A note on singular points of bundle homomorphisms from a tangent distribution into a vector bundle of the same rank

TL;DR

This paper investigates the singularities of bundle homomorphisms from tangent distributions to vector bundles of the same rank, especially those induced by Morin maps and contact structures, using Hamilton vector fields.

Contribution

It characterizes fundamental singularities of such bundle homomorphisms, particularly in the context of contact structures, providing new insights into their geometric properties.

Findings

Conditions for fundamental singularities identified

Characterization of singularities via Hamilton vector fields

Application to contact structures

Abstract

We consider bundle homomorphisms between tangent distributions and vector bundles of the same rank. We study the conditions for fundamental singularities when the bundle homomorphism is induced from a Morin map. When the tangent distribution is the contact structure, we characterize singularities of the bundle homomorphism by using the Hamilton vector fields.

Equations

$${\lambda }_{\phi }=\det M_{\phi }.$$

$$(M_{\phi }(p){)}_{u_i}(p)=0$$

$${\tilde{\eta }}^2{\lambda }_{\phi }=a({\eta }_{\phi }a{\eta }_{\phi }{\lambda }_{\phi }+a{\eta }_{\phi }^2{\lambda }_{\phi }+{\eta }_{\phi }b{\lambda }_{\phi })+ba{\eta }_{\phi }{\lambda }_{\phi }+ab{\eta }_{\phi }{\lambda }_{\phi }+b^2{\lambda }_{\phi }.$$

$${\tilde{\eta }}^2{\lambda }_{\phi }(p)=a^2{\eta }_{\phi }^2{\lambda }_{\phi }(p),$$

$${\eta }_{\phi }^3{\lambda }_{\phi }(p)\ne 0\text{and}\mathrm{rank}\left(\begin{array}{ccc}{e}_1{\lambda }_{\phi }& \cdots & e_{r-1}{\lambda }_{\phi }\\ e_1{\eta }_{\phi }{\lambda }_{\phi }& \cdots & e_{r-1}{\eta }_{\phi }{\lambda }_{\phi }\end{array}\right)(p)=2.$$

$${\tilde{\eta }}^3{\lambda }_{\phi }=$$

$$\displaystyle\hskip 28.45274pt+a\Big{(}\eta_{\varphi}^{2}a\,\eta_{\varphi}\lambda_{\varphi}+2\eta_{\varphi}a\,\eta_{\varphi}^{2}\lambda_{\varphi}+a\,\eta_{\varphi}^{3}\lambda_{\varphi}+\eta_{\varphi}^{2}b\lambda_{\varphi}\Big{)}$$

$$\displaystyle\hskip 28.45274pt+\eta_{\varphi}ba\,\eta_{\varphi}\lambda_{\varphi}+ba\,\eta_{\varphi}^{2}\lambda_{\varphi}+\eta_{\varphi}a\,b\eta_{\varphi}\lambda_{\varphi}+a\,\eta_{\varphi}b\eta_{\varphi}\lambda_{\varphi}+\eta_{\varphi}b^{2}\lambda_{\varphi}\bigg{)}+b*,$$

$${\tilde{\eta }}^3{\lambda }_{\phi }(p)=a^3{\eta }_{\phi }^3{\lambda }_{\phi }.$$

$$\Big{(}e_{i}\tilde{\eta}\lambda_{\varphi}\Big{)}(p)=\Big{(}e_{i}(a\,\eta_{\varphi}\lambda_{\varphi}+b\lambda_{\varphi})\Big{)}(p)=\Big{(}e_{i}a\,\eta_{\varphi}\lambda_{\varphi}+a\,e_{i}\eta_{\varphi}\lambda_{\varphi}+e_{i}b\lambda_{\varphi}\Big{)}(p).$$

$$\displaystyle\Big{(}e_{i}a\,\eta_{\varphi}\lambda_{\varphi}+a\,e_{i}\eta_{\varphi}\lambda_{\varphi}+e_{i}(k\lambda_{\varphi})\Big{)}(p)$$

$$=a(p)e_i{\eta }_{\phi }{\lambda }_{\phi }(p)+k(p)e_i{\lambda }_{\phi }(p).$$

$$\mathrm{rank}\left(\begin{array}{ccc}{e}_1{\lambda }_{\phi }& \cdots & e_{r-1}{\lambda }_{\phi }\\ e_1\tilde{\eta }{\lambda }_{\phi }& \cdots & e_{r-1}\tilde{\eta }{\lambda }_{\phi }\end{array}\right)(p)$$

$$=$$

$$\mu (t)=\left(\begin{array}{cc}{\gamma }_1(t)& a(t)\\ {\gamma }_2(t)& b(t)\end{array}\right).$$

$$\mu (0)=0,{\mu }^{\prime }(0)\ne 0.$$

$$3a_v(p){\lambda }_{uv}(p)+{\lambda }_{vvv}(p)\ne 0.$$

$${\gamma }_1^{\prime \prime }(0)=-\frac{2a_v(p){\lambda }_{uv}(p)+{\lambda }_{vvv}(p)}{{\lambda }_{uv}(p)}.$$

$${\mu }^{\prime }(0)={\gamma }_1^{\prime \prime }(0)-a^{\prime }(0)=-\frac{3a_v(p){\lambda }_{uv}(p)+{\lambda }_{vvv}(p)}{{\lambda }_{uv}(p)}\ne 0.$$

$$\{\phi \in \Gamma (E)|\text{any/ }p\in S\text{ is fold-like, cusp-like or swallowtail-like/}\}$$

$$\{f\in \Gamma (E)|j^{k}f\text{ is transverse to/ }K\}$$

$$Z=\{j^3\phi (p)\in J^3(M,E)|\phi (p)=O\}.$$

$$D=$$

$$W_1=$$

$$\displaystyle\hskip 108.12047pt\eta\eta\det\varphi(p)=0,\ \eta\eta\eta\det\varphi(p)=0\Big{\}},$$

$$W_2=$$

$$\displaystyle\hskip 108.12047pt\eta\eta\det\varphi(p)=0,\ \operatorname{rank}d(\det\varphi,\eta\det\varphi)(p)=1\Big{\}}.$$

$$\mathcal{O}=\{\phi \in \Gamma (E)|j^3\phi \text{ is transverse to }Z,D,W_1\text{ and }{W}_2\}.$$

$$j^3\phi (p)=\left(\begin{array}{cc}{j}^{3}a(p)& j^{3}b(p)\\ j^{3}c(p)& j^{3}d(p)\end{array}\right),$$

$$\displaystyle W_{1}=\Biggl{\{}\left.{\begin{pmatrix}j^{3}a(p)&j^{3}b(p)\\ j^{3}c(p)&j^{3}d(p)\end{pmatrix}}\,\right|\,p\in U,\ h_{1}(p)=h_{2}(p)=h_{3}(p)=h_{4}(p)=0\Biggr{\}},$$

$$\displaystyle W_{2}=\Biggl{\{}\left.{\begin{pmatrix}j^{3}a(p)&j^{3}b(p)\\ j^{3}c(p)&j^{3}d(p)\end{pmatrix}}\,\right|\,p\in U,\ h_{1}(p)=h_{2}(p)=h_{3}(p)=h_{5}(p)=0\Biggr{\}},$$

$$h_5=(ad-bc{)}_u(ad-bc{)}_{vw}-(ad-bc{)}_v(ad-bc{)}_{uw}.$$

$$\left(\begin{array}[]{c|cccc}{\cal M}&\\ \hline\cr*&d&-c&-b&a\end{array}\right),\quad{\cal M}=\left(\begin{array}[]{cccccccccccccccc}d&-c&-b&a&&&&&&&&\\ &*&*&*&d&-c&-b&a&&&&\\ &*&*&*&*&*&*&*&d&-c&-b&a\\ \end{array}\right)$$

$$\left(\begin{array}[]{c|ccccccccc}{\cal M}&&&&&&&\\ \hline\cr*&d(h_{1})_{v}&-c(h_{1})_{v}&-b(h_{1})_{v}&a(h_{1})_{v}&d(h_{1})_{u}&-c(h_{1})_{u}&-b(h_{1})_{u}&a(h_{1})_{u}\\ \end{array}\right),$$

Full text

1 Introduction

In [5, 7], the notion of coherent tangent bundle is introduced. It is a bundle homomorphism between the tangent bundle and a vector bundle with the same rank with a kind of metric. This is a generalization of fronts and $C^{\infty}$-maps between the same dimensional manifolds. Singular points of bundle homomorphisms $\varphi:TM\to E$ are points where $\varphi(p):T_{p}M\to E_{p}$ is not a bijection. In [5, 7], differential geometric invariants of singularities of bundle homomorphisms are defined and investigated. On the other hand, in [8], topological properties of singular sets of bundle homomorphisms without metric are studied. See [1] for another kind of application of coherent tangent bundle. In this paper, we consider rank $r(<m)$ tangent distributions instead of the tangent bundles of $m$-dimensional manifolds. Since $r<m$, the singularities appearing on the bundle homomorphisms are slightly different from the case $\varphi:TM\to E$, where $\dim M=\operatorname{rank}E=m$, and the case $\varphi:TM\to E$, where $\dim M=\operatorname{rank}E=r$ either.

Let $D_{1}$ be a rank $r$ tangent distribution on an $m$-dimensional manifold $M$. Let $N$ be an $r$ dimensional manifold, and $f:M\to N$ a map. Then a bundle homomorphism $\varphi=df:D_{1}\to f^{*}TN$ is induced from $f$. Singularities of $\varphi$ should be related to $D_{1}$ and $f$. In this paper, we stick to our interest into the low dimensional case, we study the relationships when $f$ is a Morin map, and $D_{1}$ is the foliation or the contact structure when $m=3$, $r=2$.

2 Bundle homomorphisms and their singular point

2.1 Singular points of bundle homomorphisms

With the terminology of [7], we give definition of singular points of bundle homomorphisms. Let $M$ be an $m$-dimensional manifold, and let $D_{1}$ be a rank $r$ ($r<m)$ tangent distribution of $M$ namely, a subbundle of $TM$. Let $D_{2}$ be a rank $r$ vector bundle over $M$, and let $\varphi:D_{1}\to D_{2}$ be a bundle homomorphism. If the rank of the linear map $\varphi_{p}:(D_{1})_{p}\to(D_{2})_{p}$ is less than $r$, then $p\in M$ is called singular point of $\varphi$. We denote by $S$ the set of singular points of $\varphi$. If the rank of $\varphi_{p}$ is $r-1$, then $p$ is called a corank one singular point.

Lemma 2.1**.**

If $p$ is a corank one singular point of $\varphi$. Then there exists a neighborhood $U$ of $p$ and a section $\eta_{\varphi}\in\varGamma(D_{1})$ such that if $q\in S\cap U$ then $(\eta_{\varphi})_{q}$ is a generator of the kernel of $\varphi_{q}$.

Proof.

By taking frames of $D_{1},D_{2}$, we consider $\varphi$ as a matrix $M_{\varphi}$ near $p$. Since $\operatorname{rank}M_{\varphi}(p)=r-1$, only one eigenvalue of $M_{\varphi}(p)$ is zero and the others are not zero. Thus the eigenvalue having minimum absolute value among the eigenvalues of $M_{\varphi}$ is uniquely determined, and is a real valued $C^{\infty}$ function near $p$. Hence corresponding eigenvector $\eta_{\varphi}$ is also well-defined. We have the desired section identifying $\eta_{\varphi}$ as a section. ∎

We call $\eta_{\varphi}$ the null section of $\varphi$. We set

[TABLE]

We call $p\in S$ is non-degenerate if $d\lambda_{\varphi}(p)\neq 0$. The notions of the null section and the non-degeneracy is introduced in [2].

Lemma 2.2**.**

Non-degenerate singular points are of corank one.

Proof.

Let $p$ be a non-degenerate singular point. We assume that $\operatorname{rank}M_{\varphi}(p)<r-1$. Then any $r-1$ rows of $M_{\varphi}(p)$ are linearly dependent. Thus

[TABLE]

holds for any $1\leq i\leq m$, where $(u_{1},\ldots,u_{m})$ is a coordinate system near $p$, and $(~{})_{u_{i}}=\partial/\partial u_{i}$. This is a contradiction. ∎

Since $S=\{\lambda_{\varphi}(p)=0\}$, $S$ is a codimension one submanifold near a non-degenerate singular point. With the terminology of [6], we give the following definition:

Definition 2.3**.**

We call a singular point $p\in S$ is a fold-like singular point if it is corank one, and $\eta_{\varphi}\lambda_{\varphi}(p)\neq 0$. We call $p\in S$ is a cusp-like singular point if $p$ is non-degenerate and $\eta_{\varphi}\lambda_{\varphi}(p)=0$ and $\eta_{\varphi}^{2}\lambda_{\varphi}(p)\neq 0$. We call $p\in S$ is a swallowtail-like singular point if $p$ is non-degenerate, and $\eta_{\varphi}\lambda_{\varphi}(p)=\eta_{\varphi}^{2}\lambda_{\varphi}(p)=0$ and $\operatorname{rank}d(\lambda_{\varphi},\eta_{\varphi}\lambda_{\varphi},\eta_{\varphi}^{2}\lambda_{\varphi})=3$ at $p$.

Here, $\xi f$ stands for the directional derivative of a function $f$ by the vector field $\xi$, and $\xi^{i}f$ stands for the $i$ times directional derivative by $\xi$.

Lemma 2.4**.**

The definitions of fold-like, cusp-like and swallowtail-like singular points do not depend on the choice of the frames of $D_{1}$, $D_{2}$, nor on the choice of the null section.

Proof.

We change the frames of $D_{1}$ by a matrix $C_{1}$, and change the frames of $D_{2}$ by a matrix $C_{2}$. Then $M_{\varphi}$ is changed to $C_{2}^{-1}M_{\varphi}C_{1}$. Thus the independence of the choice of frames are clear. We show the independence of the choice of the null section, and the case of fold-like singular points are also clear, since $\eta_{\varphi}\lambda_{\varphi}(p)$ is a directional derivative of $(\eta_{\varphi})_{p}$. Furthermore, the independence of the non-degeneracy is also clear. We set $\tilde{\eta}=a\eta_{\varphi}+b$, where $a$ is a non-zero function, and $b$ is a vector field which vanishes on $S$. Let $p\in S$ be a non-degenerate singular point. We assume that $d\lambda_{\varphi}(p)\neq 0$ and $\eta_{\varphi}\lambda_{\varphi}(p)=0$. Then we have

[TABLE]

Since $b=0$ on $S$, $b\lambda_{\varphi}=0$ on $S$, and since $\eta_{\varphi}\lambda_{\varphi}(p)=0$, it holds that $\eta_{\varphi}b\lambda_{\varphi}(p)=0$. Thus

[TABLE]

we see $\tilde{\eta}^{2}\lambda_{\varphi}(p)=0$ is equivalent to $\eta_{\varphi}^{2}\lambda_{\varphi}(p)=0$.

Next we assume $d\lambda_{\varphi}(p)\neq 0$ and $\eta_{\varphi}\lambda_{\varphi}(p)=\eta_{\varphi}^{2}\lambda_{\varphi}(p)=0$. We chose a frame $\{e_{1},\ldots,e_{r-1},\eta_{\varphi}\}$ of $D_{1}$. Then the condition for swallowtail-like singular point is equivalent to

[TABLE]

Then we have

[TABLE]

where $*$ stand for a function. Since $b\eta_{\varphi}\lambda_{\varphi}$ and $b^{2}\lambda_{\varphi}$ vanish on $S$, and by $\eta_{\varphi}\lambda_{\varphi}(p)=0$, it holds that $\eta_{\varphi}b\eta_{\varphi}\lambda_{\varphi}(p)=\eta_{\varphi}b^{2}\lambda_{\varphi}(p)=0$. We show $\eta_{\varphi}^{2}b\lambda_{\varphi}(p)=0$. Since $b=0$ on $S=\{\lambda_{\varphi}=0\}$, and $d\lambda_{\varphi}(p)\neq 0$, there exists a function $k$ such that $b\lambda_{\varphi}=k\lambda_{\varphi}$. Since $\eta_{\varphi}\lambda_{\varphi}(p)=\eta_{\varphi}^{2}\lambda_{\varphi}(p)=0$, we see $\eta_{\varphi}^{2}b\lambda_{\varphi}(p)=0$. Hence

[TABLE]

On the other hand, we have

[TABLE]

By the above, $b\lambda_{\varphi}=k\lambda_{\varphi}$ holds, and hence the right hand side of (2.1) is

[TABLE]

Thus

[TABLE]

shows the assertion. ∎

2.2 Geometric interpretation of singularities

We give geometric interpretation of singularities of bundle homomorphisms. If $p\in S$ is a non-degenerate singular point, then $S$ is a codimension one submanifold. Thus $T_{p}S\subset TM$ can be defined. Let us set $m=3$ and $r=2$. Then we have the following.

Proposition 2.5**.**

If $p\in S$ is a fold-like singular point, then $\eta_{p}\not\in T_{p}S$. If $p\in S$ is a cusp-like singular point, then $S_{2}=\{p\in S\,|\,\eta_{p}\in T_{p}S\}$ is one-dimensional submanifold of $S$, and $\eta_{p}\not\in T_{p}S_{2}$.

Proof.

Since $S=\{\lambda_{\varphi}=0\}$, the first assertion is obvious. By non-degeneracy, $\eta_{\varphi}\lambda_{\varphi}(p)=0$ and $\eta_{\varphi}^{2}\lambda_{\varphi}(p)\neq 0$, it holds that $d(\lambda_{\varphi},\eta_{\varphi}\lambda_{\varphi})(p)\neq 0$. Since $S_{2}=\{\lambda_{\varphi}=\eta_{\varphi}\lambda_{\varphi}=0\}$, $S_{2}$ is a one-dimensional submanifold of $S$. The last assertion is obvious from $\eta_{\varphi}^{2}\lambda_{\varphi}(p)\neq 0$. ∎

If $p$ is a fold-like singular point, and $(D_{1})_{p}=T_{p}S$, then $(\eta_{\varphi})_{p}\in T_{p}S$. Thus $(D_{1})_{p}\neq T_{p}S$. Let $p$ be a cusp-like singular point. If $e_{1}\lambda_{\varphi}=e_{2}\lambda_{\varphi}=0$ at $p$, then $(D_{1})_{p}=T_{p}S$. In this case, we call $p$ cusp-like singular point of tangent type. If $(e_{1}\lambda_{\varphi},e_{2}\lambda_{\varphi})\neq(0,0)$ at $p$, then $(D_{1})_{p}$ is transversal to $T_{p}S$. In this case, we call $p$ cusp-like singular point of transverse type. The picture of $S$ and $D_{1}$ can be drawn in Figure 1.

If $p\in S$ is a swallowtail-like singular point, then $S_{2}$ is one-dimensional submanifold of $S$. Let $(u,v)$ be a coordinate system near $p$ of $S$. Let $\gamma(t)=(\gamma_{1}(t),\gamma_{2}(t))$ $(\gamma(0)=p)$ be a parameterization of $S_{2}$ with respect to $(u,v)$, and let $\eta_{\gamma(t)}=a(t)\partial_{u}+b(t)\partial_{v}$. Then we have the following.

Proposition 2.6**.**

Let $p\in S$ is a swallowtail-like singular point. We set

[TABLE]

Under the above notation, it holds that

[TABLE]

Proof.

Let $(u,v)$ be a coordinate system of $S$ satisfying $\partial_{v}=\eta_{p}$. If we assume that $(\eta\lambda_{\varphi})_{u}=0$, then since $S=\{\lambda_{\varphi}=0\}$, it holds that $(\lambda_{\varphi})_{u}=0$. This contradicts to $\operatorname{rank}(\lambda_{\varphi},\eta_{\varphi}\lambda_{\varphi},\eta_{\varphi}^{2}\lambda_{\varphi})(p)=3$. Since $(\eta\lambda_{\varphi})_{u}\neq 0$, we have a parametrization of $\gamma$ as $\gamma(t)=(\gamma_{1}(t),t)$. Since $\eta_{\varphi}^{2}\lambda_{\varphi}(p)=0$, we have $\gamma_{1}^{\prime}(0)=0$. On the other hand, we may take $\eta_{\gamma(t)}=a(t)\partial_{u}+\partial_{v}$ $(a(0)=0)$. Then $\mu(t)=\gamma_{1}^{\prime}(t)-a(t)$.

Since $\eta_{\varphi}^{3}\lambda_{\varphi}(p)\neq 0$ and $a(0)=0$, $(\lambda_{\varphi})_{u}(p)=0$, we have

[TABLE]

On the other hand, since $\eta_{\varphi}\lambda_{\varphi}(\gamma_{1}(v),v))=0$, we have

[TABLE]

By (2.2) and (2.3), we have

[TABLE]

∎

Like as the case of cusp-like singular point, swallowtail-like singular point has tangent and transverse types. If $e_{1}\lambda_{\varphi}=e_{2}\lambda_{\varphi}=0$ at $p$, then $(D_{1})_{p}=T_{p}S$. In this case, we call $p$ swallowtail-like singular point of tangent type. If $(e_{1}\lambda_{\varphi},e_{2}\lambda_{\varphi})\neq(0,0)$ at $p$, then $(D_{1})_{p}$ is transversal to $T_{p}S$. In this case, we call $p$ swallowtail-like singular point of transverse type (Figure 2). Ignoring arrangements of $D_{1}$, relationship of $S,S_{2}$ and $\eta_{\varphi}$ is similar to that of the Morin singularities of $(\boldsymbol{R}^{3},0)\to(\boldsymbol{R}^{3},0)$ ([6]).

3 Generic singularities

We show if $m=3$ and $r=2$, then the generic singularities of $\varphi$ is fold-like, cusp-like and swallowtail-like singular points. The bundle homomorphism $\varphi$ can be regarded as a section of the homomorphism bundle $\operatorname{Hom}(D_{1},D_{2})$. We set $E=\operatorname{Hom}(D_{1},D_{2})$. Since the set of sections $\varGamma(E)$ is a subset of $C^{\infty}(M,E)$, we derive the Whitney $C^{\infty}$ topology to $\varGamma(E)$.

Proposition 3.1**.**

Under the above settings, the set

[TABLE]

is dense.

For the proof of Proposition 3.1, we need jet transversality theorem for vector bundle sections. Let $J^{k}(\varGamma(E))$ be the subbundle of $J^{k}(M,E)$ consisting of all $k$-jets of sections. Let $j^{k}:\varGamma(E)\to C^{\infty}(M,J^{k}(\varGamma(E)))$ be the jet-extension.

Proposition 3.2**.**

Let $M$ be a manifold and let $K$ be a submanifold of $J^{k}(\varGamma(E))$. Then the set

[TABLE]

is residual in $\varGamma(E)$, and open dense if $K$ is closed.

This is shown [9, Theorem 2.6], for sections of the tangent bundle. However the proof uses the local triviality of the tangent bundle, so the same proof works for the case interchanging the tangent bundle to a general vector bundle $E$.

Proof of Proposition 3.1.

We set

[TABLE]

Then $Z$ is independent of the choice of frames, and a closed submanifold of codimension $4$, and $J^{3}(M,E)\setminus Z$ is an open submanifold. Next, we set

[TABLE]

Then $D$ is independent of the choice of frames, and are closed submanifolds of $J^{3}(M,E)\setminus Z$ of codimension $4$. Next we consider

[TABLE]

Then $W_{1},W_{2}$ are independent of the choice of frames, and if they are closed submanifolds of $J^{3}(M,E)\setminus(Z\cup D)$ of codimension $4$. By Proposition 3.2,

[TABLE]

is a residual subset of $\varGamma(E)$. So is dense. On the other hand, since $\dim M=3$, $j^{3}\varphi$ is transverse to $Z$, $D$, $W_{1}$ and $W_{2}$ is equivalent to $j^{3}\varphi(M)\cap(Z\cup D\cup W_{1}\cup W_{2})=\emptyset$. Thus, for any $\varphi\in{\cal O}$ has only fold-like, cusp-like and swallowtail-like singular points as singular points. Thus the proof is reduced to showing the following lemma. ∎

Lemma 3.3**.**

The sets $W_{1},W_{2}$ are closed submanifolds of $J^{3}(M,E)\setminus(Z\cup D)$ of codimension $4$.

Proof.

Let $p\in M$ and take a coordinate neighborhood $U$ near $p$. It is enough to show that $W_{1},W_{2}$ are closed submanifolds in $J^{3}(U,E|_{U})\setminus(Z\cup D)$. Since $W_{1},W_{2}$ are independent of the choice of coordinate system, we choose a coordinate system $(u,v,w)$ on $U$ satisfying $\partial_{w}=\eta$. Let

[TABLE]

where $a,b,c,d$ are functions. Then in $J^{3}(U,E|_{U})\setminus(Z\cup D)$,

[TABLE]

where $h_{1}=ad-bc$, $h_{2}=(ad-bc)_{w}$, $h_{3}=(ad-bc)_{ww}$, $h_{4}=(ad-bc)_{www}$, and

[TABLE]

We define two functions $H_{i}:J^{3}(U,E|_{U})\setminus(Z\cup D)\to\boldsymbol{R}^{4}$ $(i=1,2)$ by $H_{1}=(h_{1},h_{2},h_{3},h_{4})$, $H_{2}=(h_{1},h_{2},h_{3},h_{5})$. Then it is sufficient to show that $(0,0,0,0)$ is a regular value of each $H_{1}$ and $H_{2}$. We calculate the derivative of $H_{1}$ with respect to the $16$ coordinates of $J^{3}(U,E|_{U})$ corresponding to the zero, first, second and third derivatives by $\partial_{w}$ of $a,b,c,d$. The matrix representation of them is

[TABLE]

where the blank entries are [math]. Since $(a,b,c,d)\neq(0,0,0,0)$, we have the assertion for $H_{1}$. Next we calculate the derivative of $H_{2}$ with respect to the $20$ coordinates of $J^{3}(U,E|_{U})$ corresponding to the zero, first, second derivatives by $\partial_{w}$ of $a,b,c,d$ and corresponding to the derivatives by $\partial_{u}$, $\partial_{w}$, and $\partial_{v}$, $\partial_{w}$ of $a,b,c,d$. The matrix representation of them is

[TABLE]

Since $(a,b,c,d)\neq(0,0,0,0)$, and $(h_{1})_{u}(p)=(h_{1})_{v}(p)=h_{2}(p)=0$ means $d\det\varphi(p)=(0,0,0)$, we have the assertion for $H_{2}$. ∎

4 Morin singularities from a manifold with tangent distribution

Let $D_{1}$ be a rank $r$ tangent distribution on $M$, and let $N$ be an $r$-dimensional manifold, and $f:M\to N$ a map. Setting $D_{2}=f^{*}TN$ and $\varphi:D_{1}\to D_{2}$ by

[TABLE]

we obtain a bundle homomorphism between $D_{1}$ and $D_{2}$. We call the above $\varphi$ a bundle homomorphism induced by $f$. In this section, assuming $f$ be a Morin singularity, we consider relationships of $\varphi$, $D_{1}$ and $f$ in the case of $m=3$, $r=2$. Moreover, we assume that $M$ is an open neighborhood of [math] in $\boldsymbol{R}^{3}$, $N$ is an open neighborhood of [math] in $\boldsymbol{R}^{2}$, and $f:(\boldsymbol{R}^{3},0)\to(\boldsymbol{R}^{2},0)$.

4.1 Morin singularities

We give a belief review on the Morin singularities of $(\boldsymbol{R}^{3},0)\to(\boldsymbol{R}^{2},0)$. The map-germ $f:(\boldsymbol{R}^{3},0)\to(\boldsymbol{R}^{2},0)$ is called a definite fold (respectively, a indefinite fold) if it is ${\cal A}$-equivalent to the map-germ $(u,v,w)\mapsto(u,v^{2}+w^{2})$ (respectively, $(u,v^{2}-w^{2})$) at [math]. Two map-germs $f,g:(\boldsymbol{R}^{m},0)\to(\boldsymbol{R}^{n},0)$ are ${\cal A}$-equivalent if there exist diffeomorphism-germs $\sigma:(\boldsymbol{R}^{m},0)\to(\boldsymbol{R}^{m},0)$ and $\tau:(\boldsymbol{R}^{n},0)\to(\boldsymbol{R}^{n},0)$ such that $\tau\circ f\circ\sigma^{-1}=g$. The map-germ $f:(\boldsymbol{R}^{3},0)\to(\boldsymbol{R}^{2},0)$ is called a cusp if it is ${\cal A}$-equivalent to the map-germ $(u,v,w)\mapsto(u,v^{2}+w^{3}+uw)$. Definite fold, indefinite fold and cusp are called Morin singularities, and it is known that generic singularities appearing on maps from a $3$-manifold to a $2$-manifold are only Morin singularities. A characterization of Morin singularities is given as follows: Let $f:(\boldsymbol{R}^{3},0)\to(\boldsymbol{R}^{2},0)$ be a map-germ and $\operatorname{rank}df_{0}=1$. Then there exists a pair of vector fields $\{\xi,\eta_{1},\eta_{2}\}$ such that

[TABLE]

where $S(f)$ is the set of singular points of $f$. We set

[TABLE]

Then $f$ at [math] is a definite fold (respectively, indefinite fold) if and only if $\det H(0)>0$ (respectively, $\det H(0)<0$). We assume that $\operatorname{rank}H(0)=1$, then there exists a vector field $\theta=a_{1}\eta_{1}+a_{2}\eta_{2}$ on $S(f)$ such that $\left\langle{\theta_{0}}\right\rangle=\ker H(0)$. Then $f$ at [math] is a cusp if and only if $\theta H(0)\neq 0$. See [4] in detail.

4.2 Conditions for singularities

We take a frame $\{e_{1},e_{2}\}$ of $D_{1}$. We regard $e_{1}$, $e_{2}$ as vector fields. We consider the conditions of singular points of fold-like, cusp-like and swallowtail-like singular points under the assumption that $f$ is regular, fold and cusp since these are generic singular points.

When $f$ is regular at [math], and $D_{1}\not\subset\ker df_{0}$, then $\varphi$ is non-singular. When $f$ is singular at [math], and $D_{1}\subset\ker df_{0}$, then $\varphi$ is of rank zero at [math]. Since we are stick to rank one singular points of $\varphi$, we assume that $D_{1}\cap\ker df_{0}$ is one-dimensional. By changing frame, we may assume that $e_{1}f(0)\neq 0$. The bundle homomorphism $\varphi$ can be represented by the matrix

[TABLE]

by $\{e_{1},e_{2}\}$ and the trivial frame on $\boldsymbol{R}^{2}$. Since $\operatorname{rank}\varphi=1$ at [math], we take a null section $\eta_{\varphi}$, and set

[TABLE]

The following proposition holds.

Proposition 4.1**.**

The singular point $p$ of $\varphi$ is fold-like singular point if and only if $\det(e_{1}f,\eta_{\varphi}^{2}f)\neq 0$ at $p$. A non-degenerate singular point $p$ is cusp-like singular point $($respectively, swallowtail-like singular point$)$ if and only if $\det(e_{1}f,\eta_{\varphi}^{2}f)=0$, and $\det(e_{1}f,\eta_{\varphi}^{3}f)\neq 0$ at $p$ $($respectively, $\det(e_{1}f,\eta_{\varphi}^{2}f)=\det(e_{1}f,\eta_{\varphi}^{3}f)=0$, $\det(e_{1}f,\eta_{\varphi}^{4}f)\neq 0$, and d\det\big{(}\det(e_{1}f,\eta_{\varphi}f),\det(e_{1}f,\eta_{\varphi}^{2}f),\det(e_{1}f,\eta_{\varphi}^{3}f)\big{)}\neq 0 at $p)$.

Proof.

Since $\eta_{\varphi}f(p)=0$, it is obvious that the assertion for the fold-like singular point. Let $p$ be a non-degenerate singular point, and $\eta_{\varphi}\lambda(p)=0$. Since $\eta_{\varphi}f=0$ on $S=\{\lambda_{\varphi}=0\}$, and $p$ is non-degenerate, there exists a vector valued function $g$ such that $\eta_{\varphi}f=\lambda_{\varphi}g$. Then by the assumption $\eta_{\varphi}\lambda(p)=0$, $\eta_{\varphi}^{2}f(p)=0$. Hence $\eta_{\varphi}^{2}\lambda_{\varphi}(p)\neq 0$ is equivalent to $\det(e_{1}f,\eta_{\varphi}^{3}f)(p)\neq 0$. This proves the assertion for the cusp-like singular point. Next we assume that $p$ be a non-degenerate singular point, and $\eta_{\varphi}\lambda(p)=\eta_{\varphi}^{2}\lambda(p)=0$. Then by the same reason as above, we have $\eta_{\varphi}^{2}f(p)=\eta_{\varphi}^{3}f(p)=0$. Thus we see that $\eta_{\varphi}^{3}\lambda_{\varphi}(p)\neq 0$ is equivalent to $\det(e_{1}f,\eta_{\varphi}^{4}f)(p)\neq 0$, and $\det(\lambda_{\varphi},\eta_{\varphi}\lambda_{\varphi},\eta^{2}_{\varphi}\lambda_{\varphi})(p)\neq 0$ is equivalent to d\det\big{(}\det(e_{1}f,\eta_{\varphi}f),\det(e_{1}f,\eta_{\varphi}^{2}f),\det(e_{1}f,\eta_{\varphi}^{3}f)\big{)}(p)\neq 0. This proves the assertion. ∎

4.3 Restriction of singularities of $\varphi$ by singular types of $f$

We assume that $f$ at [math] is a definite fold singular point. Then $\operatorname{rank}(e_{1}f,e_{2}f,e_{3}f)=1$ on $S(f)$, where $\{e_{1},e_{2},e_{3}\}$ is a frame of $T\boldsymbol{R}^{3}$. Thus there exist functions $k_{1},k_{2}$ such that $e_{2}f=k_{1}e_{1}f,e_{3}f=k_{2}e_{1}f$ on $S(f)$. Taking extensions of $k_{1},k_{2}$ on $\boldsymbol{R}^{3}$, we set

[TABLE]

and also set

[TABLE]

Then we see that $\eta_{2}$ is a null section of $\varphi$, and $\lambda_{2}$ is the same as $\lambda_{\varphi}$. Since $f$ is definite fold,

[TABLE]

In particular, $\eta_{2}\lambda_{2}\neq 0$. Thus $\varphi$ is fold-like at [math] if $\operatorname{rank}\varphi(0)=1$.

Next we assume that $f$ at [math] is a cusp singular point. Then we take $k_{1},k_{2},\eta_{2},\eta_{3}$ and $\lambda_{2},\lambda_{3}$ as above. We assume that $\varphi$ is not fold-like, namely, $\eta_{2}\lambda_{2}(0)=0$. Then since $f$ is cusp,

[TABLE]

Since $\eta_{3}\lambda_{2}(0)=\eta_{2}\lambda_{3}(0)$, it holds that $\eta_{3}\lambda_{2}(0)=0$. Hence the kernel of $H$ is $\theta=\eta_{1}$ at [math]. Then $f$ is cusp if and only if

[TABLE]

Thus $\varphi$ is non-degenerate and not fold-like at [math], then $\varphi$ is cusp-like at [math].

4.4 The case $D_{1}$ is a foliation

In this section, we assume $D_{1}$ is a foliation. By taking a coordinate system $(x,y,z)$ on $\boldsymbol{R}^{3}$, we may assume $D_{1}=\left\langle{e_{1},e_{2}}\right\rangle=\left\langle{\partial_{x},\partial_{y}}\right\rangle$. Let $L(x,y)$ be the leaf which contains the origin, namely, $L(x,y)=f(x,y,0)$. We have the following proposition.

Proposition 4.2**.**

Under the above setting, the following holds:* (1) $\varphi$ is fold-like if and only if $L$ is fold. (2) if $\varphi$ is non-degenerate, then $\varphi$ is cusp-like if and only if $L$ is cusp. (3) if $\varphi$ satisfies that $\operatorname{rank}d(\lambda_{\varphi},\eta_{\varphi}\lambda_{\varphi})(0)=2$, then $\varphi$ is swallowtail-like if and only if $L$ is swallowtail.*

A map-germ $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{2},0)$ is called a fold if $f$ is ${\cal A}$-equivalent to the map-germ $(u,v)\mapsto(u,v^{2})$ at [math]. Furthermore, a map-germ $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{2},0)$ is called a cusp (respectively, swallowtail) if $f$ is ${\cal A}$-equivalent to $(u,v)\mapsto(u,v^{3}+uv)$ at [math] (respectively, $(u,v)\mapsto(u,v^{4}+uv)$ at [math]). Criteria for these singularities are obtained as follows: Let $f:(\boldsymbol{R}^{2},0)\to(\boldsymbol{R}^{2},0)$ be a map-germ. We set $\lambda=\det J$, where $J$ is the Jacobian matrix of $f$. A singular point $p\in S(f)$ is non-degenerate if $d\lambda(p)\neq 0$. Then the following holds.

Fact 4.3**.**

[10, 6, 3]* A singular point $p$ is fold if $\eta\lambda(p)\neq 0$. Moreover, a non-degenerate singular point $p$ is cusp $($respectively, swallowtail$)$ if $\eta\lambda(p)=0$ and $\eta^{2}\lambda(p)\neq 0$ $($respectively, $\eta\lambda(p)=\eta^{2}\lambda(p)=0$ and $\eta^{3}\lambda(p)\neq 0)$.*

Proof of Proposition 4.2.

We may assume that $f_{x}(0,0)\neq 0$. Then there exists a function $k_{1}(x,y,z)$ such that if $p\in S$, then

[TABLE]

Take an extension of $k_{1}$ on $U$, we take a null section

[TABLE]

On the other hand, there exists a function $l(x,y)$ such that if $q\in S(L)$, then

[TABLE]

Take an extension of $l$ on $U\cap\{z=0\}$, we take a null vector field of $L$

[TABLE]

Set $\lambda_{L}(x,y)=\det(e_{1}f(x,y,0),\eta_{L}f(x,y,0))$. Then since $\lambda_{\varphi}(x,y,0)=\lambda_{L}(x,y)$, and $\eta_{\varphi}(x,y,0)=\eta_{L}(x,y)$, we see

[TABLE]

The assertion is obvious by Fact 4.3 and (4.1). ∎

4.5 The case $D_{1}$ is a contact structure

In this section, we assume $D_{1}$ is a contact structure. Since the Hamilton vector field $X$ of $\lambda_{\varphi}$ is contained in $D_{1}$ on $S$, we consider the relationship with the behavior of $X$ and the singularities of $\varphi$. We may assume $D_{1}=\left\langle{e_{1},e_{2}}\right\rangle=\left\langle{\partial_{x},\partial_{y}-x\partial_{z}}\right\rangle$ without loss of generality. Since $\varphi$ can be expressed by $(f_{x},f_{y}-xf_{z})$,

[TABLE]

The Hamilton vector field $X$ of $\lambda_{\varphi}$ is

[TABLE]

Since $S=\{\lambda_{\varphi}=0\}$ holds, $X_{p}\in D_{1}$ is equivalent to $p\in S$. We have the following theorem.

Theorem 4.4**.**

If $\varphi$ has a corank one singular point at $p$, under the above setting, $p\in S$ is fold-like if and only if

[TABLE]

are linearly independent, where $\eta_{\varphi}$ is a null section of $\varphi$.

Proof.

Since $\varphi$ is a corank one singular point at $p$, there exist functions $k_{1},k_{2}$ on $S$ such that $(k_{1},k_{2})\neq(0,0)$ and $k_{1}e_{1}f+k_{2}e_{2}f=0$. Expanding $k_{1},k_{2}$ to a neighborhood of $p$, we can take a null section $\eta_{\varphi}=k_{1}e_{1}+k_{2}e_{2}$. Then

[TABLE]

shows the assertion. ∎

By Theorem 4.4, on the set of non-fold-like singular points, $X$ is parallel to the null vector field, by Propositions 2.5 and 2.6, we have the following corollary.

Corollary 4.5**.**

If $p\in S$ is a cusp-like singular point, then $X_{p}\not\in T_{p}S_{2}$. If $p\in S$ is a swallowtail-like singular point. Then

[TABLE]

where $\gamma(t)=(\gamma_{1}(t),\gamma_{2}(t))$ $(\gamma(0)=p)$ is a parameterization of $S_{2}$, and $\eta_{\gamma(t)}=a(t)\partial_{u}+b(t)\partial_{v}$, and

[TABLE]