+tO(t)\right)

where $O(t)$ is a smooth function of $t$ . Thus by (3.6) and (3.7), we get at $t=0$ :

	$\displaystyle\tilde{\kappa}_{g}$	$\displaystyle=\frac{\rho_{0}^{\frac{k-1}{k}}k^{1/k}}{k}\dfrac{\det(\rho,\rho^{\prime},\nu)}{\rho_{0}^{3}}=\dfrac{k^{1/k}}{k}\,\dfrac{\det(\rho,\rho^{\prime},\nu)}{\rho_{0}^{2+1/k}}$
		$\displaystyle=\dfrac{k^{1/k}(k-1)!^{2+1/k}}{k\,(k-1)!\,k!}\dfrac{\det(\hat{\gamma}^{(k)},\hat{\gamma}^{(k+1)},\nu)}{\|\hat{\gamma}^{(k)}\|^{2+1/k}}=\dfrac{k!^{1/k}}{k^{2}}\dfrac{\det(\hat{\gamma}^{(k)},\hat{\gamma}^{(k+1)},\nu)}{\|\hat{\gamma}^{(k)}\|^{2+1/k}},$

	$\displaystyle\tilde{\kappa}_{n}$	$\displaystyle=\frac{\rho_{0}^{(k-1)/k}}{k^{(k-1)/k}}\,\frac{\left\langle{\rho^{\prime}},{\nu}\right\rangle}{\rho_{0}^{2}}=\frac{k^{1/k}}{k}\frac{\left\langle{\rho^{\prime}},{\nu}\right\rangle}{\rho_{0}^{1+1/k}}$
		$\displaystyle=\frac{k^{1/k}(k-1)!^{1+1/k}}{k\,k!}\frac{\left\langle{\hat{\gamma}^{(k+1)}},{\nu}\right\rangle}{\|\hat{\gamma}^{(k)}\|^{1+1/k}}=\frac{k!^{1/k}}{k^{2}}\frac{\left\langle{\hat{\gamma}^{(k+1)}},{\nu}\right\rangle}{\|\hat{\gamma}^{(k)}\|^{1+1/k}}$

and

	$\displaystyle\tilde{\tau}_{g}$	$\displaystyle=\frac{\rho_{0}^{(k-1)/k}}{k^{(k-1)/k}}\dfrac{\det(\rho,\nu,\nu^{\prime})}{\rho_{0}^{2}}=\dfrac{k^{1/k}}{k}\dfrac{\det(\rho,\nu,\nu^{\prime})}{\rho_{0}^{1+1/k}}$
		$\displaystyle=\dfrac{k^{1/k}(k-1)!^{1+1/k}}{k(k-1)!}\dfrac{\det(\hat{\gamma}^{(k)},\nu,\nu^{\prime})}{\|\hat{\gamma}^{(k)}\|^{1+1/k}}=\dfrac{k!^{1/k}}{k}\dfrac{\det(\hat{\gamma}^{(k)},\nu,\nu^{\prime})}{\|\hat{\gamma}^{(k)}\|^{1+1/k}},$

which show the assertion. ∎

Similar with the case of plane curves, these invariants can be interpreted as follows. Under the same assumption above, we set $e=\rho/|\rho|$ , $\nu=\nu(\hat{\gamma})$ and $b=-e\times\nu$ . Then $\{e,\nu,b\}$ is a frame along $\hat{\gamma}$ . We define $\kappa_{1},\kappa_{2},\kappa_{3}$ by

{\begin{pmatrix}e^{\prime}\\ b^{\prime}\\ \nu^{\prime}\end{pmatrix}}={\begin{pmatrix}0&\kappa_{1}&\kappa_{2}\\ -\kappa_{1}&0&\kappa_{3}\\ -\kappa_{2}&-\kappa_{3}&0\end{pmatrix}}{\begin{pmatrix}e\\ b\\ \nu\end{pmatrix}},

(3.11)

where ${}^{\prime}=d/dt$ is the differentiation by the $1/k$ -arc-length parameter. With the above notation, we get the following:

Proposition 3.4.

If $t$ is the $1/k$ -arc-length parameter, then

\kappa_{1}=k\tilde{\kappa}_{g},\ \kappa_{2}=k\tilde{\kappa}_{n}\ \ \text{and}\ \ \kappa_{3}=k\tilde{\tau}_{g}

holds, for any $t$ .

Proof.

The $1/k$ -arc-length parameter $t$ satisfies $|\hat{\gamma}^{\prime}(t)|=k|t^{k-1}|$ . Then $|\rho(t)|=k$ and $e=\rho/k$ . So, putting $\tilde{s}=t$ at (3.7) and using (3.6) , it holds that

	$\displaystyle\kappa_{1}$	$\displaystyle=\left\langle{e^{\prime}},{b}\right\rangle=\dfrac{1}{k^{2}}\det(\rho,\rho^{\prime},\nu)=k\tilde{\kappa}_{g},$
	$\displaystyle\kappa_{2}$	$\displaystyle=\left\langle{e^{\prime}},{\nu}\right\rangle=\dfrac{1}{k}\left\langle{\rho^{\prime}},{\nu}\right\rangle=k\tilde{\kappa}_{n}\,,$
	$\displaystyle\kappa_{3}$	$\displaystyle=-\left\langle{\nu^{\prime}},{b}\right\rangle=\dfrac{1}{k}\det(\rho,\nu,\nu^{\prime})=k\tilde{\tau}_{g}.$

Thus the assertion holds. ∎

3.3. Behaviors of $\kappa_{g},\kappa_{n}$ and $\tau_{g}$ passing through an $m$ -type edge

In this section we shall study the orders of the geodesic and normal curvatures and the geodesic torsion of a curve passing through an $m$ -type edge, concluding on boundedness. Describing the condition, we use the curvature of such curve. Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be an $m$ -type edge, $m\geq 2$ , and $\gamma:(\boldsymbol{R},0)\to(\boldsymbol{R}^{2},0)$ be a regular curve such that $\gamma^{\prime}(0)$ is a null vector of $f$ at $0$ . Let $(u,v)$ be a coordinate system which gives the form (2.5), and $\tilde{\gamma}(t)=(u(t),v(t))$ be a parametrization of $\gamma$ where the coordinate system on the target space is $(u,v)$ , and the orientation of $\tilde{\gamma}$ agrees the direction of $v$ at $0$ . Since such coordinate system is unique (unique up to $(u,v)\mapsto(u,-v)$ if $m$ is even), the order of contact of $\tilde{\gamma}$ with the $v$ -axis at $0$ and the curvature $\tilde{\kappa}$ of $\tilde{\gamma}$ is well-defined as a curve on $f$ . We call such order of contact the order of contact with the normalized null direction, and we call $\tilde{\kappa}$ the curvature written in the normal form. If $\tilde{\gamma}(t)=(t^{l}c(t),t)$ ( $c(0)\neq 0$ ), then the order of contact with the normalized null direction is $l$ , and $\tilde{\kappa}^{(l-2)}(0)=-l!c(0)$ and $\tilde{\kappa}^{(l-1)}(0)=-(l+1)!c^{\prime}(0)$ hold.

Theorem 3.5.

Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be an $m$ -type edge, $m\geq 2$ , and $\gamma:(\boldsymbol{R},0)\to(\boldsymbol{R}^{2},0)$ be a regular curve with order of contact $l\geq 2$ with the null direction of $f$ at $0$ and $\tilde{\kappa}$ the curvature of $\gamma$ written in the normal form of $f$ . Then, it holds that:

(1) The case $l\geq m$ .
For $\kappa_{g}$ ,

•

if $m<l\leq 2m$ , then $\operatorname{ord}\kappa_{g}=l-2m$ ;

•

if $l>2m$ , then $\operatorname{ord}\kappa_{g}\geq 1$ , and $\operatorname{ord}\kappa_{g}=1$ is equivalent to

\begin{dcases}(l-1)!\kappa_{t}(0)\omega_{m,m+1}(0)-m!(m+1)!\tilde{\kappa}^{(l-2)}(0)\neq 0&(\text{if $l=2m+1$}),\\ \kappa_{t}(0)\omega_{m,m+1}(0)\neq 0&(\text{if $l>2m+1$});\end{dcases}

•

if $l=m$ , then $\operatorname{ord}\kappa_{g}\geq 1-m$ , and $\operatorname{ord}\kappa_{g}=1-m$ if and only if $\tilde{\kappa}^{(l-1)}(0)\neq 0$ .

For $\kappa_{n}$ , it holds that $\operatorname{ord}\kappa_{n}\geq 1-m$ , and $\operatorname{ord}\kappa_{n}=1-m$ if and only if $\omega_{m,m+1}(0)\neq 0$ . For $\tau_{g}$ ,

•

if $l<2m$ , then $\operatorname{ord}\tau_{g}\geq l-2m+1$ , and $\operatorname{ord}\tau_{g}=l-2m+1$ is equivalent to

\begin{dcases}\omega_{m,m+1}(0)\neq 0&(\text{if $l<2m-1$}),\\ m(l-1)!\kappa_{t}(0)+{(m-1)!^{2}}\,\tilde{\kappa}^{(l-2)}(0)\,\omega_{m,m+1}(0)\neq 0&(\text{if $l=2m-1$});\end{dcases}

•

if $l\geq 2m$ , then $\operatorname{ord}\tau_{g}\geq 0$ , and $\operatorname{ord}\tau_{g}=0$ if and only if $\kappa_{t}(0)\neq 0$ .

(2) The case $m/2<l<m$ .
For this case, it holds that $\operatorname{ord}\kappa_{g}=m-2l$ , $\operatorname{ord}\kappa_{n}\geq m-2l+1$ , and $\operatorname{ord}\kappa_{n}=m-2l+1$ is equivalent to

\begin{dcases}\omega_{m,m+1}(0)\neq 0&(\text{if $l>(m+1)/2$}),\\ (m+1)!(m-l+1)\kappa_{\nu}(0)(\tilde{\kappa}^{(l-2)})^{2}(0)+2l!^{2}\omega_{m,m+1}(0)\neq 0&(\text{if $l=(m+1)/2$}).\end{dcases}

For $\tau_{g}$ , it holds that $\operatorname{ord}\tau_{g}\geq 1-l$ , and $\operatorname{ord}\tau_{g}=1-l$ is equivalent to $\omega_{m,m+1}(0)\neq 0$ .

(3) The case $l\leq m/2$ .
In this case, it holds that $\operatorname{ord}\kappa_{g}\geq 0$ , and $\operatorname{ord}\kappa_{g}=0$ is equivalent to

\begin{dcases}\kappa_{s}(0)\neq 0&(\text{if $l<m/2$}),\\ m!\kappa_{s}(0)(\tilde{\kappa}^{(l-2)})^{2}(0)+2l!^{2}\neq 0&(\text{if $l=m/2$}).\end{dcases}

For $\kappa_{n}$ it holds that $\operatorname{ord}\kappa_{n}\geq 0$ , and $\operatorname{ord}\kappa_{n}=0$ if and only if $\kappa_{n}(0)\neq 0$ . For $\tau_{g}$ , it holds that $\operatorname{ord}\tau_{g}\geq 1-l$ , and $\operatorname{ord}\tau_{g}=1-l$ if and only if $\omega_{m,m+1}(0)\neq 0$ .

If $m$ is even and $(u,v)$ be a coordinate system which gives the form (2.5), then $(u,-v)$ also gives (2.5). In this case, changing $(u,v)$ to $(u,-v)$ , the signs of $\tilde{\kappa}$ and $\omega_{m,m+1}$ reverse, and them of $\kappa_{t}$ and $\kappa_{s}$ do not change. So, when $m$ is even, neither the condition

	$\displaystyle(l-1)!\kappa_{t}(0)\omega_{m,m+1}(0)-m!(m+1)!\tilde{\kappa}^{(l-2)}(0)$	$\displaystyle\neq 0,$
	$\displaystyle m(l-1)!\kappa_{t}(0)+(m+1)!^{2}\,\tilde{\kappa}^{(l-2)}(0)\,\omega_{m,m+1}(0)$	$\displaystyle\neq 0\quad\text{nor}$
	$\displaystyle m!\kappa_{s}(0)(\tilde{\kappa}^{(l-2)})^{2}(0)+2l!^{2}$	$\displaystyle\neq 0$

changes under the coordinate change $(u,v)$ to $(u,-v)$ .

Proof.

Let $\hat{\gamma}=f\circ\gamma$ . One can assume that $f$ is given by the form (2.5) and, since $\partial v$ is a null vector of $f$ , one can take $\gamma(t)=(x(t),t)$ , with $x(0)=x^{\prime}(0)=0$ . Then $x(t)$ is of order $l$ and we set $\gamma(t)=(t^{l}c(t),t)$ ( $c(0)\neq 0$ ).

In the normal form (2.5), since $b_{m}(0,0)=0$ , further we may assume $f$ is given by $f(u,v)=(u,u^{2}a(u)/2+v^{m}/m!,u^{2}b_{0}(u)/2+(v^{m}/m!)(ub_{m1}(u)+vb_{m2}(u,v)))$ . We recall that $\kappa_{s}(0)=a(0),\kappa_{\nu}(0)=b(0)$ and $\kappa_{t}(0)=(b_{m})_{u}(0,0)$ . Furthermore, it holds that $\kappa_{t}(0)=b_{m1}(0)$ , $\omega_{m,m+1}=(m+1)b_{m2}(0,0)$ , $\tilde{\kappa}^{(l-2)}(0)=-l!c(0)$ and $\tilde{\kappa}^{(l-1)}(0)=-(l+1)!c^{\prime}(0)$ . We set $\varphi$ by $f_{v}=v^{m-1}\varphi/(m-1)!$ . Then $\tilde{\nu}_{2}=f_{u}\times\varphi$ gives a non-zero normal vector field to $f$ .

(1). Assume $l\geq m$ . By (2.5) we get $\hat{\gamma}=t^{m}\tilde{\rho}$ , where

	$\displaystyle\tilde{\rho}(t)$	$\displaystyle=(t^{l-m}c(t),g_{2}(t),tg_{3}(t)),$
	$\displaystyle g_{2}(t)$	$\displaystyle=\dfrac{m!t^{2l-m}a(t)c(t)^{2}+2}{2m!},$
	$\displaystyle g_{3}(t)$	$\displaystyle=\dfrac{m!t^{2l-m-1}b_{0}(t)c(t)^{2}+2b_{m2}(t)+2t^{l-1}b_{m1}(t)c(t)}{2m!}.$

Then $\hat{\gamma}=t^{m-1}\rho$ , where $\rho=m\tilde{\rho}+t\tilde{\rho}^{\prime}$ . Note that $\rho(0)\neq 0$ . Setting $\nu_{2}(t)=\tilde{\nu}_{2}(\gamma(t))$ , we can show that $\nu_{2}(t)=(t^{m}d(t),te(t),1)$ , where

$\displaystyle d(t)$	$\displaystyle=\dfrac{1}{2mm!}\bigg{(}-2mb_{m1}+2mt^{2l-m}ab_{m1}c^{2}m!-2mt^{l-m}b_{0}cm!$	(3.12)
	$\displaystyle\hskip 56.9055pt+2t^{1+l-m}ab_{m2}cm!+2mt^{1+l-m}ab_{m2}cm!+mt^{3l-m}b_{m1}c^{3}m!a^{\prime}$
	$\displaystyle\hskip 56.9055pt+t^{1+2l-m}b_{m2}c^{2}m!a^{\prime}+mt^{1+2l-m}b_{m2}c^{2}m!a^{\prime}-mt^{2l-m}c^{2}m!b_{0}^{\prime}$
	$\displaystyle\hskip 56.9055pt+2t^{2+l-m}acm!b_{m2,v}+t^{2+2l-m}c^{2}m!a^{\prime}b_{m2,v}-2mt^{l}cb^{\prime}_{m1}-2mtb_{m2,u}\bigg{)},$
$\displaystyle e(t)$	$\displaystyle=\dfrac{-1}{m}\Big{(}mt^{l-1}b_{m1}c+(1+m)b_{m2}+tb_{m2,v}\Big{)}.$

We abbreviate the variable, namely $a=a(t)$ , $b_{m2}=b_{m2}(\gamma(t))$ for instance, and $(b_{m2})_{v}=b_{m2,v}$ . Here, we see

		$\displaystyle g_{2}(0)=\dfrac{1}{m!},\ g_{2}^{\prime}(0)=0,\ g_{3}(0)=\dfrac{b_{m2}(0)}{m!},\ d(0)=\dfrac{-b_{m1}(0)}{m!}\ (\text{if}\ m<l),$
		$\displaystyle d(0)=\dfrac{-m!b_{0}(0)c(0)-b_{m1}(0)}{m!}\ (\text{if}\ m=l),\ e(0)=-\dfrac{(m+1)b_{m2}(0)}{m}.$

To see the rational order of the invariants $\kappa_{g},\kappa_{n}$ , $\tau_{g}$ at $0$ , we may use $\nu_{2}(t)$ instead of $\nu\circ\gamma(t)$ in (3.6). Since $g_{2}^{\prime}(0)=0$ , we can write $g_{2}^{\prime}=t\tilde{g}$ . We see

	$\displaystyle\rho$	$\displaystyle=(lt^{l-m}c+t^{l-m+1}c^{\prime},mg_{2}+t^{2}\tilde{g}_{2},(m+1)tg_{3}+t^{2}g_{3}^{\prime}),$		(3.13)
	$\displaystyle\rho^{\prime}$	$\displaystyle=\begin{dcases}\Big{(}l(l-m)t^{l-m-1}c+t^{l-m}O(1),\\ \hskip 56.9055pttO(1),(m+1)g_{3}+tO(1)\Big{)}\quad(l>m)\\ \Big{(}(m+1)c^{\prime}+tO(1),\\ \hskip 56.9055pttO(1),(m+1)g_{3}+tO(1)\Big{)}\ (m=l),\end{dcases}$		(3.14)

where $O(1)$ means a smooth function depending on $t$ . Then we see $|\rho,\rho^{\prime},\nu_{2}|$ , where $|\cdot|=\det(\cdot)$ is, for $l>m$ :

{\begin{vmatrix}lt^{l-m}c+t^{l-m+1}O(1)&l(l-m)t^{l-m-1}c+t^{l-m}O(1)&t^{m}d\\ mg_{2}+tO(1)&tO(1)&te\\ (m+1)tg_{3}+t^{2}O(1)&(m+1)g_{3}+tO(1)&1\end{vmatrix}}.

(3.15)

If $2m-l+1>0$ , then (3.15) is $t^{l-m-1}A_{1}(t)$ , where

A_{1}(0)={\begin{vmatrix}0&l(l-m)c(0)&0\\ mg_{2}(0)&0&0\\ 0&(m+1)g_{3}(0)&1\end{vmatrix}}=-\dfrac{l(l-m)}{(m-1)!}c(0).

If $2m-l+1=0$ , then (3.15) is $t^{m}A_{2}(t)$ , where

	$\displaystyle A_{2}(0)$	$\displaystyle={\begin{vmatrix}0&l(m+1)c(0)&d(0)\\ mg_{2}(0)&0&0\\ 0&(m+1)g_{3}(0)&1\end{vmatrix}}=-m(m+1)g_{2}(0)(lc-dg_{3})(0)$
		$\displaystyle=-\dfrac{m+1}{(m-1)!}\left(\dfrac{b_{m1}b_{m2}}{(m!)^{2}}+lc\right)(0).$

If $2m-l+1<0$ , then $l>m$ and (3.15) is $t^{m}A_{3}(t)$ , where

	$\displaystyle A_{3}(0)$	$\displaystyle={\begin{vmatrix}0&0&d(0)\\ mg_{2}(0)&0&0\\ 0&(m+1)g_{3}(0)&1\end{vmatrix}}=m(m+1)d(0)g_{2}(0)g_{3}(0)$
		$\displaystyle=-\dfrac{m(1+m)}{(m!)^{3}}b_{m1}(0)b_{m2}(0).$

If $m=l$ , by (3.13) and (3.14), we see the assertion, once $\operatorname{ord}|t|^{m-1}=m-1$ and $\operatorname{ord}|\rho|^{3}=0$ . These shows the assertion for $\kappa_{g}$ .

Since one can easily see that $\left\langle{\rho^{\prime}},{\nu_{2}}\right\rangle=(m+1)b_{m2}/m!$ at $0$ , the assertion for $\kappa_{n}$ is proved. Next we see $|\rho,\nu_{2},\nu_{2}^{\prime}|$ is

{\begin{vmatrix}lt^{l-m}c+t^{l-m+1}O(1)&t^{m}d&mt^{m-1}d+t^{m}O(1)\\ mg_{2}+tO(1)&te&e+tO(1)\\ (m+1)tg_{3}+t^{2}O(1)&1&0\end{vmatrix}}.

(3.16)

If $2m-l-1>0$ , then (3.16) is $t^{l-m}B_{1}(t)$ , where

B_{1}(0)={\begin{vmatrix}lc(0)&0&0\\ mg_{2}(0)&0&e(0)\\ 0&1&0\end{vmatrix}}=-lc(0)e(0)=\dfrac{l(m+1)}{m}b_{m2}(0)c(0).

If $2m-l-1=0$ , then $m<l$ and (3.16) is $t^{m-1}B_{2}(t)$ , where

	$\displaystyle B_{2}(0)$	$\displaystyle={\begin{vmatrix}lc(0)&0&md(0)\\ mg_{2}(0)&0&e(0)\\ 0&1&0\end{vmatrix}}=(-lce+m^{2}dg_{2})(0)$
		$\displaystyle=-\dfrac{b_{m1}(0)}{(m-1)!^{2}}+\dfrac{l(m+1)b_{m2}(0)c(0)}{m}.$

If $2m-l-1<0$ , then $m<l$ , and (3.16) is $t^{m-1}B_{3}(t)$ , where

\displaystyle B_{3}(0)

\displaystyle={\begin{vmatrix}0&0&md(0)\\ mg_{2}(0)&0&e(0)\\ 0&1&0\end{vmatrix}}=m^{2}d(0)g_{2}(0)=-\dfrac{b_{m1}(0)}{(m-1)!^{2}}\,.

This show the assertion for $\tau_{g}$ .

(2) and (3). We assume $l<m$ and we shall use the same notation of case (1). Setting $\nu_{2}(t)=\tilde{\nu}_{2}(\gamma(t))$ , we can show that $\nu_{2}(t)=(t^{l}d(t),te(t),1)$ , where

	$\displaystyle d(t)$	$\displaystyle=\dfrac{1}{2mm!}\bigg{(}-2mt^{m-l}b_{m1}+2mm!t^{l}ab_{m1}c^{2}-2mm!b_{0}c+2m!tab_{m2}c+2mm!tab_{m2}c$
		$\displaystyle\hskip 56.9055pt+mm!t^{2l}b_{m1}c^{3}a^{\prime}+m!t^{l+1}b_{m2}c^{2}a^{\prime}+mm!t^{l+1}b_{m2}c^{2}a^{\prime}-mt^{l}c^{2}m!b_{0}^{\prime}$
		$\displaystyle\hskip 56.9055pt+2m!t^{2}acb_{m2,v}+m!t^{l+2}c^{2}a^{\prime}b_{m2,v}-2mt^{m}cb^{\prime}_{m1}-2mt^{m-l+1}b_{m2,u}\bigg{)}$

and $e$ is the same as in (3.12). We assume $l\leq m/2$ . Then $\hat{\gamma}=t^{l}\tilde{\rho}$ , where $\tilde{\rho}(t)=(c(t),t^{l}g_{2}(t)$ , $t^{l}g_{3}(t))$ and

	$\displaystyle g_{2}(t)$	$\displaystyle=\dfrac{2t^{m-2l}+m!a(t)c(t)^{2}}{2m!},$
	$\displaystyle g_{3}(t)$	$\displaystyle=\dfrac{2t^{m-l}b_{m1}(t)c(t)+2t^{m-2l+1}b_{m2}(\gamma(t))+m!b_{0}(t)c(t)^{2}}{2m!}\,.$

Then $\hat{\gamma}^{\prime}=t^{l}\rho$ , where $\rho=l\tilde{\rho}+t\tilde{\rho}^{\prime}$ , with $\rho(0)\neq 0$ . Since $\hat{\gamma}$ has multiplicity $l$ , we need replace $m-1$ in equations (3.6) by $l-1$ . Here, we see

		$\displaystyle g_{2}(0)=a(0)c(0)^{2}/2\ (\text{if }l<m/2),\ g_{2}(0)=a(0)c(0)^{2}/2+1/m!\ (\text{if }m=2l),$
		$\displaystyle g_{3}(0)=b_{0}(0)c(0)^{2}/2,\ e(0)=-(1+m)b_{m2}(0)/m\,.$

To see the order, we may use $\nu_{2}(t)$ instead of $\nu\circ\gamma(t)$ in (3.6). We see

	$\displaystyle\rho$	$\displaystyle=\Big{(}lc+tc^{\prime},t^{l}(2lg_{2}+tg_{2}^{\prime}),t^{l}(2lg_{3}+tg_{3}^{\prime})\Big{)},$		(3.17)
	$\displaystyle\rho^{\prime}$	$\displaystyle=\Big{(}(l+1)c^{\prime}+tO(1),t^{l-1}(2l^{2}g_{2}+tO(1)),t^{l-1}(2l^{2}g_{3}+tO(1))\Big{)}.$		(3.17)

By applying the formula

{\begin{vmatrix}x_{11}&x_{12}&x_{13}\\ kx_{21}&x_{22}&x_{23}\\ kx_{31}&x_{32}&x_{33}\end{vmatrix}}={\begin{vmatrix}x_{11}&kx_{12}&kx_{13}\\ x_{21}&x_{22}&x_{23}\\ x_{31}&x_{32}&x_{33}\end{vmatrix}},

for $k=t^{l-1}$ , we see $|\rho,\rho^{\prime},\nu_{2}|$ is

	$\displaystyle{\begin{vmatrix}lc+tO(1)&(l+1)c^{\prime}+tO(1)&t^{l}d\\ t^{l}(2lg_{2}+tO(1))&t^{l-1}(2l^{2}g_{2}+tO(1))&te\\ t^{l}(2lg_{3}+tO(1))&t^{l-1}(2l^{2}g_{3}+tO(1))&1\end{vmatrix}}$
$\displaystyle=$	$\displaystyle{\begin{vmatrix}lc+tO(1)&t^{l-1}(l+1)c^{\prime}+tO(1)&t^{2l-1}d\\ t(2lg_{2}+tO(1))&t^{l-1}(2l^{2}g_{2}+tO(1))&te\\ t(2lg_{3}+tO(1))&t^{l-1}(2l^{2}g_{3}+tO(1))&1\end{vmatrix}}$	(3.18)
$\displaystyle=$	$\displaystyle t^{l-1}C_{1}(t)\quad\left(C_{1}(t)={\begin{vmatrix}lc+tO(1)&(l+1)c^{\prime}+tO(1)&t^{2l-1}d\\ t(2lg_{2}+tO(1))&2l^{2}g_{2}+tO(1)&te\\ t(2lg_{3}+tO(1))&2l^{2}g_{3}+tO(1)&1\end{vmatrix}}\right).$

Then $C_{1}(0)=2l^{3}c(0)g_{2}(0)$ . This shows the assertion for $\kappa_{g}$ . By (3.17), we see $\left\langle{\rho^{\prime}},{\nu_{2}}\right\rangle=t^{l-1}(2l^{2}g_{3}+tO(1))$ and $|\rho,\nu_{2},\nu_{2}^{\prime}|(0)=l(m+1)c(0)b_{m2}/m$ . This shows the assertions for $\kappa_{n}$ and $\tau_{g}$ .

Next we assume $l>m/2$ . In this case, $\hat{\gamma}=t^{l}(c(t),t^{m-l}g_{2}(t),t^{m-l+1}g_{3}(t))$ . We set $\tilde{\rho}(t)=(c(t),t^{m-l}g_{2}(t),t^{m-l+1}g_{3}(t))$ and

	$\displaystyle g_{2}(t)$	$\displaystyle=\dfrac{2+m!\,t^{2l-m}a(t)c(t)^{2}}{2m!},$
	$\displaystyle g_{3}(t)$	$\displaystyle=\dfrac{2t^{l-1}b_{m1}(t)c(t)+2b_{m2}(\gamma(t))+m!\,t^{2l-m-1}b_{0}(t)c(t)^{2}}{2m!}.$

Here, it holds that

		$\displaystyle g_{2}(0)=1/m!,\ e(0)=-(1+m)b_{m2}(0)/m,$
		$\displaystyle g_{3}(0)=b_{m2}(0)/m!\ (\text{if }2l-m-1>0),$
		$\displaystyle g_{3}(0)=b_{0}(0)c(0)^{2}/2+b_{m2}(0)/m!\ (\text{if }m=2l-1).$

It holds that $\hat{\gamma}^{\prime}=t^{l-1}\rho$ , with $\rho=l\tilde{\rho}+t\tilde{\rho}^{\prime}$ and $\rho(0)\neq 0$ . We see

$\displaystyle\rho(t)$	$\displaystyle=\Big{(}lc+tc^{\prime},t^{m-l}(mg_{2}+tg_{2}^{\prime}),t^{m-l+1}((m+1)g_{3}+tg_{3}^{\prime})\Big{)},$	(3.19)
$\displaystyle\rho^{\prime}(t)$	$\displaystyle=\Big{(}(l+1)c^{\prime}+tO(1),t^{m-l-1}(m(m-l)g_{2}+tO(1)),$
	$\displaystyle\hskip 142.26378ptt^{m-l}((m+1)(m-l+1)g_{3}+tO(1))\Big{)}.$

By the similar method to (3.18), we see $|\rho,\rho^{\prime},\nu_{2}|$ is

		$\displaystyle{\begin{vmatrix}lc+tO(1)&(1+l)c^{\prime}+tO(1)&t^{l}d\\ t^{m-l}(mg_{2}+tO(1))&t^{m-l-1}(m(m-l)g_{2}+tO(1))&te\\ t^{m-l+1}((m+1)g_{3}+tO(1))&t^{m-l}((m+1)(m-l+1)g_{3}+tO(1))&1\end{vmatrix}}$
	$\displaystyle=$	$\displaystyle{\begin{vmatrix}lc+tO(1)&t^{m-l-1}((1+l)c^{\prime}+tO(1))&t^{m-1}d\\ t(mg_{2}+tO(1))&t^{m-l-1}(m(m-l)g_{2}+tO(1))&te\\ t^{2}((m+1)g_{3}+tO(1))&t^{m-l}((m+1)(m-l+1)g_{3}+tO(1))&1\end{vmatrix}}$
	$\displaystyle=$	$\displaystyle t^{m-l-1}C_{2}(t),$
	$\displaystyle C_{2}(t)=$	$\displaystyle{\begin{vmatrix}lc+tO(1)&(1+l)c^{\prime}+tO(1)&t^{m-1}d\\ t(mg_{2}+tO(1))&m(m-l)g_{2}+tO(1)&te\\ t^{2}((m+1)g_{3}+tO(1))&t((m+1)(m-l+1)g_{3}+tO(1))&1\end{vmatrix}}.$

Then $C_{2}(0)=l(m-l)mc(0)g_{2}(0)=l(m-l)c(0)/(m-1)!$ and, replacing $m-1$ by $l-1$ in equations (3.6), this shows the assertion for $\kappa_{g}$ . By (3.17), we see $\left\langle{\rho^{\prime}},{\nu_{2}}\right\rangle=t^{m-l}C_{3}(t)$ , where $C_{3}(t)=m(m-l)g_{2}(t)e_{2}(t)+(m+1)(m-l+1)g_{3}(t)+tO(1)$ . It holds that

C_{3}(0)=\begin{dcases}\dfrac{(m+1)b_{m2}(0)}{m!}\ &(m<2l-1),\\ (m+1)\left(\dfrac{lb_{0}(0)c(0)^{2}}{2(l!)^{2}}+\dfrac{b_{m2}(0)}{m!}\right)&(m=2l-1).\end{dcases}

and $|\rho,\nu_{2},\nu_{2}^{\prime}|(0)=-lc(0)e(0)=l(m+1)c(0)b_{m2}(0)/m$ . This shows the assertions for $\kappa_{n}$ and $\tau_{g}$ . ∎

In particular, we have the following corollary on boundedness directly obtained from Theorem 3.5.

Corollary 3.6.

Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be an $m$ -type edge with $m\geq 2$ , and $\gamma:(\boldsymbol{R},0)\to(\boldsymbol{R}^{2},0)$ be a regular curve with order of contact $l\geq 2$ with the null direction of $f$ at $0$ .

(1)
The case $l\geq m$ . For $\kappa_{g}$ ,
- •
  
  if $l\geq 2m$ , then $\kappa_{g}$ is bounded at $0$ ;
- •
  
  if $m<l<2m$ , then $\kappa_{g}$ is unbounded at $0$ ;
- •
  
  if $m=l$ and $\tilde{\kappa}^{(l-1)}(0)\neq 0$ , then $\kappa_{g}$ is unbounded at $0$ .
For $\kappa_{n}$ , if $\omega_{m,m+1}(0)\neq 0$ , then $\kappa_{n}$ is unbounded at $0$ . For $\tau_{g}$ ,
- •
  
  if $m\leq l<2m-1$ and $\omega_{m,m+1}(0)\neq 0$ , then $\tau_{g}$ is unbounded at $0$ ;
- •
  
  if $l=2m-1$ and $m(l-1)!\kappa_{t}(0)+{(m-1)!^{2}}\,\tilde{\kappa}^{(l-2)}(0)\,\omega_{m,m+1}(0)\neq 0$ , then $\tau_{g}$ is bounded at $0$ ;
- •
  
  if $l>2m-1$ , then $\tau_{g}$ is bounded at $0$ .
(2)

The case $m/2<l<m$ . In this case, $\kappa_{g}$ is unbounded at $0$ . If $l=(m+1)/2$ , then $\kappa_{n}$ is bounded at $0$ . If $m>l>(m+1)/2$ and $\omega_{m,m+1}(0)\neq 0$ , then $\kappa_{n}$ is unbounded at $0$ . If $\omega_{m,m+1}(0)\neq 0$ , then $\tau_{g}$ is unbounded at $0$ .
(3)

The case $l\leq m/2$ . In this case, $\kappa_{g}$ and $\kappa_{n}$ are bounded at $0$ . If $\omega_{m,m+1}(0)\neq 0$ , then $\tau_{g}$ is unbounded at $0$ .

We consider the case that $f\colon(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ is a cuspidal edge. By definition, it is a $(2,3)$ -edge, in particular, a $2$ -type edge. Then by Theorem 3.5, the following assertion holds.

Corollary 3.7.

Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{3},0)$ be a cuspidal edge, and let $\gamma:(\boldsymbol{R},0)\to(\boldsymbol{R}^{2},0)$ be a regular curve with order of contact $l\geq 2$ with the null direction of $f$ at $0$ and $\tilde{\kappa}$ the curvature of $\gamma$ written in the normal form of $f$ . Then, it holds that:

For $\kappa_{g}$ ,

•

if $l=2$ , then $\operatorname{ord}\kappa_{g}\geq-1$ , and $\operatorname{ord}\kappa_{g}=-1$ if and only if $\tilde{\kappa}^{(l-1)}(0)\neq 0$ .
•

if $l=3$ or $4$ , then $\operatorname{ord}\kappa_{g}=l-4$ ;

•

if $l\geq 5$ , then $\operatorname{ord}\kappa_{g}\geq 1$ , and $\operatorname{ord}\kappa_{g}=1$ is equivalent to

\begin{dcases}(l-1)!\kappa_{t}(0)\omega_{2,3}(0)-12\tilde{\kappa}^{(l-2)}(0)\neq 0&(\text{if $l=5$}),\\ \kappa_{t}(0)\omega_{2,3}(0)\neq 0&(\text{if $l>5$});\end{dcases}

For $\kappa_{n}$ , it holds that $\operatorname{ord}\kappa_{n}=-1$ .

For $\tau_{g}$ ,

•

if $l=2$ or $3$ , then $\operatorname{ord}\tau_{g}\geq l-3$ , and $\operatorname{ord}\tau_{g}=l-3$ is equivalent to

\begin{dcases}\omega_{2,3}(0)\neq 0&(\text{if $l<3$}),\\ 2(l-1)!\kappa_{t}(0)+\tilde{\kappa}^{(l-2)}(0)\,\omega_{2,3}(0)\neq 0&(\text{if $l=3$});\end{dcases}

•

if $l\geq 4$ , then $\operatorname{ord}\tau_{g}\geq 0$ , and $\operatorname{ord}\tau_{g}=0$ if and only if $\kappa_{t}(0)\neq 0$ .

Proof.

Since $\omega_{2,3}$ corresponds to the cuspidal curvature $\kappa_{c}$ and it does not vanish at $0$ ([12, Proposition 3.11]), we have the assertion by Theorem 3.5. ∎

About the boundedness, we have the following corollary immideately from Theorem 3.7.

Corollary 3.8.

Under the same assumption of Corollary 3.7, we have the following:

(1)
For the geodesic curvature $\kappa_{g}$ ,
- •
  
  if $l\geq 4$ , then $\kappa_{g}$ is bounded at $0$ ;
- •
  
  if $l=3$ , then $\kappa_{g}$ is unbounded at $0$ ;
- •
  
  if $l=2$ and $\tilde{\kappa}^{\prime}(0)\neq 0$ , then $\kappa_{g}$ is unbounded at $0$ .
(2)

The normal curvature $\kappa_{n}$ is unbounded at $0$ .
(3)
For the geodesic torsion $\tau_{g}$ ,
- •
  
  if $l=2$ , then $\tau_{g}$ is unbounded at $0$ ;
- •
  
  if $l=3$ and $4\kappa_{t}(0)+\,\tilde{\kappa}^{\prime}(0)\,\kappa_{c}(0)\neq 0$ , then $\tau_{g}$ is bounded at $0$ ;
- •
  
  if $l\geq 4$ , then $\tau_{g}$ is bounded at $0$ ,
where $\kappa_{c}$ is the cuspidal curvature (cf. [12]) corresponding to $\omega_{2,3}$ .

Note that $\operatorname{ord}\kappa_{g}\geq-1$ for $l\geq 2$ is pointed out in [2, Proposition 2.19].

We observe that although in the above results we could not guarantee that the three invariants are bounded at the same time near a singular point, it is easy to find an example where it happens: taking $f=(u,\frac{v^{2}}{2},v^{5})$ and $\gamma(t)=(t^{4},t)$ , it holds that $m=2,l=4$ , $\operatorname{ord}\kappa_{g}=0,\operatorname{ord}\kappa_{n}=1,\operatorname{ord}\tau_{g}=3$ (see Figure 1). Thus these three invariants are bounded at $0$ (cf. Corollary 3.6). For the cuspidal edge $f(u,v)=(u,v^{2},v^{3})$ and the same $\gamma$ , we see that $\kappa_{g}$ and $\tau_{g}$ are bounded, but $\kappa_{n}$ is unbounded at $0$ (cf. Corollary 3.8). Figure 2 shows the graphs of these invariants near $0$ .

Refer to caption — Figure 1. The graphs of $\kappa_{g}$ (left), $\kappa_{n}$ (middle) and $\tau_{g}$ (right) of the curve $\hat{\gamma}(t)=f(\gamma(t))$ , where $f=(u,\frac{v^{2}}{2},v^{5})$ and $\gamma(t)=(t^{4},t)$ .

Acknowledgements

The authors thank the referee, Yuki Hattori and Atsufumi Honda for valuable comments and suggestions.

Appendix A Generalized biases for plane curve

Let $\gamma:(\boldsymbol{R},0)\to(\boldsymbol{R}^{2},0)$ be a curve-germ of $(m,n)$ -type which is given by the form (2.4) in the $xy$ -plane $(\boldsymbol{R}^{2},0)$ . The terms $a_{i}$ $(i=2,\ldots,\lfloor n/m\rfloor)$ measures the bias of $\gamma$ near singular point. We call $a_{i+1}$ the $(m,im)$ -bias $(i=2,\ldots,\lfloor n/m\rfloor)$ of $\gamma$ at $0$ , and it is denoted by $\beta_{m,im}$ . We call $b(0)$ is called the $(m,n)$ -cuspidal curvature as in [7], and it is denoted by $r_{m,n}$ .

If $m$ and $n$ are even, then it is a half part of a curve of $(m/2,n/2)$ -type, we consider the following cases: (1) both $m,n$ are odd, (2) $m$ is odd and $n$ is even, and (3) $m$ is even and $n$ is odd. Moreover, let $a_{k}$ denotes the first non-zero term of $a_{i}$ $(i=2,\ldots,\lfloor n/m\rfloor)$ . We consider the case $(1)$ and $(2)$ . Then $\gamma$ passes through the origin tangent to the $x$ -axis. In the case $(1)$ , if $k$ is odd, it also passes across the $x$ -axis. If $k$ is even, it approaches to the origin from one side of $x$ -axis and goes away into the same side of $x$ -axis, and if there does not exist such $k$ (namely, the bias is zero), it passes through the $x$ -axis. In the case $(2)$ , if the bias is zero, it approaches to the origin from one side of $x$ -axis and goes away into the same side of $x$ -axis. Figure 3 shows the images of the curves $\gamma_{1}:t\mapsto(t^{3},a_{1}t^{6}+a_{2}t^{9}+t^{11})$ with $(a_{1},a_{2})=(1,0),(0,1),(0,0)$ from left to right. Figure 4 shows the images of the curves $\gamma_{2}:t\mapsto(t^{3},a_{1}t^{6}+a_{2}t^{9}+t^{14})$ with $(a_{1},a_{2})=(1,0),(0,1),(0,0)$ from left to right.

We consider the case $(3)$ . Then $\gamma$ approaches the origin from a direction of the $x$ -axis and making a cusp, and back to the same direction. If $k$ is both odd and even, it approaches to the origin from one side of $x$ -axis and goes away into the same side of $x$ -axis. If the bias is zero, it passes through the $x$ -axis. Figure 5 shows the images of the curves $\gamma_{3}:t\mapsto(t^{4},a_{1}t^{8}+a_{2}t^{12}+t^{13})$ with $(a_{1},a_{2})=(1,0),(0,1),(0,0)$ from left to right.

Example A.1.

Let $\gamma$ be a curve-germ $\mathcal{A}^{3}$ -equivalent to $(t^{3},0)$ . We set

\tilde{a}_{i}=\dfrac{\gamma^{(3)}(0)\cdot\gamma^{(i)}(0)}{i!|\gamma^{(3)}(0)|},\quad\tilde{b}_{i}=\dfrac{\det(\gamma^{(3)}(0),\gamma^{(i)}(0))}{i!|\gamma^{(3)}(0)|}.

One can calculate the invariants up to $10$ degrees as follows. The $(3,4)$ -cuspidal curvature $r_{3,4}$ is

r_{3,4}=\dfrac{\tilde{b}_{4}}{\tilde{a}_{3}^{4/3}}.

(A.1)

If $r_{3,4}\neq 0$ , i.e., $\tilde{b}_{4}\neq 0$ , then $\gamma$ is $\mathcal{A}$ -equivalent to $(t^{3},t^{4})$ . We assume $\tilde{b}_{4}=0$ . Then the $(3,5)$ -cuspidal curvature $r_{3,5}$ is

r_{3,5}=\dfrac{\tilde{b}_{5}}{\tilde{a}_{3}^{5/3}}.

(A.2)

If $r_{3,5}\neq 0$ , i.e., $\tilde{b}_{5}\neq 0$ , then $\gamma$ is $\mathcal{A}$ -equivalent to $(t^{3},t^{5})$ . We assume $\tilde{b}_{5}=0$ . Then the $(3,6)$ -bias $\beta_{3,6}$ and the $(3,7)$ -cuspidal curvature $r_{3,7}$ are

	$\displaystyle\beta_{3,6}$	$\displaystyle=\dfrac{\tilde{b}_{6}}{\tilde{a}_{3}^{2}},$		(A.3)
	$\displaystyle r_{3,7}$	$\displaystyle=\dfrac{-7\tilde{a}_{4}\tilde{b}_{6}+2\tilde{a}_{3}\tilde{b}_{7}}{2\tilde{a}_{3}^{10/3}}.$		(A.4)

If $r_{3,7}\neq 0$ , i.e., $-7\tilde{a}_{4}\tilde{b}_{6}+2\tilde{a}_{3}\tilde{b}_{7}\neq 0$ , then $\gamma$ is $\mathcal{A}^{7}$ -equivalent to $(t^{3},t^{7})$ . We assume $r_{3,7}=0$ , i.e., $\tilde{b}_{7}=7\tilde{a}_{4}\tilde{b}_{6}/(2\tilde{a}_{3})$ . Then the $(3,8)$ -cuspidal curvature $r_{3,8}$ is

r_{3,8}=\dfrac{-35\tilde{a}_{4}^{2}\tilde{b}_{6}+2\tilde{a}_{3}(-28\tilde{a}_{5}\tilde{b}_{6}+5\tilde{a}_{3}\tilde{b}_{8})}{10\tilde{a}_{3}^{14/3}}.

(A.5)

If $r_{3,8}\neq 0$ , then $\gamma$ is $\mathcal{A}^{8}$ -equivalent to $(t^{3},t^{8})$ . We assume $r_{3,8}=0$ , i.e., $\tilde{b}_{8}=7(5\tilde{a}_{4}^{2}+8\tilde{a}_{3}\tilde{a}_{5})\tilde{b}_{6}/(10\tilde{a}_{3}^{2})$ . Then the $(3,9)$ -bias $\beta_{3,9}$ and the $(3,10)$ -cuspidal curvature $r_{3,10}$ are

	$\displaystyle\beta_{3,9}$	$\displaystyle=-\dfrac{63\tilde{a}_{4}\tilde{a}_{5}\tilde{b}_{6}+42\tilde{a}_{3}\tilde{a}_{6}\tilde{b}_{6}-5\tilde{a}_{3}^{2}\tilde{b}_{9}}{5\tilde{a}_{3}^{5}},$		(A.6)
	$\displaystyle r_{3,10}$	$\displaystyle=\dfrac{(10\tilde{a}_{3}^{3}\tilde{b}_{10}+945\tilde{a}_{4}^{2}\tilde{a}_{5}\tilde{b}_{6}-42\tilde{a}_{3}(3\tilde{a}_{5}^{2}-10\tilde{a}_{4}\tilde{a}_{6})\tilde{b}_{6}-15\tilde{a}_{3}^{2}(8\tilde{a}_{7}\tilde{b}_{6}+5\tilde{a}_{4}\tilde{b}_{9}))}{10\tilde{a}_{3}^{19/3}}.$		(A.7)

Proof of Example A.1.

By rotating $\gamma$ in $\boldsymbol{R}^{3}$ , we can write

\gamma(t)=\left(\sum_{i=3}^{10}\dfrac{\tilde{a}_{i}}{i!}t^{i},\ \sum_{i=4}^{10}\dfrac{\tilde{b}_{i}}{i!}t^{i}\right)+O(10).

(A.8)

We set

\varphi(t)=t\left(6\sum_{i=3}^{10}\dfrac{\tilde{a}_{i}}{i!}t^{i-3}\right)^{1/3},

and the inverse function of $s=\varphi(t)$ as $t=\psi(s)$ . We set $\psi(s)=\sum_{i=1}^{10}\psi_{i}s^{i}/i!+O(10)$ . Then we have

	$\displaystyle\psi_{1}$	$\displaystyle=1/\tilde{a}_{3}^{1/3},$
	$\displaystyle\psi_{2}$	$\displaystyle=-\tilde{a}_{4}/(6\tilde{a}_{3}^{5/3}),$
	$\displaystyle\psi_{3}$	$\displaystyle=(5\tilde{a}_{4}^{2}-4\tilde{a}_{3}\tilde{a}_{5})/(40\tilde{a}_{3}^{3}),$
	$\displaystyle\psi_{4}$	$\displaystyle=(-175\tilde{a}_{4}^{3}+252\tilde{a}_{3}\tilde{a}_{4}\tilde{a}_{5}-72\tilde{a}_{3}^{2}\tilde{a}_{6})/(1080\tilde{a}_{3}^{13/3}),$
	$\displaystyle\psi_{5}$	$\displaystyle=(13475\tilde{a}_{4}^{4}-27720\tilde{a}_{3}\tilde{a}_{4}^{2}\tilde{a}_{5}+10080\tilde{a}_{3}^{2}\tilde{a}_{4}\tilde{a}_{6}+432\tilde{a}_{3}^{2}(14\tilde{a}_{5}^{2}-5\tilde{a}_{3}\tilde{a}_{7}))/(45360\tilde{a}_{3}^{17/3}),$
	$\displaystyle\psi_{6}$	$\displaystyle=(-1575\tilde{a}_{4}^{5}+4200\tilde{a}_{3}\tilde{a}_{4}^{3}\tilde{a}_{5}-1680\tilde{a}_{3}^{2}\tilde{a}_{4}^{2}\tilde{a}_{6}+96\tilde{a}_{3}^{2}\tilde{a}_{4}(-21\tilde{a}_{5}^{2}+5\tilde{a}_{3}\tilde{a}_{7})$
		$\displaystyle\hskip 28.45274pt+16\tilde{a}_{3}^{3}(42\tilde{a}_{5}\tilde{a}_{6}-5\tilde{a}_{3}\tilde{a}_{8}))/(2240\tilde{a}_{3}^{7}),$
	$\displaystyle\psi_{7}$	$\displaystyle=(475475\tilde{a}_{4}^{6}-1556100\tilde{a}_{3}\tilde{a}_{4}^{4}\tilde{a}_{5}+655200\tilde{a}_{3}^{2}\tilde{a}_{4}^{3}\tilde{a}_{6}-42120\tilde{a}_{3}^{2}\tilde{a}_{4}^{2}(-28\tilde{a}_{5}^{2}+5\tilde{a}_{3}\tilde{a}_{7})$
		$\displaystyle\hskip 28.45274pt+3240\tilde{a}_{3}^{3}\tilde{a}_{4}(-182\tilde{a}_{5}\tilde{a}_{6}+15\tilde{a}_{3}\tilde{a}_{8})$
		$\displaystyle\hskip 28.45274pt-1296\tilde{a}_{3}^{3}(91\tilde{a}_{5}^{3}-60\tilde{a}_{3}\tilde{a}_{5}\tilde{a}_{7}+5\tilde{a}_{3}(-7\tilde{a}_{6}^{2}+\tilde{a}_{3}\tilde{a}_{9})))/(233280\tilde{a}_{3}^{25/3}),$
	$\displaystyle\psi_{8}$	$\displaystyle=(-155520\tilde{a}_{10}\tilde{a}_{3}^{6}+11(-4447625\tilde{a}_{4}^{7}+17243100\tilde{a}_{3}\tilde{a}_{4}^{5}\tilde{a}_{5}-7497000\tilde{a}_{3}^{2}\tilde{a}_{4}^{4}\tilde{a}_{6}$
		$\displaystyle\hskip 28.45274pt+2570400\tilde{a}_{3}^{2}\tilde{a}_{4}^{3}(-7\tilde{a}_{5}^{2}+\tilde{a}_{3}\tilde{a}_{7})-45360\tilde{a}_{3}^{3}\tilde{a}_{4}^{2}(-238\tilde{a}_{5}\tilde{a}_{6}+15\tilde{a}_{3}\tilde{a}_{8})$
		$\displaystyle\hskip 28.45274pt+15552\tilde{a}_{3}^{4}(-98\tilde{a}_{5}^{2}\tilde{a}_{6}+20\tilde{a}_{3}\tilde{a}_{6}\tilde{a}_{7}+15\tilde{a}_{3}\tilde{a}_{5}\tilde{a}_{8})$
		$\displaystyle\hskip 28.45274pt+5184\tilde{a}_{3}^{3}\tilde{a}_{4}(833\tilde{a}_{5}^{3}-420\tilde{a}_{3}\tilde{a}_{5}\tilde{a}_{7}+5\tilde{a}_{3}(-49\tilde{a}_{6}^{2}+5\tilde{a}_{3}\tilde{a}_{9}))))/(6998400\tilde{a}_{3}^{29/3}),$
	$\displaystyle\psi_{9}$	$\displaystyle=(17920\tilde{a}_{10}\tilde{a}_{4}\tilde{a}_{3}^{6}+2480625\tilde{a}_{4}^{8}-11113200\tilde{a}_{3}\tilde{a}_{4}^{6}\tilde{a}_{5}+4939200\tilde{a}_{3}^{2}\tilde{a}_{4}^{4}(3\tilde{a}_{5}^{2}+\tilde{a}_{4}\tilde{a}_{6})$
		$\displaystyle\hskip 28.45274pt-70560\tilde{a}_{3}^{3}\tilde{a}_{4}^{2}(84\tilde{a}_{5}^{3}+140\tilde{a}_{4}\tilde{a}_{5}\tilde{a}_{6}+25\tilde{a}_{4}^{2}\tilde{a}_{7})+4032\tilde{a}_{3}^{4}(84\tilde{a}_{5}^{4}+840\tilde{a}_{4}\tilde{a}_{5}^{2}\tilde{a}_{6}$
		$\displaystyle\hskip 28.45274pt+600\tilde{a}_{4}^{2}\tilde{a}_{5}\tilde{a}_{7}+25\tilde{a}_{4}^{2}(14\tilde{a}_{6}^{2}+5\tilde{a}_{4}\tilde{a}_{8}))+2560\tilde{a}_{3}^{6}(12\tilde{a}_{7}^{2}+21\tilde{a}_{6}\tilde{a}_{8}+14\tilde{a}_{5}\tilde{a}_{9})$
		$\displaystyle\hskip 28.45274pt-4480\tilde{a}_{3}^{5}(72\tilde{a}_{5}^{2}\tilde{a}_{7}+\tilde{a}_{5}(84\tilde{a}_{6}^{2}+90\tilde{a}_{4}\tilde{a}_{8})+5\tilde{a}_{4}(24\tilde{a}_{6}\tilde{a}_{7}+5\tilde{a}_{4}\tilde{a}_{9})))/(89600\tilde{a}_{3}^{11}),$
	$\displaystyle\psi_{10}$	$\displaystyle=13(-16865646875\tilde{a}_{4}^{9}+85717170000\tilde{a}_{3}\tilde{a}_{4}^{7}\tilde{a}_{5}+19595520\tilde{a}_{10}\tilde{a}_{3}^{6}(-10\tilde{a}_{4}^{2}+3\tilde{a}_{3}\tilde{a}_{5})$
		$\displaystyle\hskip 28.45274pt-38710980000\tilde{a}_{3}^{2}\tilde{a}_{4}^{6}\tilde{a}_{6}+2844072000\tilde{a}_{3}^{2}\tilde{a}_{4}^{5}(-49\tilde{a}_{5}^{2}+5\tilde{a}_{3}\tilde{a}_{7})$
		$\displaystyle\hskip 28.45274pt-1422036000\tilde{a}_{3}^{3}\tilde{a}_{4}^{4}(-70\tilde{a}_{5}\tilde{a}_{6}+3\tilde{a}_{3}\tilde{a}_{8})$
		$\displaystyle\hskip 28.45274pt+372314880\tilde{a}_{3}^{4}\tilde{a}_{4}^{2}(-154\tilde{a}_{5}^{2}\tilde{a}_{6}+20\tilde{a}_{3}\tilde{a}_{6}\tilde{a}_{7}+15\tilde{a}_{3}\tilde{a}_{5}\tilde{a}_{8})$
		$\displaystyle\hskip 28.45274pt+206841600\tilde{a}_{3}^{3}\tilde{a}_{4}^{3}(385\tilde{a}_{5}^{3}-132\tilde{a}_{3}\tilde{a}_{5}\tilde{a}_{7}+\tilde{a}_{3}(-77\tilde{a}_{6}^{2}+5\tilde{a}_{3}\tilde{a}_{9}))$
		$\displaystyle\hskip 28.45274pt-1119744\tilde{a}_{3}^{4}\tilde{a}_{4}(10241\tilde{a}_{5}^{4}-7980\tilde{a}_{3}\tilde{a}_{5}^{2}\tilde{a}_{7}+150\tilde{a}_{3}^{2}(4\tilde{a}_{7}^{2}+7\tilde{a}_{6}\tilde{a}_{8})$
		$\displaystyle\hskip 28.45274pt+70\tilde{a}_{3}\tilde{a}_{5}(-133\tilde{a}_{6}^{2}+10\tilde{a}_{3}\tilde{a}_{9}))+186624\tilde{a}_{3}^{5}(22344\tilde{a}_{5}^{3}\tilde{a}_{6}-10080\tilde{a}_{3}\tilde{a}_{5}\tilde{a}_{6}\tilde{a}_{7}$
		$\displaystyle\hskip 28.45274pt-3780\tilde{a}_{3}\tilde{a}_{5}^{2}\tilde{a}_{8}+5\tilde{a}_{3}(-392\tilde{a}_{6}^{3}+135\tilde{a}_{3}\tilde{a}_{7}\tilde{a}_{8}+105\tilde{a}_{3}\tilde{a}_{6}\tilde{a}_{9})))/(1763596800\tilde{a}_{3}^{37/3}).$

Substituting $t=\psi(s)$ into $\gamma(t)$ , and by a straightforward calculation, we see $\gamma(\psi(s))=(s^{3}/6,r_{3,4}s^{4}/4!)+O(4)$ , and we have (A.1). Under the condition $r_{3,4}=0$ , we have $\gamma(\psi(s))=(s^{3}/6,r_{3,5}s^{5}/5!)+O(5)$ , and we have (A.2). We assume $r_{3,4}=r_{3,5}=0$ , Then we see $\gamma(\psi(s))=(s^{3}/6,\beta_{3,6}s^{6}/6!+\beta_{3,7}s^{7}/7!)+O(7)$ , and we have (A.3) and (A.4). We assume $r_{3,7}=0$ . Then we see $\gamma(\psi(s))=(s^{3}/6,\beta_{3,6}s^{6}/6!+\beta_{3,8}s^{8}/8!)+O(8)$ , and we have (A.5). We assume $r_{3,8}=0$ . Then we see $\gamma(\psi(s))=(s^{3}/6,\beta_{3,6}s^{6}/6!+\beta_{3,9}s^{9}/9!+r_{3,10}s^{10}/10!)+O(10)$ , and we have (A.6) and (A.7). ∎

Example A.2.

Let $\gamma$ be a curve-germ $\mathcal{A}^{m+1}$ -equivalent to $(t^{m},t^{m+1})$ . We set

\gamma(t)=\left(\sum_{i=m}^{m+1}\dfrac{a_{i}}{i!}t^{i},\ \sum_{i=m}^{m+1}\dfrac{b_{i}}{i!}t^{i}\right)+O(m+1)\quad((a_{m},b_{m})\neq(0,0)).

(A.9)

Then by a standard rotation $A$ in $\boldsymbol{R}^{2}$ and a parameter change

t\mapsto\bar{a}^{-1/m}\left(t-\dfrac{\bar{a}_{m+1}}{m(m+1)a^{(m+1)/m}}t^{2}\right),

we see

A\gamma(t)=\left(\dfrac{t^{m}}{m!},\dfrac{r_{m,m+1}}{(m+1)!}t^{m+1}\right),\quad\left(r_{m,m+1}=\dfrac{\bar{b}_{m+1}}{\bar{a}_{m}^{(m+1)/m}}\right).

Thus the $(m,m+1)$ -cuspidal curvature is $r_{m,m+1}$ . Here, $\bar{a}_{i}$ and $\bar{b}_{i}$ are

\bar{a}_{i}=\dfrac{\gamma^{(m)}(0)\cdot\gamma^{(i)}(0)}{i!|\gamma^{(m)}(0)|},\quad\bar{b}_{i}=\dfrac{\det(\gamma^{(m)}(0),\gamma^{(i)}(0))}{i!|\gamma^{(m)}(0)|}.

References

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	$\displaystyle\frac{\|\xi{f}\|^{(m+1)/m}\det(\xi{f},\eta^{m}{f},\eta^{m+1}f)}{\|\xi{f}\times\eta^{m}{f}\|^{(2m+1)/m}((m-1)!)^{1/m}}(u,0)=$	$\displaystyle\frac{\|f_{u}\|^{(m+1)/m}\det(f_{u},\psi,\psi_{v})}{\|f_{u}\times\psi\|^{(2m+1)/m}}(u,0)$
	$\displaystyle=$	$\displaystyle\frac{\|f_{u}\|^{(m+1)/m}\det(f_{u},f_{v^{m}},f_{v^{m+1}})}{\|f_{u}\times f_{v^{m}}\|^{(2m+1)/m}}(u,0).$

Boundedness of geometric invariants near a singularity which is a suspension of a singular curve

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

2. Geometry of σ\sigma-edges

Lemma 2.1.

Proof.

2.1. A sufficient condition

Proposition 2.2.

Lemma 2.3.

Proof.

Lemma 2.4.

Proof.

Lemma 2.5.

Proof.

Lemma 2.6.

Proof.

Proof of Proposition 2.2.

Corollary 2.7.

2.2. Normal form of mm or (m,n)(m,n)-type edge

Lemma 2.8.

Proof.

Proposition 2.9.

Proof.

2.3. Geometric invariants

2.3.1. Cuspidal curvatures

Proposition 2.10.

Proof.

Proposition 2.11.

Proof.

Corollary 2.12.

Proof.

Proposition 2.13.

Proof.

Lemma 2.14.

Proof.

2.3.2. Singular, normal curvatures and cuspidal torsion

Proposition 2.15.

Proof.

2.4. Boundedness of Gaussian curvature and mean curvature near an mm-type edge

Lemma 2.16.

Proof.

Theorem 2.17.

Proof.

3. Curves passing through mm-type edges

3.1. Normalized curvatures of singular curves

Proposition 3.1.

Proof.

Proposition 3.2.

Proof.

3.2. Normalized curvatures on frontals

Lemma 3.3.

Proof.

Proposition 3.4.

Proof.

3.3. Behaviors of κg,κn\kappa_{g},\kappa_{n} and τg\tau_{g} passing through an mm-type edge

Theorem 3.5.

Proof.

Corollary 3.6.

Corollary 3.7.

Proof.

Corollary 3.8.

Acknowledgements

Appendix A Generalized biases for plane curve

Example A.1.

Proof of Example A.1.

Example A.2.

References

2. Geometry of $\sigma$ -edges

2.2. Normal form of $m$ or $(m,n)$ -type edge

2.4. Boundedness of Gaussian curvature and mean curvature near an $m$ -type edge

3. Curves passing through $m$ -type edges

3.3. Behaviors of $\kappa_{g},\kappa_{n}$ and $\tau_{g}$ passing through an $m$ -type edge