Composing generic linearly perturbed mappings and immersions/injections
Shunsuke Ichiki
Daz \textsc{Baz
Let $N$ (resp., $U$) be a manifold (resp., an open subset of $\R^m$). Let $f:N\to U$ and $F:U\to \R^\ell$ be an immersion and a mapping, respectively. Generally, the composition $F\circ f$ does not necessarily yield a mapping transverse to a given subfiber-bundle of $J^1(N,\mathbb{R^\ell)$. Nevertheless, in this paper, for any $\mathcal{A^1$-invariant fiber, we show that composing generic linearly perturbed mappings of $F$ and the given immersion $f$ yields a mapping transverse to the subfiber-bundle of $J^1(N,\mathbb{R^\ell)$ with the given fiber. Moreover, we show a specialized transversality theorem on crossings of compositions of generic linearly perturbed mappings of a given mapping $F:U\to \mathbb{R^\ell$ and a given injection $f:N\to U$. Furthermore, applications of the two main theorems are given.
1612.01100.tex
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\begin{document}

\title[Composing linearly perturbed mappings 
and immersions/injections]{Composing generic linearly perturbed mappings 
and immersions/injections}
%%% \subtitle{Sub-Title of Your Article}%%% optional

\author{Shunsuke Ichiki}% Author Name (\sc should NOT be used here)
%\thanks{Research Fellow DC1 of Japan Society for the Promotion of Science }
\address{
Graduate School of Environment and Information Sciences,  
Yokohama National University, 
Yokohama 240-8501, Japan}
\email{ichiki-shunsuke-jb@ynu.jp}

%\author{Daz \textsc{Baz}}% Author Name (\sc should NOT be used here)
%\address{Address of the organization to which the second author beongs}
%\email{mail\_address\_of\_the\_second\_author}

\subjclass[2010]{Primary 57R45; Secondary 57R42}% Subject code(s)

\keywords{generic linear perturbations, transverse, immersion, injection,  
generalized distance-squared mapping}% Key word(s)


\begin{abstract}
Let $N$ (resp., $U$) be a manifold (resp., an open subset of $\R^m$). 
Let $f:N\to U$ and $F:U\to \R^\ell$ be an immersion and a mapping, respectively.  
Generally, the composition $F\circ f$ does not necessarily 
yield a mapping transverse  
to a given subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$. 
Nevertheless, in this paper, for any $\mathcal{A}^1$-invariant fiber, we show that 
composing generic linearly perturbed mappings of $F$ and the given immersion $f$ yields a mapping transverse to 
the subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$ with the given fiber. 
Moreover, we show a specialized transversality theorem on crossings 
of compositions of 
generic linearly perturbed mappings of a given mapping $F:U\to \mathbb{R}^\ell$ 
and a given injection $f:N\to U$. 
Furthermore, applications of the two main theorems are given.
\end{abstract}

\maketitle

\section{Introduction}\label{section 1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
Throughout this paper, let $\ell$, $m$ and $n$ stand for positive integers. 
In this paper, unless otherwise stated, all manifolds and mappings belong to class $C^{\infty}$ 
and all manifolds are without boundary. 
Let $\pi:\mathbb{R}^m\to \mathbb{R}^\ell$, $U$ and 
$F:U\to \mathbb{R}^\ell$ be a linear mapping, 
an open subset of $\R^m$ 
and a mapping, respectively. 

Set 
\begin{eqnarray*} 
F_\pi=F+\pi. 
\end{eqnarray*} 
Here, the mapping $\pi$ in $F_\pi=F+\pi$ is restricted to $U$. 

Let $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ 
be the space consisting of linear mappings 
of $\mathbb{R}^{m}$ into $\mathbb{R}^{\ell}$. 
Remark that \<we have \> the natural identification 
$\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})=(\mathbb{R}^m)^\ell$.  
An $n$-dimensional manifold is denoted by $N$. 
For 
a given mapping $f:N \to U$, 
a property of mappings $F_\pi \circ f:N\to \mathbb{R}^\ell$ will be said to be 
true for a {\it generic mapping} 
if there exists a subset $\Sigma$ with Lebesgue measure zero of 
$\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ 
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\Sigma$, 
the mapping $F_\pi \circ f:N\to \mathbb{R}^{\ell}$ has the property. 
In the case $F=0$, 
by John Mather, for a given embedding $f:N\to \mathbb{R}^m$, 
a generic mapping $\pi\circ f:N\to \mathbb{R}^\ell$ $(m>\ell)$ 
is investigated in the celebrated paper \cite{GP}. 
The main theorem in \cite{GP} yields many applications. 
On the other hand, 
in this paper, for a given immersion or a given injection 
$f:N\to U$, 
 a generic mapping   
$F_\pi \circ f:N\to \mathbb{R}^\ell$ 
is investigated, respectively. 

The main purpose of this paper is to show two main theorems 
(Theorems \ref{main} and \ref{main2} in Section \ref{section 2}) 
and 
\<to give some of their applications\>. 
The first main theorem 
(Theorem \ref{main}) is as follows. 
Let $f:N\to U$ (resp., $F:U\to \R^\ell$) be an immersion 
(resp., a mapping). 
Then, 
generally, the composition $F\circ f$ does not necessarily 
yield a mapping transverse 
to a given subfiber-bundle of the jet bundle $J^1(N,\mathbb{R}^\ell)$. 
Nevertheless, Theorem \ref{main} asserts that  
for any $\mathcal{A}^1$-invariant fiber, a 
generic mapping $F_\pi \circ f$ yields a mapping transverse to 
the subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$ with the given fiber. 
The second main theorem (Theorem \ref{main2}) is a specialized transversality theorem on crossings of 
a generic mapping $F_\pi \circ f$, where $f:N\to U$ is a given injection and 
$F:U\to \mathbb{R}^\ell$ is a given mapping. 

%As some applications of {\color{red}the} two main theorems in this paper, 
%for a given immersion (resp., injection) $f:N\to U$, 
%the following assertions (1)-(4) (resp., (5))
%are obtained for a generic mapping $F_\pi \circ f:N\to \mathbb{R}^\ell$. 
%Namely, their assertions {\color{red}concern} the properties obtained by generic linear perturbations 
%of the given mapping $F:U\to \mathbb{R}^\ell$. 
%All of (1)-(4) (resp., (5)) follow from the main theorem on compositions with 
%the immersion (resp., injection) in this paper  
%(Theorem \ref{main} (resp., Theorem \ref{main2}) in Section \ref{section 2}). 

For a given immersion (resp., injection) $f:N\to U$, 
the following (1)-(4) (resp., (5)) are obtained as applications of Theorem \ref{main} (resp., Theorem \ref{main2}). 

\begin{enumerate}
\item[(1)] If $(n,\ell)=(n,1)$, 
then a generic function $F_{\pi}\circ f:N\to \mathbb{R}$ is a Morse function. 
\item[(2)] If $(n,\ell)=(n,2n-1)$ and $n\geq 2$, 
then any singular point of a generic mapping $F_{\pi}\circ f:N\to \mathbb{R}^{2n-1}$ 
is a singular point of Whitney umbrella. 
\item[(3)]If $\ell \geq 2n$, 
then a generic mapping $F_{\pi}\circ f:N\to \mathbb{R}^{\ell}$ is an immersion.  
\item[(4)] A generic mapping $F_{\pi}\circ f:N\to \mathbb{R}^{\ell}$ 
has corank at most $k$ singular points 
\<(for the definition of corank at most $k$ singular points, see Subsection \ref{application1})\>, 
where $k$ is the maximum integer satisfying 
$(n-v+k)(\ell-v+k)\leq n$ $(v={\rm min}\{n,\ell\})$.
\item[(5)]If $\ell > 2n$, 
then a generic mapping $F_{\pi}\circ f:N\to \mathbb{R}^{\ell}$ is injective. 
\end{enumerate}
%Notice that in the case that $f:N\to U$ is an embedding, by combining 
%the assertion (3) and (5), the following assertion (6) also follows.
%\begin{enumerate}
%\item[(6)]If $\ell > 2n$, 
%then a generic mapping $F_{\pi}\circ f:N\to \mathbb{R}^{\ell}$ is an embedding. 
Moreover, by combining the assertions (3) and (5), 
for a given embedding $f:N\to U$, 
the following assertion (6) is obtained.
\begin{enumerate}
\item[(6)]If $\ell > 2n$ and $N$ is compact, 
then a generic mapping $F_{\pi}\circ f:N\to \mathbb{R}^{\ell}$ is an embedding. 
\end{enumerate}
%For the proofs of the assertions (1)-(6), see Section \ref{section 5}.
%
%\subsection{Remark}\label{remark 1}
%In \cite{F1}, for a given embedding $f:N\to U$, 
%a generic mapping 
%$F_\pi \circ f:N\to \mathbb{R}^\ell$ is investigated. 
%The important lemma for the proof of the main theorem in \cite{F1} 
%is the main theorem in \cite{GP} 
%(for the assertions of the main theorems in \cite{GP} and \cite{F1}, see 
%Theorem \ref{@m} and Theorem \ref{@f} in Section \ref{section 7}, respectively). 
%On the other hand, the two main theorems 
%(Theorem \ref{main} and Theorem \ref{main2} in Section \ref{section 2}) 
%in this paper are for 
%compositions with a given immersion and a given injection, 
%respectively. 
%Moreover, they are shown without use of 
%the main theorem in \cite{F1} and the main theorem in \cite{GP}. 
%However, it can be remarked that the assertion (6) in Section \ref{section 1}, 
%Corollary \ref{embedding} in Section \ref{section 5} and Corollary \ref{stable} in Section \ref{section 6} 
%are also corollaries obtained by using the main theorem in \cite{F1}. 
%Since the paper \cite{F1} is under review 
%at the time of submission of this paper, 
%the assertion (6) in Section \ref{section 1}, 
%Corollary \ref{embedding} in Section \ref{section 5} and Corollary \ref{stable} in Section \ref{section 6} 
%are written as corollaries obtained by the two main theorems 
%in Section \ref{section 2} of this paper. 
%
%For the sake of the reader's convenience, 
%the main theorem in \cite{F1} and the main theorem in \cite{GP} 
%are introduced as an appendix (see Section \ref{section 7}).  
%It can be remarked that Section \ref{section 7} is 
%introduced only for Remark \ref{remark 1}. 
%Hence, Section \ref{section 7} is not necessary except for Remark \ref{remark 1}.    
\par 
\bigskip 
%%%%%%%%%%%  
In Section \ref{section 2}, some standard definitions are reviewed, 
and the two main theorems (Theorems \ref{main} and \ref{main2}) are stated. 
Section \ref{section 3} (resp., Section \ref{section 4}) is devoted to
the proof of Theorem \ref{main} (resp., Theorem \ref{main2}). 
In Section \ref{section 5}, 
the assertions (1)-(6) above are shown. 
Moreover, in Section \ref{section 6}, as further applications, 
the two main theorems are adapted to quadratic mappings 
of $\mathbb{R}^m$ into $\mathbb{R}^\ell$ 
of a special type called ``generalized distance-squared mappings'' 
(for precise definition of generalized distance-squared mappings, see Section \ref{section 6}). 
%For the sake of reader's convenience, in Section \ref{section 7}, 
%the main theorem in \cite{F1} and the main theorem in \cite{GP} 
%are introduced for Remark in Subsection \ref{remark 1}.  
Since some corollaries in this paper 
(the assertion (6) in Section \ref{section 1}, Corollary \ref{embedding} 
in Section \ref{section 5} and 
Corollary \ref{stable} in Section \ref{section 6} )
are also obtained by using the main theorem in \cite{F1}, which is an improvement of 
the main theorem in \cite{GP}, 
for the sake of readers' convenience, \<Section \ref{section 7} explains \> 
the main theorems in \cite{F1} and \cite{GP} as an appendix. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    
\section{Preliminaries and the statements of Theorems \ref{main} and \ref{main2}}\label{section 2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

Let $N$ and $P$ be manifolds. Firstly, we recall the definition of transversality. 
\begin{definition}\label{transverse}
{\rm
Let $W$ be a submanifold of $P$. 
Let $g:N\to P$ be a mapping. 
\begin{enumerate}
\item 
We say that $g:N\to P$ is {\it transverse} to $W$ 
{\it at $q$} if $g(q)\not\in W$ or  
in the case of $g(q)\in W$, the following holds: 
\begin{eqnarray*}
dg_q(T_qN)+T_{g(q)}W=T_{g(q)}P.
\end{eqnarray*}
\item 
We say that $g:N\to P$ is {\it transverse} to $W$ 
if for any $q\in N$, 
the mapping $g$ is transverse to $W$ at $q$. 
\end{enumerate}
}
\end{definition}


%\subsection{Preliminaries and the statements of Theorem \ref{main} and its applications}
%Let $s$ be a positive integer. Define $N^{(s)}$ as follows:
%\begin{eqnarray*}
%N^{(s)}=\{(q_1,\ldots ,q_s) \mid q_i\not =q_j  (1\leq i<j\leq s) \}. 
%\end{eqnarray*}
%Let ${}_s J^r(N,P)$ be the space consisting of 
%elements $(j^r g(q_1),\ldots ,j^r g(q_s))\in J^r(N,P)^s$ 
%satisfying $(q_1,\ldots ,q_s)\in N^{(s)}$. 
%Since $N^{(s)}$ is an open submanifold of $N^s$, 
%the space ${}_s J^r(N,P)$ is also an open submanifold of $J^r(N,P)^s$.
%For a given mapping $g:N\to P$, 
%the mapping ${}_s j^rg:N^{(s)}\to {}_s J^r(N,P)$ 
%is defined by $(q_1,\ldots ,q_s) \mapsto (j^rg(q_1),\ldots ,j^rg(q_s))$. 

We say that $g : N\to P$ is {\it $\mathcal{A}$-equivalent} to $h : N\to P$ 
if there exist diffeomorphisms $\Phi:N\to N$ and 
$\Psi:P\to P$ such that $g=\Psi \circ h \circ \Phi^{-1}$. 

Let $J^r(N,P)$ be the space of 
$r$-jets of mappings of $N$ into $P$. 
For a given mapping $g:N\to P$, 
the mapping $j^r g:N\to J^r(N,P)$ 
is defined by $q \mapsto j^r g(q)$  
(for details on the space $J^r(N,P)$ or 
the mapping $j^r g:N\to J^r(N,P)$, see for example, \cite{GG}). 
%Then, we may suppose 
%that $\varphi _\lambda(q)=0\in \mathbb{R}^n$ for any $q\in N$. 

For the statement and the proof of Theorem \ref{main},
 it is sufficient to consider the case of $r=1$ and $P=\mathbb{R}^\ell$. 
Let $\{(U_\lambda ,\varphi _\lambda )\}_{\lambda \in \Lambda}$ be a coordinate neighborhood system of $N$. 
Let $\Pi :J^1(N,\mathbb{R}^\ell)$$\to N\times \mathbb{R}^\ell$ be the natural projection defined by $\Pi(j^1g(q))=(q,g(q))$. 
Let $\Phi _\lambda :\Pi^{-1}(U_\lambda \times \mathbb{R}^\ell)\to \varphi _\lambda (U_\lambda)\times \mathbb{R}^\ell \times J^1(n,\ell)$ be 
the homeomorphism defined by 
\begin{eqnarray*} 
\Phi _\lambda \left(j^1g(q)\right)=\left(\varphi _\lambda (q),g(q),j^1(\psi_{_\lambda} \circ g\circ \varphi _\lambda ^{-1}\circ \widetilde{\varphi} _\lambda)(0)\right), 
\end{eqnarray*} 
where 
$J^1(n, \ell)=\{ j^1g(0) \mid g : (\mathbb{R}^n, 0) \to  (\mathbb{R}^\ell, 0) \}$ 
and 
$\widetilde{\varphi} _\lambda : \mathbb{R}^n\to \mathbb{R}^n$ 
(resp., $\psi_{\lambda} : \mathbb{R}^m \to \mathbb{R}^m $) is the 
translation defined by 
$\widetilde{\varphi} _\lambda(0)=\varphi _\lambda (q)$ 
(resp., $\psi_{\lambda}(g(q))=0$). 
Then, $\{(\Pi^{-1}(U_\lambda \times \mathbb{R}^\ell), 
\Phi _\lambda )\}_{\lambda \in \Lambda}$ 
is a coordinate neighborhood system of $J^1(N,\mathbb{R}^\ell)$. 
A subset $X$ of $J^1(n,\ell)$ is called {\it $\mathcal{A}^1$-invariant} 
if for any $j^1g(0)\in X$, and for any two germs of diffeomorphisms  
$H : (\mathbb{R}^\ell, 0)\to (\mathbb{R}^\ell, 0)$ and 
$h : (\mathbb{R}^n, 0)\to (\mathbb{R}^n, 0)$, 
\<we have \> $j^1(H\circ g\circ h^{-1})(0)\in X$. 
Let $X$ be an $\mathcal{A}^1$-invariant submanifold of $J^1(n,\ell)$. 
Set 
\begin{eqnarray*}
X(N,\mathbb{R}^\ell)=\bigcup_{\lambda \in \Lambda}\Phi ^{-1}_\lambda \left(\varphi _\lambda (U_\lambda )\times \mathbb{R}^\ell \times X\right). 
\end{eqnarray*}
Then, the set $X(N,\mathbb{R}^\ell)$ is a subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$ 
with the fiber $X$ such that 
\begin{eqnarray*}
{\rm codim}\ X(N,\mathbb{R}^\ell)&=&{\rm dim}\ J^1(N,\mathbb{R}^\ell)- 
{\rm dim}\ X(N,\mathbb{R}^\ell) \\
&=&{\rm dim}\ J^1(n,\ell)- {\rm dim}\ X \\
&=&{\rm codim}\ X.
\end{eqnarray*}

Then, the first main theorem in this paper is the following. 
\begin{theorem}\label{main}
Let $N$ be a manifold of dimension $n$. 
Let $f$ be an immersion 
of $N$ into an open subset $U$ of $\mathbb{R}^m$. 
Let $F:U\to \mathbb{R}^\ell$ be a mapping. 
If $X$ is an $\mathcal{A}^1$-invariant submanifold of $J^1(n,\ell)$, 
then there exists a subset $\Sigma$ with Lebesgue measure zero 
of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ 
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\Sigma$, 
the mapping $j^1(F_\pi \circ f):
N\to J^1(N,\mathbb{R}^\ell)$ is transverse to the submanifold $X(N,\mathbb{R}^\ell)$.
\end{theorem}

Now, in order to state the second main theorem (Theorem \ref{main2}), 
we will prepare some definitions. 
Set $N^{(s)}=\{(q_1,\<q_2,\>\ldots ,q_s)\in N^s\mid q_i\not=q_j \ (i\not=j)\}$. 
Notice that $N^{(s)}$ is an open submanifold of $N^s$. 
For any mapping $g:N\to P$, 
let $g^{(s)}:N^{(s)}\to P^s$ be the mapping defined by 
\begin{eqnarray*}
g^{(s)}(q_1,\<q_2,\>\ldots ,q_s)=(g(q_1),\<g(q_2),\>\ldots ,g(q_s)). 
\end{eqnarray*}
Set 
$\Delta_s=\{(y,\ldots ,y) \in P^s \mid y\in P \}$. 
It is clearly seen that $\Delta_s$ is a submanifold of $P^s$ 
such that 
\begin{eqnarray*}
{\rm codim}\ \Delta_s ={\rm dim}\ P^s- 
{\rm dim}\ \Delta_s  =(s-1){\rm dim}\ P.
\end{eqnarray*}

\begin{definition}\label{normal crossings}
{\rm
Let $g$ be a mapping of $N$ into $P$. 
Then, $g$ is called a {\it  mapping with normal crossings} if  
for any positive integer $s$ $(s\geq 2)$, 
the mapping $g^{(s)}:N^{(s)}\to P^s$ is 
transverse to the submanifold $\Delta_s$. 
}
\end{definition}
For any injection $f:N\to \mathbb{R}^m$, set 
\begin{eqnarray*}
s_f={\rm max}\left\{s\ \middle| \ \forall (q_1,\<q_2,\>\ldots ,q_s)\in N^{(s)}, {\rm dim\ }\sum_{i=2}^s \mathbb{R}\overrightarrow{f(q_1)f(q_i)}=s-1 \right\}. 
\end{eqnarray*}
Since the mapping $f$ is injective, we get $2 \leq s_f$. 
Since $f(q_1),\<f(q_2),\>\ldots ,f(q_{s_f})$ are points of $\mathbb{R}^m$, 
it follows that $s_f\leq m+1$. 
Thus, remark that 
\begin{eqnarray*}
2\leq s_f \leq m+1. 
\end{eqnarray*}
%\begin{eqnarray*}
%s_f={\it max}\{s \mid for any (q_1,\ldots ,q_s)\in N^{(s)}, f(q_1),\ldots ,f(q_s) are in general position\}
%\end{eqnarray*}
Then, the second main theorem in this paper is the following. 
\begin{theorem}\label{main2}
Let $N$ be a manifold of dimension $n$. 
Let $f$ be an injection of $N$ into an open subset 
$U$ of $\mathbb{R}^m$. 
Let $F:U\to \mathbb{R}^\ell$ be a mapping. 
Then, there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)$ with Lebesgue measure zero such that 
for any $\pi \in \mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)-\Sigma $, \<and \> 
for any $s$ $(2\leq s \leq s_f)$, 
the mapping $(F_\pi \circ f)^{(s)}:N^{(s)}\to (\R^\ell)^s$ is transverse to the 
submanifold $\Delta_s$. 
Moreover, if the mapping $F_\pi$ 
satisfies that $| F_\pi^{-1}(y) | \leq s_f$ for any $y\in\mathbb{R}^\ell$, 
then the mapping $F_\pi \circ f:N\to \mathbb{R}^\ell$ is a mapping with normal crossings, where $|X|$ is the number of elements of the set $X$.    
\end{theorem}

The following well known result is important for the proofs of Theorems \ref{main} and \ref{main2}. 
\begin{lemma}[\cite{abra}, \cite{GP}]\label{abra}
Let $N$, $P$, $Z$ be manifolds, and 
let $W$ be a submanifold of $P$. Let $\Gamma:N\times Z\to P$ be a mapping.
If $\Gamma$ is transverse to $W$, then there exists a subset $\Sigma$ of $Z$ with 
Lebesgue measure zero such that for any $p \in Z-\Sigma $, 
the mapping 
$\Gamma_p:N\to P$ is transverse to $W$, 
where $\Gamma_p(q)=\Gamma(q,p)$. 
\end{lemma}
\subsection{Remark}
\begin{enumerate}
\item 

We explain the advantage that the domain of the mapping $F$ 
is an arbitrary open set. 
Suppose that $U=\mathbb{R}$. 
Let $F:\mathbb{R}\to \mathbb{R}$ be the mapping defined by $x\mapsto |x|$. 
Since $F$ is not differentiable at $x=0$, we \<cannot \> apply Theorems \ref{main} and \ref{main2} to 
the mapping $F:\mathbb{R}\to \mathbb{R}$. 

On the other hand, if $U=\mathbb{R}-\{0\}$, then 
Theorems \ref{main} and \ref{main2} 
can be applied to the restriction $F|_U$.
\item 
There is a case of $s_f =3$ as follows. 
If $n+1 \leq m$, $N=S^n$ and 
$f:S^n\to \mathbb{R}^m$ is the inclusion $f(x)=(x,0\ldots ,0)$, 
then it is easily seen that $s_f=3$. 
Indeed, suppose that there exists a point $(q_1, q_2, q_3)\in (S^n)^{(3)}$ 
such that ${\rm dim\ }\sum_{i=2}^3 \mathbb{R}\overrightarrow{f(q_1)f(q_i)}=1$. 
Then, since the number of the intersections of $f(S^n)$ and 
a straight line of $\mathbb{R}^m$ is at most two, 
this contradicts the assumption. 
Thus, 
we get $s_f\geq3$. 
\<From \>$S^1\times \{0\} \subset f(S^n)$, 
it follows that $s_f<4$, where $0=\underbrace{(0,\ldots ,0)}_{(m-2)-tuples}$.  
Hence, we have $s_f=3$.   
\item 

The essential idea for the proofs of Theorems \ref{main} and \ref{main2}  
is to apply Lemma \ref{abra}, and it 
is almost similar to the idea of the proofs of main results in \cite{G3}. 
Nevertheless, the two main theorems in this paper are drastically improved. 
As an effect of the improvement, 
many applications 
are obtained by the two main theorems 
(for the applications, see Sections \ref{section 5} and \ref{section 6}). 
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    
\section{Proof of Theorem \ref{main}}\label{section 3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%Remark that by $n\leq m$, 
%%it is clearly seen that for any $k$ $(k=1,\ldots ,n)$, $X$ is not empty. 
%The set $X$ is an $\mathcal{A}^1$ invariant submanifold of $J^1(n,m)$ whose 
%codimension is $k(m-n+k)$ $(n\leq m)$. Hence, set 
%\begin{eqnarray*}
%\Sigma^k(N,\mathbb{R}^m)=\bigcup_{\lambda \in \Lambda}\Phi ^{-1}_\lambda \left(\varphi _\lambda (U_\lambda )\times \mathbb{R}^m \times X\right),
%\end{eqnarray*}
%and the set $X(N,\mathbb{R}^m)$ is a submanifold of $J^1(N,\mathbb{R}^m)$ whose codimension is $k(m-n+k)$. 
%For simplicity, set $A=\underbrace{\mathbb{R}^n\times \cdots \times \mathbb{R}^n}_{n}$. 
Let $(\alpha_{ij})_{1\leq i \leq \ell, 1\leq j \leq m}$ be a representing matrix of 
a linear mapping $\pi:\mathbb{R}^m\to \mathbb{R}^\ell$. 
Set $F_{\alpha}=F_{\pi}$, and we have
\begin{eqnarray*}
F_{\alpha}(x)=
\biggl(F_{1}(x)+\sum_{j=1}^{m}\alpha_{1j}x_j, 
\<
F_{2}(x)+\sum_{j=1}^{m}\alpha_{2j}x_j, 
\>
\ldots,
F_{\ell}(x)+\sum_{j=1}^{m}\alpha_{\ell j}x_j\biggr), 
\end{eqnarray*}
where $F=(F_1, \<F_2,\>\ldots ,F_\ell)$, 
 $\alpha=(\alpha_{11},\<\alpha_{12},\>\ldots, 
 \alpha_{1m},\ldots ,
 \alpha_{\ell 1},\<\alpha_{\ell 2},\>\ldots ,
\alpha_{\ell m})\in (\mathbb{R}^m)^\ell$ and $x=(x_1,\<x_2,\>\ldots ,x_m)$. 
For \<a \> given immersion $f:N\to U$, 
\<the \> mapping $F_{\alpha}\circ f:N\to \mathbb{R}^\ell$ is 
\<given \>as follows: 
\begin{eqnarray*}
F_{\alpha}\circ f=
\biggl(F_{1}\circ f+\sum_{j=1}^{m}\alpha_{1j}f_j, 
\<F_{2}\circ f+\sum_{j=1}^{m}\alpha_{2j}f_j, \>
\ldots,
F_{\ell}\circ f+\sum_{j=1}^{m}\alpha_{\ell j}f_j\biggr),
\end{eqnarray*}
where $f=(f_1,\<f_2,\>\ldots ,f_m)$.  
Since \<we have \>the natural identification 
$\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})=(\mathbb{R}^{m})^\ell$, 
in order to prove Theorem \ref{main}, 
it is sufficient to show that 
there exists a subset $\Sigma$ 
with Lebesgue measure zero of $(\mathbb{R}^m)^{\ell}$ 
such that for any $\alpha \in (\mathbb{R}^m)^{\ell}-\Sigma$, 
the mapping $j^{1}(F_{\alpha}\circ f):
N\to J^{1}(N,\mathbb{R}^\ell)$ is transverse to 
the given submanifold $X(N,\mathbb{R}^\ell)$. 

Now, let $\Gamma:N\times (\mathbb{R}^m)^\ell \to J^1(N,\mathbb{R}^\ell)$ 
be the mapping defined by 
\[
\Gamma(q,\alpha)=j^1(F_\alpha \circ f)(q).
\]
If the mapping $\Gamma$ is transverse to the submanifold 
$X(N,\mathbb{R}^\ell)$, 
then \<from \> Lemma \ref{abra},  
it follows that there exists a subset $\Sigma$ of $(\mathbb{R}^m)^\ell$ with 
Lebesgue measure zero such that 
for any $\alpha \in (\mathbb{R}^m)^\ell-\Sigma $, 
the mapping $\Gamma_\alpha :N\to J^1(N,\mathbb{R}^\ell)$ 
($\Gamma_\alpha=j^1(F_\alpha \circ f)$) 
is transverse to the submanifold 
$X(N,\mathbb{R}^\ell)$. 
Thus, in order to finish the proof of Theorem \ref{main}, 
it is sufficient to show that if $\Gamma(\widetilde{q},\widetilde{\alpha})
\in X(N,\mathbb{R}^\ell)$, then the following $(\ast)$ holds. 
\begin{align}
d\Gamma_{(\widetilde{q},\widetilde{\alpha})}(T_{(\widetilde{q},\widetilde{\alpha})}
(N\times (\mathbb{R}^m)^\ell))
+
T_{\Gamma(\widetilde{q},\widetilde{\alpha})}X(N,\mathbb{R}^\ell)
=
T_{\Gamma(\widetilde{q},\widetilde{\alpha})}J^1(N,\mathbb{R}^\ell).\tag{$\ast $}
\end{align}
As \<in \> Section \ref{section 2}, 
let $\{(U_\lambda ,\varphi _\lambda )\}_{\lambda \in \Lambda}$ 
(resp., $\{(\Pi^{-1}(U_\lambda \times \mathbb{R}^\ell), 
\Phi _\lambda )\}_{\lambda \in \Lambda}$) 
be a coordinate neighborhood system of $N$ (resp., $J^1(N,\mathbb{R}^\ell)$). 
There exists a coordinate neighborhood 
$\left(U_{\widetilde{\lambda}}\times (\mathbb{R}^m)^\ell, \varphi_{\widetilde{\lambda}}\times id \right)$ 
containing the point $(\widetilde{q},\widetilde{\alpha})$ of 
$N\times (\mathbb{R}^m)^\ell$, 
where $id$ is the identity mapping of $(\mathbb{R}^m)^\ell$ into $(\mathbb{R}^m)^\ell$, 
and the mapping $\varphi_{\widetilde{\lambda}}\times id : 
U_{\widetilde{\lambda}}\times (\mathbb{R}^m)^\ell \to 
\varphi_{\widetilde{\lambda}}(U_{\widetilde{\lambda}})\times (\mathbb{R}^m)^\ell$ ($\subset \mathbb{R}^n\times (\mathbb{R}^m)^\ell$) 
is defined by 
$\left(\varphi_{\widetilde{\lambda}}\times id\right)(q,\alpha)=
\left(\varphi_{\widetilde{\lambda}}(q), id(\alpha)\right)$.
%\begin{eqnarray*}
%\left(\varphi_{\widetilde{\lambda}}\times id\right)(q,p)&=&
%\left(\varphi_{\widetilde{\lambda}}(q), id(p)\right)\\
%&=&\left(\varphi_{\widetilde{\lambda}}(q), p\right).
%\end{eqnarray*}
There exists a coordinate neighborhood 
$\left(\Pi^{-1}(U_{\widetilde{\lambda}} \times \mathbb{R}^\ell), 
\Phi _{\widetilde{\lambda}} \right)$ 
containing the point $\Gamma (\widetilde{q},\widetilde{\alpha})$ of 
$J^1(N,\mathbb{R}^\ell)$. 
Let $t=(t_1,\<t_2,\>\ldots ,t_n)\in \mathbb{R}^n$ be a local coordinate on 
$\varphi_{\widetilde{\lambda}}(U_{\widetilde{\lambda}})$ 
containing 
$\varphi_{\widetilde{\lambda}}(\widetilde{q})$. 
Then, the mapping $\Gamma$ is locally given by the following:
{\small 
\begin{eqnarray*}
&{}&(\Phi _{\widetilde{\lambda}} \circ \Gamma \circ 
(\varphi_{\widetilde{\lambda}}\times id )^{-1})(t,\alpha)\\
&=&(\Phi _{\widetilde{\lambda}} \circ \Gamma \circ 
(\varphi_{\widetilde{\lambda}}^{-1}\times id^{-1} ))(t,\alpha)\\
&=&(\Phi _{\widetilde{\lambda}} \circ \Gamma )
(\varphi_{\widetilde{\lambda}}^{-1}(t), \alpha )\\
&=&\Phi _{\widetilde{\lambda}}(\Gamma 
(\varphi_{\widetilde{\lambda}}^{-1}(t), \alpha ))\\
&=&\Phi _{\widetilde{\lambda}}(j^1(F_\alpha \circ f)
(\varphi_{\widetilde{\lambda}}^{-1}(t)))\\
&=&(\Phi _{\widetilde{\lambda}}\circ j^1(F_\alpha \circ f)\circ \varphi_{\widetilde{\lambda}}^{-1})(t)\\
&=&\left(t, (F_\alpha \circ f \circ \varphi_{\widetilde{\lambda}}^{-1})(t), \right.  \\%%%
&& \frac{\partial (F_{\alpha, 1}\circ f\circ \varphi_{\widetilde{\lambda}}^{-1})}
{\partial t_1}(t), 
\<
\frac{\partial (F_{\alpha, 1}\circ f\circ \varphi_{\widetilde{\lambda}}^{-1})}
{\partial t_2}(t), 
\>
\ldots ,
\frac{\partial (F_{\alpha, 1}\circ f\circ \varphi_{\widetilde{\lambda}}^{-1})}
{\partial t_n}(t),
\\ 
&& \< 
\frac{\partial (F_{\alpha, 2}\circ f\circ \varphi_{\widetilde{\lambda}}^{-1})}
{\partial t_1}(t), 
\frac{\partial (F_{\alpha, 2}\circ f\circ \varphi_{\widetilde{\lambda}}^{-1})}
{\partial t_2}(t), 
\ldots ,
\frac{\partial (F_{\alpha, 2}\circ f\circ \varphi_{\widetilde{\lambda}}^{-1})}
{\partial t_n}(t), 
\> 
\\ 
\\
&&\hspace{120pt}\cdots \cdots \cdots , \\ 
\\
&& \left. \frac{\partial (F_{\alpha, \ell}\circ f\circ \varphi_{\widetilde{\lambda}}^{-1})}
{\partial t_1}(t)
,
\<
\frac{\partial (F_{\alpha, \ell}\circ f\circ \varphi_{\widetilde{\lambda}}^{-1})}
{\partial t_2}(t), 
\>
\ldots ,
\frac{\partial (F_{\alpha, \ell}\circ f\circ \varphi_{\widetilde{\lambda}}^{-1})}
{\partial t_n}(t)
\right)\\%%%
&=&\left(t, (F_\alpha \circ f \circ \varphi_{\widetilde{\lambda}}^{-1})(t), \right. \\
&&\frac{\partial F_1\circ \widetilde{f}}{\partial t_1}(t)+
\sum_{j=1}^m \alpha_{1j}\frac{\partial \widetilde{f}_j}{\partial t_1}(t),
\<
\frac{\partial F_1\circ \widetilde{f}}{\partial t_2}(t)+
\sum_{j=1}^m \alpha_{1j}\frac{\partial \widetilde{f}_j}{\partial t_2}(t),
\>
\ldots ,
\frac{\partial F_1\circ \widetilde{f}}{\partial t_n}(t)+
\sum_{j=1}^m \alpha_{1j}\frac{\partial \widetilde{f}_j}{\partial t_n}(t),
\\ 
&& 
\<
\frac{\partial F_2\circ \widetilde{f}}{\partial t_1}(t)+
\sum_{j=1}^m \alpha_{2j}\frac{\partial \widetilde{f}_j}{\partial t_1}(t),
\frac{\partial F_2\circ \widetilde{f}}{\partial t_2}(t)+
\sum_{j=1}^m \alpha_{2j}\frac{\partial \widetilde{f}_j}{\partial t_2}(t),
\ldots ,
\frac{\partial F_2\circ \widetilde{f}}{\partial t_n}(t)+
\sum_{j=1}^m \alpha_{2j}\frac{\partial \widetilde{f}_j}{\partial t_n}(t),
\> 
\\
\\
&&\hspace{160pt}\cdots \cdots \cdots , 
\\
\\
&& \left. 
\frac{\partial F_\ell \circ \widetilde{f}}{\partial t_1}(t)+\sum_{j=1}^m \alpha_{\ell j}\frac{\partial \widetilde{f}_j}{\partial t_1}(t),
\<
\frac{\partial F_\ell \circ \widetilde{f}}{\partial t_2}(t)+\sum_{j=1}^m \alpha_{\ell j}\frac{\partial \widetilde{f}_j}{\partial t_2}(t),
\>
\ldots ,
\frac{\partial F_\ell \circ \widetilde{f}}{\partial t_n}(t)+
\sum_{j=1}^m \alpha_{\ell j}\frac{\partial \widetilde{f}_j}{\partial t_n}(t)\right), 
\end{eqnarray*} 
}
where $F_\alpha=(F_{\alpha, 1},\<F_{\alpha, 2},\> \ldots , F_{ \alpha, \ell})$ and 
$\widetilde{f}=(\widetilde{f}_1, \<\widetilde{f}_2,\> \ldots ,\widetilde{f}_m)
=(f_1\circ \varphi_{\widetilde{\lambda}}^{-1},
\<
f_2\circ \varphi_{\widetilde{\lambda}}^{-1},
\>
\ldots , f_m\circ
\varphi_{\widetilde{\lambda}}^{-1})=f\circ \varphi_{\widetilde{\lambda}}^{-1}$. 
%$\widetilde{f}_j= f_j\circ \varphi_{\widetilde{\lambda}}^{-1}$ $(1\leq j\leq m)$. 
The Jacobian matrix of the mapping $\Gamma$ at 
$(\widetilde{q},\widetilde{\alpha})$ is the following:\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray*}
J\Gamma_{(\widetilde{q},\widetilde{\alpha})}=
\left(
\begin{array}{c|ccccccccccccccc}
E_n       &    0    &         \cdots          &   \cdots        &0    \\
\hline                &   \ast      &    \cdots    &  \cdots      & \ast  \\ 
      &   {}^t(Jf_{\widetilde{q}}) & & \bigzerol &\\
 \ast    &              & {}^t(Jf_{\widetilde{q}})    &       &  \\
    &    & \bigzerol &\ddots &      \\
      &    &  &  &   {}^t(Jf_{\widetilde{q}}) \\
\end{array}
\right)_{(t, \alpha)=(\varphi_{\widetilde{\lambda}}(\widetilde{q}),\widetilde{\alpha})}, 
\end{eqnarray*}
where $E_n$ is the $n\times n$ unit matrix and 
$Jf_{\widetilde{q}}$ is the Jacobian matrix of the mapping $f$ at $\widetilde{q}$. 
Remark that \<${}^t\!A$ \> means the transposed matrix of $A$. 
Since $X(N,\mathbb{R}^\ell)$ is a subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$ 
with the fiber $X$, 
it is clearly seen that the rank of the following matrix $M_1$ is 
\<equal to \> $n+\ell+n\ell$\<:\>
\begin{eqnarray*}
M_1=
\left(
\begin{array}{c|ccccccccccccccc}
E_{n+\ell}       &    \ast    &         \cdots          &   \cdots        &\ast   \\
\hline         &   {}^t(Jf_{\widetilde{q}})  & & \bigzerol &\\
\bigzerol  &              & {}^t(Jf_{\widetilde{q}})      &       &  \\
    &    & \bigzerol &\ddots &      \\
      &    &  &  &   {}^t(Jf_{\widetilde{q}})  \\
\end{array}
\right)_{(t, \alpha)=(\varphi_{\widetilde{\lambda}}(\widetilde{q}),\widetilde{\alpha})}, 
\end{eqnarray*}
where $E_{n+\ell}$ is the $(n+\ell)\times (n+\ell)$ unit matrix. 
Notice that for any $i$ $(1\leq i \leq m\ell)$, the $(n+\ell+i)$-th 
column vector of $M_1$ \<coincides with \> 
the $(n+i)$-th column vector of $J\Gamma_{(\widetilde{q},\widetilde{\alpha})}$. 
Since 
the mapping $f$ is an immersion $(n\leq m)$, we have that the rank of the matrix 
$M_1$ 
is \<equal to \>$n+\ell+n\ell$. 
Hence, we have $(\ast)$. 
%Thus, the mapping $\Gamma$ is transverse to the submanifold $X(N,\mathbb{R}^\ell)$. 

\hfill $\Box$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem \ref{main2}}\label{section 4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $(\alpha_{ij})_{1\leq i \leq \ell, 1\leq j \leq m}$ be a representing matrix of 
a linear mapping $\pi:\mathbb{R}^m\to \mathbb{R}^\ell$. 
Set $F_{\alpha}=F_{\pi}$, and we have
\begin{eqnarray*}
F_{\alpha}(x)=
\biggl(F_{1}(x)+\sum_{j=1}^{m}\alpha_{1j}x_j, 
\<
F_{2}(x)+\sum_{j=1}^{m}\alpha_{2j}x_j, 
\>
\ldots,
F_{\ell}(x)+\sum_{j=1}^{m}\alpha_{\ell j}x_j\biggr), 
\end{eqnarray*}
where $F=(F_1, \<F_2,\> \ldots ,F_\ell)$, 
$\alpha=(\alpha_{11},\<\alpha_{12},\>\ldots, \alpha_{1m},\ldots ,
\alpha_{\ell 1},\<\alpha_{\ell 2},\>\ldots ,
\alpha_{\ell m})\in (\mathbb{R}^m)^\ell$ and $x=(x_1,\<x_2,\>\ldots ,x_m)$. 
For \<a \> given injection $f:N\to U$, 
\<the \> mapping $F_{\alpha}\circ f:N\to \mathbb{R}^\ell$ is 
\<given \> as follows: 
\begin{eqnarray*}
F_{\alpha}\circ f=
\biggl(F_{1}\circ f+\sum_{j=1}^{m}\alpha_{1j}f_j, 
\<
F_{2}\circ f+\sum_{j=1}^{m}\alpha_{2j}f_j, 
\>
\ldots,
F_{\ell}\circ f+\sum_{j=1}^{m}\alpha_{\ell j}f_j\biggr),
\end{eqnarray*}
where $f=(f_1,\<f_2,\>\ldots ,f_m)$.  
Since \<we have \> the natural identification 
$\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})=(\mathbb{R}^{m})^\ell$, 
in order to show that 
there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)$ with Lebesgue measure zero such that 
for any $\pi \in \mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)-\Sigma $, 
\<and \> 
for any $s$ $(2\leq s \leq s_f)$, 
the mapping $(F_\pi \circ f)^{(s)}:N^{(s)}\to (\R^\ell)^s$ is transverse to the 
submanifold $\Delta_s$,  
it is sufficient to show that 
there exists a subset $\Sigma$ of $(\mathbb{R}^m)^{\ell}$ 
with Lebesgue measure zero 
such that for any $\alpha \in (\mathbb{R}^m)^{\ell}-\Sigma$, 
\<and \> 
for any $s$ $(2\leq s \leq s_f)$, 
the mapping $(F_{\alpha}\circ f)^{(s)}:
N^{(s)}\to (\mathbb{R}^\ell)^s$ is transverse to $\Delta_s$.  

Now, let $s$ be a positive integer satisfying $2\leq s \leq s_f$.  
Let $\Gamma : N^{(s)}\times (\mathbb{R}^m)^\ell \to (\mathbb{R}^{\ell})^s$ be the mapping defined by 
\begin{eqnarray*}
\Gamma(q_1,\<q_2,\>\ldots ,q_s,\alpha)=
\left( (F_\alpha \circ f)(q_1), \< (F_\alpha \circ f)(q_2),\>\ldots, 
(F_\alpha \circ f)(q_s)\right).
\end{eqnarray*}
If for any positive integer $s$ $(2\leq s \leq s_f)$, 
the mapping $\Gamma$ is transverse to $\Delta_s$, then \<from \> Lemma \ref{abra}, 
 it follows that for any positive integer $s$ $(2\leq s \leq s_f)$, 
 there exists a subset $\Sigma_s$ of $(\mathbb{R}^m)^\ell$ 
 with Lebesgue measure zero 
such that for any $\alpha  \in (\mathbb{R}^m)^{\ell}-\Sigma_s$,  
the mapping $\Gamma_\alpha : 
N^{(s)}\to (\mathbb{R}^\ell)^s$ $(\Gamma_\alpha=(F_\alpha\circ f)^{(s)})$ 
is transverse to $\Delta_s$. 
Then, set $\Sigma=\bigcup_{i=2}^{s_f}\Sigma_i$. 
It is clearly seen that $\Sigma$ is a subset of  $(\mathbb{R}^m)^\ell$ 
with Lebesgue measure zero. 
Therefore, it follows that  
for any $\alpha \in (\mathbb{R}^m)^{\ell}-\Sigma $, 
\<and \> 
for any $s$ $(2\leq s \leq s_f)$, 
the mapping $\Gamma_\alpha : 
N^{(s)}\to (\mathbb{R}^\ell)^s$ $(\Gamma_\alpha=(F_\alpha\circ f)^{(s)})$ 
is transverse to $\Delta_s$. 

Hence, in order to show that 
there exists a subset $\Sigma$ of $(\mathbb{R}^m)^{\ell}$ 
with Lebesgue measure zero 
such that for any $\alpha \in (\mathbb{R}^m)^{\ell}-\Sigma$, 
\<and \> 
for any $s$ $(2\leq s \leq s_f)$, 
the mapping $(F_{\alpha}\circ f)^{(s)}:
N^{(s)}\to (\mathbb{R}^\ell)^s$ is transverse to $\Delta_s$, 
it is sufficient to show that 
for any positive integer $s$ $(2\leq s \leq s_f)$,   
if $\Gamma(\widetilde{q}, \widetilde{\alpha})\in \Delta_s $ 
$(\widetilde{q}=(\widetilde{q}_1,
\<
\widetilde{q}_2,
\>
\ldots \widetilde{q}_s))$, 
then the following $(\ast\ast)$ holds. 
\begin{align}
d\Gamma_{(\widetilde{q}, \widetilde{\alpha})}(T_{(\widetilde{q}, \widetilde{\alpha})}(N^{(s)}\times (\mathbb{R}^m)^\ell))+
T_{\Gamma(\widetilde{q}, \widetilde{\alpha})}\Delta_s
=T_{\Gamma(\widetilde{q}, \widetilde{\alpha})}(\mathbb{R}^\ell)^s.
\tag{$\ast\ast$}
\end{align}
Let $\{(U_\lambda ,\varphi _\lambda )\}_{\lambda \in \Lambda}$ be a coordinate  neighborhood system of $N$. 
There exists a coordinate neighborhood 
$(U_{\widetilde{\lambda}_1}\times \<U_{\widetilde{\lambda}_2}\times \>
\cdots \times U_{\widetilde{\lambda}_s} \times (\mathbb{R}^m)^\ell, 
\varphi_{\widetilde{\lambda}_1}\times 
\<\varphi_{\widetilde{\lambda}_2}\times 
\>
\cdots \times \varphi_{\widetilde{\lambda}_s}\times id )$ 
containing the point $(\widetilde{q}, \widetilde{\alpha})$ 
of $N^{(s)}\times (\mathbb{R}^m)^\ell$, 
where $id$ is the identity mapping of $(\mathbb{R}^m)^\ell$ into $(\mathbb{R}^m)^\ell$, 
and the mapping 
$\varphi_{\widetilde{\lambda}_1}\times 
\<
\varphi_{\widetilde{\lambda}_2}\times 
\>
\cdots \times \varphi_{\widetilde{\lambda}_s}\times id  : 
U_{\widetilde{\lambda}_1}\times 
\<
U_{\widetilde{\lambda}_2}\times 
\>
\cdots \times U_{\widetilde{\lambda}_s} \times (\mathbb{R}^m)^\ell 
 \to (\mathbb{R}^n)^s\times (\mathbb{R}^m)^\ell$ is defined by 
$(\varphi_{\widetilde{\lambda}_1}\times 
\<
\varphi_{\widetilde{\lambda}_2}\times 
\>
\cdots \times \varphi_{\widetilde{\lambda}_s}\times id )(q_1,\<q_2,\>\ldots ,q_s,\alpha)=
(\varphi_{\widetilde{\lambda}_1}(q_1),  
\<
\varphi_{\widetilde{\lambda}_2}(q_2),  
\>
\ldots ,
\varphi_{\widetilde{\lambda}_s}(q_s), id(\alpha ))$.
Let $t_i=(t_{i1},\<t_{i2},\>\ldots ,t_{in})$ be a local coordinate 
containing $\varphi_{\widetilde{\lambda}_i}(\widetilde{q}_i)$ $(1\leq i \leq s)$.  
Then, the mapping $\Gamma$ is locally given by the following:
\begin{eqnarray*}
&&\Gamma \circ \left(\varphi_{\widetilde{\lambda}_1}\times 
\<
\varphi_{\widetilde{\lambda}_2}\times 
\>
\cdots \times \varphi_{\widetilde{\lambda}_s}\times id\right)^{-1}
(t_1,\<t_2,\>\ldots ,t_s,\alpha)
\\
&=&\Gamma \circ \left(\varphi_{\widetilde{\lambda}_1}^{-1}\times 
\<
\varphi_{\widetilde{\lambda}_2}^{-1}\times 
\>
\cdots \times 
\varphi_{\widetilde{\lambda}_s}^{-1}\times id^{-1}\right)
(t_1,\<t_2,\>\ldots ,t_s,\alpha)
\\
&=&\Gamma \left(\varphi_{\widetilde{\lambda}_1}^{-1}(t_1),
\<
\varphi_{\widetilde{\lambda}_2}^{-1}(t_2),
\>
\ldots ,   
\varphi_{\widetilde{\lambda}_s}^{-1}(t_s), \alpha \right)
\\
&=&\left( (F_\alpha \circ f\circ \varphi_{\widetilde{\lambda}_1}^{-1})(t_1), 
\<
(F_\alpha \circ f\circ \varphi_{\widetilde{\lambda}_2}^{-1})(t_2), 
\>
\ldots ,
(F_\alpha \circ f\circ \varphi_{\widetilde{\lambda}_s}^{-1})(t_s) 
\right)
\\
&=&\left(
F_1\circ \widetilde{f}(t_1)+\sum_{j=1}^m\alpha_{1j}\widetilde{f}_j(t_1), 
\<
F_2\circ \widetilde{f}(t_1)+\sum_{j=1}^m\alpha_{2j}\widetilde{f}_j(t_1), 
\>
\ldots ,
F_\ell \circ \widetilde{f}(t_1)+\sum_{j=1}^m\alpha_{\ell j}\widetilde{f}_j(t_1), \right.\\
&& 
\<
F_1\circ \widetilde{f}(t_2)+\sum_{j=1}^m\alpha_{1j}\widetilde{f}_j(t_2), 
F_2\circ \widetilde{f}(t_2)+\sum_{j=1}^m\alpha_{2j}\widetilde{f}_j(t_2), 
\ldots ,
F_\ell \circ \widetilde{f}(t_2)+\sum_{j=1}^m\alpha_{\ell j}\widetilde{f}_j(t_2), 
\>  
\\
\\
&&\hspace{150pt}\cdots \cdots \cdots , 
\\
\\
&& \left. 
F_1\circ \widetilde{f}(t_s)+\sum_{j=1}^m\alpha_{1j}\widetilde{f}_j(t_s),
\<
F_2\circ \widetilde{f}(t_s)+\sum_{j=1}^m\alpha_{2j}\widetilde{f}_j(t_s),
\>
\ldots ,
F_\ell \circ \widetilde{f}(t_s)+\sum_{j=1}^m\alpha_{\ell j}\widetilde{f}_j(t_s)
\right), 
\end{eqnarray*}  
where
$\widetilde{f}(t_i)=(\widetilde{f}_1(t_i),
\<
\widetilde{f}_2(t_i),
\>
\ldots ,\widetilde{f}_m(t_i))
=(f_1\circ \varphi_{\widetilde{\lambda}_i}^{-1}(t_i),
\<
f_2\circ \varphi_{\widetilde{\lambda}_i}^{-1}(t_i),
\>
\ldots ,
f_m\circ \varphi_{\widetilde{\lambda}_i}^{-1}(t_i))$ $(1\leq i\leq s)$. 
For simplicity, set $t=(t_1,\<t_2,\>\ldots ,t_s)$ and 
$z=(\varphi_{\widetilde{\lambda}_1}\times 
\<
\varphi_{\widetilde{\lambda}_2}\times 
\>
\cdots \times \varphi_{\widetilde{\lambda}_s})
(\widetilde{q}_1,\<\widetilde{q}_2,\>\ldots ,\widetilde{q}_s)$. 

The Jacobian matrix of the mapping $\Gamma$ at 
$(\widetilde{q}, \widetilde{\alpha})$ is the following: 
\begin{eqnarray*}
J\Gamma_{(\widetilde{q}, \widetilde{\alpha})}=
\left(
\begin{array}{c|cccccccccccccc}
&{\bf b}_{}(t_1)& & &\bigzerol \\
&&{\bf b}_{}(t_1)&& \\
&\bigzerol &   &  \ddots   &    \\ 
&&   &         &{\bf b}_{}(t_1)\\ 
 &{\bf b}_{}(t_2)&  & &   \bigzerol       \\
&& {\bf b}_{}(t_2) & &\\
\ast&\bigzerol&   &  \ddots   &    \\ 
&  & &&{\bf b}_{}(t_2)     \\
 & \vdots &\vdots &\vdots& \vdots  \\
  & \vdots &\vdots &\vdots& \vdots  \\
&{\bf b}_{}(t_s)&  & &   \bigzerol       \\
&& {\bf b}_{}(t_s) & &\\
&\bigzerol&   &  \ddots   &    \\ 
&  & &&{\bf b}_{}(t_s)     \\
\end{array}
\right)_{(t,\alpha)=(z, \widetilde{\alpha})},
\end{eqnarray*}
where 
${\bf b}_{ }(t_i)=(
\widetilde{f}_1(t_i),
\<\widetilde{f}_2(t_i),\>
\ldots ,\widetilde{f}_m(t_i))$. 
By the construction of 
$T_{\Gamma(\widetilde{q}, \widetilde{\alpha})}\Delta_s$, 
in order to show $(\ast\ast)$, it is sufficient to 
show that the rank of the following matrix $M_2$ is \<equal to \> $\ell s$\<:\> 
\begin{eqnarray*}
M_2=
\left(
\begin{array}{c|cccccccccccccc}
&{\bf b}_{}(t_1)& & &\bigzerol \\
E_\ell  &&{\bf b}_{}(t_1)&& \\
&\bigzerol &   &  \ddots   &    \\ 
&&   &         &{\bf b}_{}(t_1)\\ 
 \hline &{\bf b}_{}(t_2)&  & &   \bigzerol       \\E_\ell
 && {\bf b}_{}(t_2) & &\\
&\bigzerol&   &  \ddots   &    \\ 
&  & &&{\bf b}_{}(t_2)     \\
 \hline  \vdots & \vdots &\vdots &\vdots& \vdots  \\
\vdots& \vdots &\vdots &\vdots& \vdots  \\
 \hline &{\bf b}_{}(t_s)&  & &   \bigzerol       \\
E_\ell&& {\bf b}_{}(t_s)& &\\
&\bigzerol&   &  \ddots   &    \\ 
&  & &&{\bf b}_{}(t_s)   \\
\end{array}
\right)_{t=z}.
\end{eqnarray*}
There exists an 
$\ell s\times \ell s$ regular matrix $Q_1$ 
such that 
\begin{eqnarray*}
Q_1M_2=
\left(
\begin{array}{c|cccccccccccccc}
&{\bf b}_{}(t_1)& & &\bigzerol \\
E_\ell  &&{\bf b}_{}(t_1)&& \\
&\bigzerol &   &  \ddots   &    \\ 
&&   &         &{\bf b}_{}(t_1)\\ 
 \hline &{\bf b}_{}(t_2)-{\bf b}_{}(t_1)&  & &   \bigzerol       \\
\bigzerol&& {\bf b}_{}(t_2) -{\bf b}_{}(t_1)& &\\
&\bigzerol&   &  \ddots   &    \\ 
&  & &&{\bf b}_{}(t_2) -{\bf b}_{}(t_1)    \\
 \hline  \vdots & \vdots &\vdots &\vdots& \vdots  \\
\vdots& \vdots &\vdots &\vdots& \vdots  \\
 \hline &{\bf b}_{}(t_s)-{\bf b}_{}(t_1)&  & &   \bigzerol       \\
\bigzerol&& {\bf b}_{}(t_s) -{\bf b}_{}(t_1)& &\\
&\bigzerol&   &  \ddots   &    \\ 
&  & &&{\bf b}_{}(t_s)  -{\bf b}_{}(t_1)   \\
\end{array}
\right)_{t=z}. 
\end{eqnarray*}
Namely, 
\begin{eqnarray*}
Q_1M_2=
\left(
\begin{array}{c|cccccccccccccc}
&{\bf b}_{}(t_1)& & &\bigzerol \\
E_\ell  &&{\bf b}_{}(t_1)&& \\
&\bigzerol &   &  \ddots   &    \\ 
&&   &         &{\bf b}_{}(t_1)\\ 
\hline \\ \\[-7mm] &  \overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_2)}  && &   \bigzerol       \\
\bigzerol&& \overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_2)}  & &\\
&\bigzerol&   &  \ddots   &    \\ 
&  & &&  \overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_2)}   \\
 \hline  \vdots & \vdots &\vdots &\vdots& \vdots  \\
\vdots& \vdots &\vdots &\vdots& \vdots  \\
\hline \\ \\[-7mm]
 &  \overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_s)}   &  & &   \bigzerol       \\
\bigzerol&& \overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_s)}   & &\\
&\bigzerol&   &  \ddots   &    \\ 
&  & &&   \overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_s)}    \\
\end{array}
\right)_{t=z},
\end{eqnarray*}
where 
$\overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_i)} =(
\widetilde{f}_1(t_i)-\widetilde{f}_1(t_1),
\<
\widetilde{f}_2(t_i)-\widetilde{f}_2(t_1),
\>
\ldots ,\widetilde{f}_m(t_i)-\widetilde{f}_m(t_1) 
)$ $(2\leq i \leq s)$. 
There exists an 
$(\ell+m\ell)\times (\ell+m\ell)$ regular matrix $Q_2$ 
such that 
\begin{eqnarray*}
Q_1M_2Q_2=
\left(
\begin{array}{c|cccccccccccccc}
&& & & \\
E_\ell  &  &\bigzerol & & \\
&  &   &    &    \\ 
&&   &         &  \\ 
\hline \\ \\[-7mm] &  \overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_2)}  && &   \bigzerol       \\
\bigzerol&& \overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_2)}  & &\\
&\bigzerol&   &  \ddots   &    \\ 
&  & &&  \overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_2)}   \\
 \hline  \vdots & \vdots &\vdots &\vdots& \vdots  \\
\vdots& \vdots &\vdots &\vdots& \vdots  \\
\hline \\ \\[-7mm]
 &  \overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_s)}   &  & &   \bigzerol       \\
\bigzerol&& \overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_s)}   & &\\
&\bigzerol&   &  \ddots   &    \\ 
&  & &&   \overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_s)}    \\
\end{array}
\right)_{t=z}\<.\>
\end{eqnarray*}
\<From \> $s-1\leq s_f-1$ and the definition of $s_f$, it follows that 
\begin{eqnarray*}
{\rm dim} \sum_{i=2}^s\mathbb{R}\overrightarrow{\widetilde{f}(t_1)\widetilde{f}(t_i)}=s-1,
\end{eqnarray*}
where $t=z$. Thus, 
by the construction of the matrix 
$Q_1M_2Q_2$ and $s-1\leq m$, 
we have that the rank of the matrix 
$Q_1M_2Q_2$ 
is \<equal to \> $\ell s$. 
Hence, the rank of the matrix $M_2$ 
must be \<equal to \> $\ell s$. 
Therefore, we have $(\ast\ast)$. 
Thus, there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)$ with Lebesgue measure zero such that 
for any $\pi \in \mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)-\Sigma $, 
\<and \> 
for any $s$ $(2\leq s \leq s_f)$, 
the mapping $(F_\pi \circ f)^{(s)}:N^{(s)}\to (\R^\ell)^s$ is transverse to the 
submanifold $\Delta_s$. 

Moreover, suppose that 
the mapping $F_\pi$ 
satisfies that $| F_\pi^{-1}(y) | \leq s_f$ for any $y\in\mathbb{R}^\ell$. 
Since $f:N\to \mathbb{R}^m$ is injective, 
it follows that $| (F_\pi \circ f)^{-1}(y) | \leq s_f$ for any $y\in\mathbb{R}^\ell$. 
Hence, it follows that for any positive integer $s$ $(s_f+1\leq s)$, 
$(F_\pi \circ f)^{(s)}(N^{(s)})\bigcap \Delta_s=\emptyset$.  
Namely, for any positive integer $s$ $(s_f+1\leq s)$, 
the mapping $(F_\pi \circ f)^{(s)}$ is transverse to $\Delta_s$. 
Thus, the mapping $F_\pi \circ f:N\to \mathbb{R}^\ell$ is a mapping with normal crossings. 

\hfill $\Box$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    
\section{Applications of Theorems \ref{main} and \ref{main2}}\label{section 5}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
In Subsection \ref{application1} (resp., Subsection \ref{application2}), 
applications of Theorem \ref{main} (resp., Theorem \ref{main2}) 
are stated and proved. 
In Subsection \ref{application2}, 
the applications obtained by combining Theorems \ref{main} and  \ref{main2}  
are also given. 
\subsection{Applications of Theorem \ref{main}}\label{application1}

Set 
\begin{eqnarray*} 
\Sigma ^k=\left\{j^1g(0)\in J^1(n,\ell)\mid {\rm corank\ }Jg(0)=k\right\}, 
\end{eqnarray*}
where ${\rm corank\ }Jg(0)={\rm min} \{n,\ell \}-{\rm rank\ } Jg(0)$ 
and $k=1,\<2,\>\ldots ,{\rm min}\{n, \ell\}$. 
Then, $\Sigma ^k$ is an $\mathcal{A}^1$-invariant submanifold of $J^1(n,\ell)$. 
Set 
\begin{eqnarray*}
\Sigma^k(N,\mathbb{R}^\ell)=\bigcup_{\lambda \in \Lambda}\Phi ^{-1}_\lambda \left(\varphi _\lambda (U_\lambda )\times \mathbb{R}^\ell \times \Sigma ^k\right), 
\end{eqnarray*}
where the mappings $\Phi _\lambda$ and $\varphi _\lambda $ are defined 
in Section \ref{section 2}.  
Then, the set $\Sigma ^k(N,\mathbb{R}^\ell)$ is a subfiber-bundle of 
$J^1(N,\mathbb{R}^\ell)$ with the fiber $\Sigma^k$ such that 
\begin{eqnarray*}
{\rm codim}\ \Sigma ^k(N,\mathbb{R}^\ell)&=&{\rm dim}\ J^1(N,\mathbb{R}^\ell)- 
{\rm dim}\ \Sigma ^k(N,\mathbb{R}^\ell) \\
&=&(n-v+k)(\ell-v+k), 
\end{eqnarray*}
where $v={\rm min}\{n, \ell\}$. 
(\<For \> details on $\Sigma^k$ and $\Sigma^k(N,\mathbb{R}^\ell)$, see for example 
\cite{GG}, \<pp.\,\>60--61).
 
As applications of Theorem \ref{main}, we have 
the following Proposition \ref{submain}, 
Corollaries  
\ref{Morse function}, \ref{WU}, \ref{immersion} and \ref{corank}. 

\begin{proposition}\label{submain}
Let $N$ be a manifold of dimension $n$. 
Let $f$ be an immersion 
of $N$ into an open subset $U$ of $\mathbb{R}^m$. 
Let $F:U\to \mathbb{R}^\ell$ be a mapping. 
Then, there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ with Lebesgue measure zero  
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\Sigma$, 
the mapping $j^1(F_\pi \circ f):
N\to J^1(N,\mathbb{R}^{\ell})$ is transverse to the submanifold 
$\Sigma ^k(N,\mathbb{R}^\ell)$ for any positive integer $k$ satisfying $1\leq k\leq v$. 
Especially, in the case of $\ell \geq2$, we have 
$k_0+1\leq v$ and it follows that the mapping $j^1(F_\pi \circ f)$ satisfies that 
$j^1(F_\pi \circ f)(N)\bigcap \Sigma^k(N,\mathbb{R}^\ell)=\emptyset$ 
for any positive integer $k$ satisfying $k_0+1\leq k \leq v$, 
where $k_0$ is the maximum integer 
satisfying 
$(n-v+k_0)(\ell-v+k_0)\leq n$ $(v={\rm min}\{n,\ell\})$. 
\end{proposition}

%\subsection{Proof of  Proposition \ref{corank}}
{\it Proof.}\qquad 
By Theorem \ref{main}, for any positive integer $k$ satisfying $1\leq k\leq v$, 
there exists a subset $\widetilde{\Sigma}_k$ of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ 
with Lebesgue measure zero 
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\widetilde{\Sigma}_k$, 
the mapping $j^1(F_\pi \circ f):
N\to J^1(N,\mathbb{R}^{\ell})$ is transverse to $\Sigma^k(N,\mathbb{R}^\ell)$. 
Set $\Sigma=\bigcup_{k=1}^v\widetilde{\Sigma}_k$. 
Then, it is clearly seen that $\Sigma$ is a subset of 
$\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ with Lebesgue measure zero. 
Hence, it follows that there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ 
with Lebesgue measure zero  
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\Sigma$, 
the mapping $j^1(F_\pi \circ f):
N\to J^1(N,\mathbb{R}^{\ell})$ is transverse to the submanifold 
$\Sigma ^k(N,\mathbb{R}^\ell)$ for any positive integer $k$ satisfying $1\leq k\leq v$. 

Now, we will consider the case of $\ell \geq 2$.  
Firstly, we will show that $k_0+1\leq v$ in the case. 
Suppose that $v\leq k_0$. 
Then, by $(n-v+k_0)(\ell-v+k_0)\leq n$, we have $n\ell \leq n$. 
This contradicts the assumption $\ell \geq 2$. 

Secondly, we will show that in the case of $\ell \geq 2$, 
the mapping $j^1(F_\pi \circ f): 
N\to J^1(N,\mathbb{R}^{\ell})$ satisfies that 
$j^1(F_\pi \circ f)(N)\bigcap \Sigma^k(N,\mathbb{R}^\ell)=\emptyset$ 
for any positive integer $k$ satisfying $k_0+1\leq k\leq v$. 
Suppose that there exist a positive integer $k$ $(k_0+1\leq k \leq v)$ 
and a point $q\in N$ such that 
$j^1(F_\pi \circ f)(q)\in \Sigma^k(N,\mathbb{R}^\ell)$. 
Since the mapping $j^1(F_\pi \circ f):
N\to J^1(N,\mathbb{R}^{\ell})$ is transverse to $\Sigma^k(N,\mathbb{R}^\ell)$ 
at the point $q$, the following holds: 
\begin{eqnarray*}
d(j^1(F_{\pi}\circ f))_{q}(T_{q}N)
+T_{j^1(F_{\pi}\circ f)(q)}\Sigma ^{k}(N,\mathbb{R}^{\ell})
=T_{j^1(F_{\pi}\circ f)(q)}J^1(N,\mathbb{R}^{\ell}).
\end{eqnarray*}
Hence, we have
\begin{eqnarray*}
{}&&{}{\rm dim}\ d(j^1(F_{\pi}\circ f))_{q}(T_{q}N)\\
&\geq &{\rm dim}\ T_{j^1(F_{\pi}\circ f)(q)}J^1(N,\mathbb{R}^{\ell})-
{\rm dim}\ T_{j^1(F_{\pi}\circ f)(q)}\Sigma ^{k}(N,\mathbb{R}^{\ell})\\
&=&{\rm codim}\ T_{j^1(F_{\pi}\circ f)(q)}\Sigma ^{k}(N,\mathbb{R}^{\ell}).
\end{eqnarray*}
Thus, we get $n\geq (n-v+k)(\ell-v+k)$. 
Since the given integer $k_0$ is the maximum integer 
satisfying 
$n\geq(n-v+k_0)(\ell-v+k_0)$, 
it follows that $k\leq k_0$. 
This contradicts the assumption $k_0+1\leq k$. 

\hfill 
$\Box$ 
\subsubsection{Remark}\label{remark k_0}
\begin{enumerate}
\item
In Proposition \ref{submain}, by $(n-v+k_0)(\ell-v+k_0)\leq n$, it is clearly seen that  $k_0\geq0$.  
\item 
In Proposition \ref{submain}, in the case of $\ell=1$, we have $k_0+1>v$. 
Indeed, in the case, by $v=1$, 
we get $(n-1+k_0)k_0\leq n$. Hence, we have $k_0=1$.  
\end{enumerate}
\par 
\bigskip 
\bigskip

A mapping $g:N\to \mathbb{R}$ is called a {\it Morse function} 
if all of the singularities of the mapping $g$ are nondegenerate 
(for details on Morse function\<s\>, see for example, \cite{GG}, \<p.\,\>63). 
In the case of $(n,\ell)=(n,1)$, we have the following. 

\begin{corollary}\label{Morse function}
Let $N$ be a manifold of dimension $n$. 
Let $f$ be an immersion 
of $N$ into an open subset $U$ of $\mathbb{R}^m$. 
Let $F:U\to \mathbb{R}$ be a mapping. 
Then, there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R})$  with Lebesgue measure zero 
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R})-\Sigma$, 
the mapping $F_\pi \circ f:
N\to \mathbb{R}$ is a Morse function. 
\end{corollary}
%\subsection{Proof of Proposition \ref{Morse function}}
{\it Proof.}\qquad 
By Proposition \ref{submain}, 
there exists a subset $\Sigma$ with Lebesgue measure zero 
of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R})$ 
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R})-\Sigma$, 
the mapping $j^1(F_\pi \circ f):
N\to J^1(N, \mathbb{R})$ is transverse to the submanifold $\Sigma ^1(N,\mathbb{R})$. 
Hence, if $q\in N$ is a singular point of the mapping $F_\pi \circ f$, then 
the point $q$ is nondegenerate. 

\hfill 
$\Box$ 
\par 
\bigskip 

For a given mapping $g:N\to \mathbb{R}^{2n-1}$ $(n\geq2)$, 
a singular point $q\in N$ is called a {\it singular point of Whitney umbrella} 
if there exist two germs of diffeomorphisms  
$H:(\mathbb{R}^\ell, g(q)) \to (\mathbb{R}^\ell , 0)$ and 
 $h:(N, q) \to (N, 0)$ 
such that 
$H\circ g \circ h^{-1}(x_1,\<x_2,\>\ldots ,x_n)=(x_1^2,x_1x_2,\ldots ,x_1x_n, x_2, \ldots ,x_{n})$, 
where $(x_1,\<x_2,\>\ldots ,x_n)$ is a local coordinate \<around \> the point 
$h(q)=0\in \mathbb{R}^n$. 
In the case of $(n,\ell)=(n,2n-1)$ $(n\geq2)$, we have the following. 
\begin{corollary}\label{WU}
Let $N$ be a manifold of dimension $n$ $(n\geq2)$. 
Let $f$ be an immersion 
of $N$ into an open subset $U$ of $\mathbb{R}^m$. 
Let $F:U\to \mathbb{R}^{2n-1}$ be a mapping. 
Then, there exists a subset $\Sigma$ with Lebesgue measure zero 
of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{2n-1})$ 
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{2n-1})-\Sigma$, 
any singular point of the mapping $F_\pi \circ f:
N\to \mathbb{R}^{2n-1}$ is 
a singular point of Whitney umbrella. 
\end{corollary}
%\subsection{Proof of Proposition \ref{WU}}
{\it Proof.}\qquad 
By\<, \> for example, \cite{GG}, \<p.\,\>179, we see that 
a point $q\in N$ is a singular point of Whitney umbrella of the mapping $F_\pi \circ f$ 
if 
$j^1(F_\pi \circ f)(q)\in \Sigma^1(N,\mathbb{R}^{2n-1})$ and the mapping 
$j^1(F_\pi \circ f)$ is transverse to the submanifold 
$\Sigma^1(N,\mathbb{R}^{2n-1})$ at $q$. 
Set $\ell=2n-1$ and $v=n$ in Proposition \ref{submain}. 
Then, it is clearly seen that we have $k_0=1$ in Proposition \ref{submain}. 
Hence, there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{2n-1})$ 
with Lebesgue measure zero 
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{2n-1})-\Sigma$, 
the mapping $F_\pi \circ f:
N\to \mathbb{R}^{2n-1}$ is transverse to 
$\Sigma ^k(N,\mathbb{R}^{2n-1})$ for any positive integer $k$ satisfying $1\leq k\leq n$, 
and the mapping satisfies that 
$j^1(F_\pi \circ f)(N)\bigcap \Sigma^k(N,\mathbb{R}^{2n-1})=\emptyset$ 
for any positive integer $k$ satisfying $2\leq k \leq n$. 
Thus, if a point $q\in N$ is a singular point of the mapping $F_\pi \circ f$, then 
it follows that 
$j^1(F_\pi \circ f)(q)\in \Sigma^1(N,\mathbb{R}^{2n-1})$ and
$j^1(F_\pi \circ f)$ is transverse to 
$\Sigma^1(N,\mathbb{R}^{2n-1})$ at $q$. 

\hfill 
$\Box$ 
\par 
\bigskip  

In the case of $\ell \geq2n$,  
the \<immersion \> property of a given mapping $f:N\to U$ 
is preserved by composing generic linearly perturbed mappings as follows:  
\begin{corollary}\label{immersion}
Let $N$ be a manifold of dimension $n$. 
Let $f$ be an immersion 
of $N$ into an open subset $U$ of $\mathbb{R}^{m}$. 
Let $F:U\to \mathbb{R}^{\ell}$ be a mapping $(\ell \geq2n)$. 
Then, there exists a subset $\Sigma$ of 
$\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ with Lebesgue measure zero 
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\Sigma$, 
the mapping $F_\pi \circ f:
N\to \mathbb{R}^{\ell}$ is an immersion. 
\end{corollary}
%\subsection{Proof of  Proposition \ref{immersion}}
{\it Proof.}\qquad 
It is clearly seen that the mapping $F_\pi \circ f:N\to \mathbb{R}^\ell$ 
is an immersion 
if and only if   
$j^1(F_\pi \circ f)(N)\bigcap \bigcup_{k=1}^n\Sigma^k(N,\mathbb{R}^\ell)=
 \emptyset$.
Set $v=n$ and $\ell \geq2n$ in Proposition \ref{submain}. 
Then, it is clearly seen that $k_0\leq0$. By Remark \ref{remark k_0}, 
we get $k_0=0$. 
Hence, 
there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ with Lebesgue measure zero 
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\Sigma$, 
the mapping $j^1(F_\pi \circ f):
N\to J^1(N,\mathbb{R}^{\ell})$ satisfies that 
$j^1(F_\pi \circ f)(N)\bigcap \Sigma^k(N,\mathbb{R}^\ell)=\emptyset$ 
for any positive integer $k$ $(1\leq k \leq n)$. 

\hfill 
$\Box$ 
\par 
\bigskip 
A mapping $g:N\to \mathbb{R}^\ell$ 
{\it has corank  at most k singular points} 
if  
\begin{eqnarray*}
\sup\left\{{\rm corank\ }dg_q\mid q\in N \right\}=k, 
\end{eqnarray*}
where ${\rm corank\ }dg_q={\rm min} \{n,\ell \}-{\rm rank\ } dg_q$. 
By Proposition \ref{submain}, we have the following corollary.  
\begin{corollary}\label{corank}
Let $N$ be a manifold of dimension $n$. 
Let $f$ be an immersion 
of $N$ into an open subset $U$ of $\mathbb{R}^m$. 
Let $F:U\to \mathbb{R}^\ell$ be a mapping. 
Let $k_0$ be the maximum integer 
satisfying 
$(n-v+k_0)(\ell-v+k_0)\leq n$ $(v={\rm min}\{n,\ell\})$.
Then, there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ 
with Lebesgue measure zero  
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\Sigma$, 
the mapping $F_\pi \circ f:
N\to \mathbb{R}^\ell$ has corank at most $k_0$ singular points. 
\end{corollary}
\par 
\bigskip 

\subsection{Applications of Theorem \ref{main2}}\label{application2}
\begin{proposition}\label{normal}
Let $N$ be a manifold of dimension $n$. 
Let $f$ be an injection of $N$ into an open subset  
$U$ of $\mathbb{R}^m$. 
Let $F:U\to \mathbb{R}^\ell$ be a mapping. 
If $(s_f-1)\ell>n s_f$, then 
there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)$ 
with Lebesgue measure zero such that 
for any $\pi \in \mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)-\Sigma $, 
the mapping $F_\pi \circ f:N\to \mathbb{R}^\ell$ is a mapping with normal crossings 
satisfying $(F_\pi \circ f)^{(s_f)}(N^{(s_f)})\bigcap\Delta _{s_f}=\emptyset$. 
\end{proposition}
{\it Proof.}\qquad 
By Theorem \ref{main2}, 
there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)$ with Lebesgue measure zero such that 
for any $\pi \in \mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)-\Sigma $, 
\<and \> 
for any $s$ $(2\leq s \leq s_f)$, 
the mapping $(F_\pi \circ f)^{(s)}:N^{(s)}\to (\R^\ell)^s$ is transverse to the 
submanifold $\Delta_s$. 
Hence, in order to show Proposition \ref{normal},  
it is sufficient to show that 
for any $\pi \in \mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)-\Sigma $, 
the mapping $(F_\pi \circ f)^{(s_f)}$ satisfies that 
$(F_\pi \circ f)^{(s_f)}(N^{(s_f)})\bigcap \Delta_{s_f}=\emptyset$. 

Suppose that there exists an element 
$\pi \in \mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)-\Sigma $ 
such that there exists a point $q\in N^{(s_f)}$ 
satisfying $(F_{\pi}\circ f)^{(s_f)}(q)\in \Delta_{s_f}$. 
Since $(F_{\pi}\circ f)^{(s_f)}$ is transverse to $\Delta_{s_f}$, 
we have the following: 
\begin{eqnarray*}
d((F_{\pi}\circ f)^{(s_f)})_{q}(T_{q}N^{(s_f)})
+T_{(F_{\pi}\circ f)^{(s_f)}(q)}\Delta_{s_f}
=T_{(F_{\pi}\circ f)^{(s_f)}(q)}(\mathbb{R}^\ell)^{s_f}.
\end{eqnarray*}
Hence, we have
%\begin{eqnarray*}
%{\rm dim}\ d(\Gamma_{p_0})_{q_0}(T_{q_0}N)
%+{\rm dim}\ T_{\Gamma_{p_0}(q_0)}\Sigma ^{k'}(N,\mathbb{R}^m)\\
%\geq {\rm dim}\ T_{\Gamma_{p_0}(q_0)}J^1(N,\mathbb{R}^m),
%\end{eqnarray*}
%the following holds:
\begin{eqnarray*}
{}&&{}{\rm dim}\ d((F_{\pi}\circ f)^{(s_f)})_{q}(T_{q}N^{(s_f)})\\
&\geq &{\rm dim}\ T_{(F_{\pi}\circ f)^{(s_f)}(q)}(\mathbb{R}^\ell)^{s_f}-
{\rm dim}\ T_{(F_{\pi}\circ f)^{(s_f)}(q)}\Delta_{s_f}\\
&=&{\rm codim}\ T_{(F_{\pi}\circ f)^{(s_{f})}(q)}\Delta_{s_f}.
\end{eqnarray*}
Thus, we get $ns_f\geq (s_f-1)\ell$. 
This contradicts the assumption $(s_f-1)\ell>ns_f$. 

\hfill 
$\Box$ 
\par 
\bigskip  
In the case of $\ell >2n$, 
the \<injection \> property of a given mapping $f:N\to U$ 
is preserved by composing generic linearly perturbed mappings as follows:  
\begin{corollary}\label{injective}
Let $N$ be a manifold of dimension $n$. 
Let $f$ be an injection of $N$ into an open subset  
$U$ of $\mathbb{R}^m$. 
Let $F:U\to \mathbb{R}^\ell$ be a mapping. 
If $\ell>2n$, then 
there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)$ 
with Lebesgue measure zero such that 
for any $\pi \in \mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)-\Sigma $, 
the mapping $F_\pi \circ f:N\to \mathbb{R}^\ell$ is injective. 
\end{corollary}
{\it Proof.}\qquad 
Since $s_f\geq 2$ and $\ell>2n$, it is easily seen that the dimension pair $(n, \ell)$ 
satisfies the assumption $(s_f-1)\ell>ns_f$ of Proposition \ref{normal}. 
Indeed, \<from \> $\ell>2n$, it follows that $(s_f-1)\ell>2n(s_f-1)$. 
By $s_f\geq 2$, we get $2n(s_f-1)\geq ns_f$. 
 
Hence, 
by Proposition \ref{normal}, 
there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)$ 
with Lebesgue measure zero such that 
for any $\pi \in \mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)-\Sigma $, 
the mapping $(F_\pi \circ f)^{(2)} : N^{(2)} \to (\mathbb{R}^\ell)^2$ is 
transverse to $\Delta_2$. 
In order to show Corollary \ref{injective}, 
it is sufficient to show that 
the mapping $(F_\pi \circ f)^{(2)}$ satisfies that 
$(F_\pi \circ f)^{(2)}(N^{(2)})\bigcap \Delta_2 =\emptyset$. 

Suppose that there exists a point $q\in N^{(2)}$ such that 
$(F_\pi \circ f)^{(2)}(q)\in \Delta_2$. 
Then, we have the following: 
\begin{eqnarray*}
d((F_{\pi}\circ f)^{(2)})_{q}(T_{q}N^{(2)})
+T_{(F_{\pi}\circ f)^{(2)}(q)}\Delta_{2}
=T_{(F_{\pi}\circ f)^{(2)}(q)}(\mathbb{R}^\ell)^{2}.
\end{eqnarray*}
Hence, we have
\begin{eqnarray*}
{}&&{}{\rm dim}\ d((F_{\pi}\circ f)^{(2)})_{q}(T_{q}N^{(2)})\\
&\geq &{\rm dim}\ T_{(F_{\pi}\circ f)^{(2)}(q)}(\mathbb{R}^\ell)^{2}-
{\rm dim}\ T_{(F_{\pi}\circ f)^{(2)}(q)}\Delta_{2}\\
&=&{\rm codim}\ T_{(F_{\pi}\circ f)^{(2)}(q)}\Delta_{2}.
\end{eqnarray*}
Thus, we get $2n\geq \ell$. 
This contradicts the assumption $\ell>2n$. 

\hfill 
$\Box$ 
\par 
\bigskip  

By combining Corollaries \ref{immersion} and \ref{injective}, 
we have the following. 
\begin{corollary}\label{injective immersion}
Let $N$ be a manifold of dimension $n$. 
Let $f$ be an injective immersion of $N$ into an open subset  
$U$ of $\mathbb{R}^m$. 
Let $F:U\to \mathbb{R}^\ell$ be a mapping. 
If $\ell>2n$, then 
there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)$ 
with Lebesgue measure zero such that 
for any $\pi \in \mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)-\Sigma $, 
the mapping $F_\pi \circ f:N\to \mathbb{R}^\ell$ is an injective immersion. 
\end{corollary}

\subsubsection{Remark}
%\begin{enumerate}
%\item
In Corollary \ref{injective immersion}, suppose that the mapping 
$F_\pi \circ f:N\to \mathbb{R}^\ell$ is proper. 
Then, \<an \> injective immersion  $F_\pi \circ f$ 
\<is necessarily an embedding 
(see \cite{GG}, p.\,11). \> Thus, we get the following. 
\begin{corollary}\label{embedding}
Let $N$ be a compact manifold of dimension $n$. 
Let $f$ be an embedding of $N$ into an open subset  
$U$ of $\mathbb{R}^m$. 
Let $F:U\to \mathbb{R}^\ell$ be a mapping. 
If $\ell>2n$, then 
there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)$ 
with Lebesgue measure zero such that 
for any $\pi \in \mathcal{L}(\mathbb{R}^m, \mathbb{R}^\ell)-\Sigma $, 
the mapping $F_\pi \circ f:N\to \mathbb{R}^\ell$ is an embedding. 
\end{corollary}
%\item
%In Theorem \ref{main2} or Proposition \ref{normal}, for instance, if $N$ is a sphere and $f$ is the identity, 
%then it is clearly seen that we get $s_f=3$.  
%\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Further applications}\label{section 6}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Introduction of generalized distance-squared mappings}

Let $i$ and $j$ be positive integers, and 
let $p_i=(p_{i1}, p_{i2}, \ldots, p_{im})$  $(1\le i\le \ell)$ 
(resp., $A=(a_{ij})_{1\le i\le \ell, 1\le j\le m}$) 
be a point of $\mathbb{R}^m$ 
(resp., an $\ell\times m$ matrix with all entries non-zero\<)\>. 
Set $p=(p_1,p_2,\ldots,p_{\ell})\in (\mathbb{R}^m)^{\ell}$. 
Let $G_{(p, A)}:\mathbb{R}^m \to \mathbb{R}^\ell$ be the mapping 
defined by 
{\small 
\[
G_{(p, A)}(x)=\left(
\sum_{j=1}^m a_{1j}(x_j-p_{1j})^2, 
\sum_{j=1}^m a_{2j}(x_j-p_{2j})^2, 
\ldots, 
\sum_{j=1}^m a_{\ell j}(x_j-p_{\ell j})^2
\right), 
\] }where $x=(x_1, x_2, \ldots, x_m)\in \mathbb{R}^m$. 
The mapping $G_{(p, A)}$ is called a {\it generalized distance-squared mapping}, 
and the $\ell$-tuple of points 
\<$p=(p_1,p_2,\ldots ,p_{\ell})\> \in (\mathbb{R}^m)^{\ell}$  
\>
is called the {\it central point} 
of the generalized distance-squared mapping $G_{(p,A)}$. 
%In this section, as an application of Theorem \ref{@f}, 
%we investigate the stability of the generalized distance-squared mappings. 
A {\it distance-squared mapping} $D_p$ 
(resp., {\it Lorentzian distance-squared mapping}
$L_p$) is the mapping $G_{(p,A)}$ 
satisfying that each entry of $A$ is \<equal to \> $1$ 
(resp., $a_{i1}=-1$ and $a_{ij}=1$ $(j\ne 1)$). 

In \cite{D} (resp., \cite{L}), 
a classification result of distance-squared mappings $D_p$ 
(resp., Lorentzian distance-squared mappings
$L_p$) is given. 

In \cite{G1}, 
a classification result of 
generalized distance-squared mappings of the plane into the plane 
is given. 
If the rank of $A$ is \<equal to \> two, 
\<then \>  
a generalized distance-squared mapping 
having a generic central point 
is a mapping of which any singular point is 
a {\color{black}{\rm fold point}} 
except one {\color{black}{\rm cusp point}}.     
%(for details on fold points and cusp points, refer to \cite{plane to plane}). 
The singular set is a rectangular hyperbola. 
If the rank of $A$ is \<equal to \>one, \<then \>
a generalized distance-squared mapping 
having a generic central point is $\mathcal{A}$-equivalent 
to 
{\color{black}{\rm the normal form of fold singularity}} $(x_1,x_2)
\mapsto (x_1,x_2^2)$. 

In \cite{G2}, 
a classification result of 
generalized distance-squared mappings 
of $\mathbb{R}^{m+1}$ into $\mathbb{R}^{2m+1}$ 
is given. 
If the rank of $A$ is \<equal to \> $m+1$, 
\<then \> 
a generalized distance-squared mapping 
having a generic central point 
is $\mathcal{A}$-equivalent to 
{\color{black}{\rm the normal form of Whitney umbrella}} 
$
(x_1,\<x_2,\>\ldots ,x_{m+1})\mapsto 
(x_1^2,x_1x_2,\ldots ,x_1x_{m+1},x_2,\ldots ,x_{m+1})
$. 
If the rank of $A$ is \<strictly smaller \> than $m+1$, 
\<then \> 
a generalized distance-squared mapping 
having a generic central point 
is $\mathcal{A}$-equivalent to the inclusion 
%as follows:
$
(x_1,\<x_2,\>\ldots ,x_{m+1})\mapsto (x_1,\<x_2,\>\ldots ,x_{m+1},0,\ldots ,0)
$. 
\par 
Namely, in \cite{D}, \cite{L}, \cite{G2} and \cite{G1}, 
the properties of generic generalized distance-squared mappings 
are investigated. 
Hence, it is natural to investigate 
the properties of compositions {\color{black}with} generic generalized distance-squared mappings. 

We have another original motivation. 
Height functions and distance-squared functions have been investigated 
in detail {\color{black}so far}, 
and they are \<useful tools \>  
in the applications of singularity theory to differential geometry 
({\color{black}for instance}, see \cite{CS}). 
The mapping in which each component is a height function is 
{\color{black}nothing but} a projection. 
Projections as well as height functions or distance-squared functions 
have been investigated so far.  
In \cite{GP}, compositions of 
{\color{black}generic} projections and embeddings are investigated.

On the other hand, 
the mapping in which each component 
is a distance-squared function is a distance-squared mapping. 
In addition, the notion of generalized distance-squared mapping is 
an extension of the distance-squared mappings. 
Therefore, it is natural to investigate compositions 
with generic generalized distance-squared mappings as well as projections. 
\par 
%\medskip 

\subsection{Applications of Theorem \ref{main} to $G_{(p, A)}:\mathbb{R}^m \to \mathbb{R}^\ell$}
\begin{proposition} 
\label{Gmain}
Let $N$ be a manifold of dimension $n$. 
Let $f:N\to \mathbb{R}^m$ be an immersion. 
Let $A=(a_{ij})_{1\leq i \leq \ell, 1\leq j \leq m}$ 
be an $\ell \times m$ matrix with all entries non-zero. 
If $X$ is an $\mathcal{A}^1$-invariant submanifold of $J^1(n,\ell)$, 
then there exists a subset $\Sigma$ of $(\mathbb{R}^m)^\ell$ 
with Lebesgue measure zero  
such that for any $p \in (\mathbb{R}^m)^\ell-\Sigma$, 
the mapping $j^1(G_{(p,A)} \circ f):
N\to J^1(N,\mathbb{R}^\ell)$ is transverse to 
the submanifold $X(N,\mathbb{R}^\ell)$.
\end{proposition}
{\it Proof.}\qquad 
Let $H:\mathbb{R}^{\ell} \to \mathbb{R}^{\ell}$ 
be \<a \> diffeomorphism of the target for deleting constant terms. 
The composition $H\circ G_{(p,A)}:\mathbb{R}^{m}\to \mathbb{R}^{\ell}$ 
is \<given \> as follows:
\begin{eqnarray*}
H\circ G_{(p, A)}(x)&=&\left(
\sum_{j=1}^m a_{1j}x_j^2-2\sum_{j=1}^m a_{1j}p_{1j}x_j, 
\<
\sum_{j=1}^m a_{2j}x_j^2-2\sum_{j=1}^m a_{2j}p_{2j}x_j, \right. 
\>
\\&&
\left. 
\ldots, 
\sum_{j=1}^m a_{\ell j}x_j^2-2\sum_{j=1}^m a_{\ell j}p_{\ell j}x_j
\right), 
\end{eqnarray*}
where $x=(x_1,\<x_2,\>\ldots ,x_m)$. 

Let $\psi :(\mathbb{R}^m)^\ell \to \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ 
be the mapping defined by 
\begin{eqnarray*}
\psi (p_{11},p_{12},\ldots ,p_{\ell m})=-2(a_{11}p_{11},a_{12}p_{12},\ldots ,
a_{\ell m}p_{\ell m}). 
\end{eqnarray*}
Remark that \<we have \> the natural identification 
$\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})=(\mathbb{R}^m)^\ell$. 
Since $a_{ij}\not =0$ for any $i$, $j$ $(1\leq i \leq \ell$, $1\leq j \leq m)$, 
it is clearly seen that $\psi$ is a $C^\infty$ diffeomorphism. 

Set $F_{i}(x)=\sum_{j=1}^m a_{ij}x_j^2$ $(1\leq i \leq \ell)$ and 
$F=(F_1,\<F_2,\>\ldots ,F_\ell)$.  
By Theorem \ref{main}, 
there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ with Lebesgue measure zero  
such that 
for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\Sigma$, 
the mapping $j^{1}(F_{\pi}\circ f):N\to J^{1}(N,\mathbb{R}^\ell)$ 
is transverse to $X(N,\mathbb{R}^\ell)$. 
Since $\psi^{-1}:\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell}) \to 
(\mathbb{R}^m)^\ell$ is a $C^\infty$ mapping, 
$\psi^{-1}(\Sigma)$ is a subset of $(\mathbb{R}^m)^\ell$ 
with Lebesgue measure zero. 
For any $p\in (\mathbb{R}^m)^\ell-\psi^{-1}(\Sigma)$, 
we have $\psi(p)\in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\Sigma$. 
Hence, for any $p\in (\mathbb{R}^m)^\ell-\psi^{-1}(\Sigma)$, 
the mapping 
$j^{1}(H\circ G_{(p, A)}\circ f):N\to J^{1}(N,\mathbb{R}^\ell)$ 
is transverse to $X(N,\mathbb{R}^\ell)$. 
Then, since $H:\mathbb{R}^{\ell} \to \mathbb{R}^{\ell}$ 
is a diffeomorphism, 
the mapping 
$j^{1}(G_{(p, A)}\circ f):N\to J^{1}(N,\mathbb{R}^\ell)$ 
is transverse to $X(N,\mathbb{R}^\ell)$. 

\hfill 
$\Box$ 

\subsubsection{Remark}\label{remarkGmain}
As applications of Proposition \ref{Gmain}, regarding generalized distance-squared mappings, 
we get analogies of  Proposition \ref{submain}, 
Corollaries \ref{Morse function}, \ref{WU}, \ref{immersion} and  \ref{corank}.  

\subsection{Applications of Theorem \ref{main2} to $G_{(p, A)}:\mathbb{R}^m \to \mathbb{R}^\ell$}
\begin{proposition}\label{Gmain2}
Let $N$ be a manifold of dimension $n$. 
Let $f : N\to \mathbb{R}^m$ be an injection. 
Let $A=(a_{ij})_{1\leq i \leq \ell, 1\leq j \leq m}$ 
be an $\ell \times m$ matrix with all entries non-zero. 
Then, there exists a subset $\Sigma$ of $(\mathbb{R}^m)^\ell$ with Lebesgue measure zero such that 
for any $p \in (\mathbb{R}^m)^\ell-\Sigma $, 
\<and \>
for any $s$ $(2\leq s \leq s_f)$, 
the mapping $(G_{(p,A)} \circ f)^{(s)}:N^{(s)}\to (\R^\ell)^s$ is transverse to the 
submanifold $\Delta_s$. 
Moreover, if the mapping $G_{(p,A)}$ satisfies that $| G_{(p,A)}^{-1}(y) | \leq s_f$ for any $y\in\mathbb{R}^\ell$, 
then the mapping $G_{(p,A)} \circ f:N\to \mathbb{R}^\ell$ is a mapping with normal crossings, where $|X|$ is the number of elements of the set $X$.    
\end{proposition}
{\it Proof.}\qquad 
The method of the proof of this proposition is 
the same as \<that \> of Proposition \ref{Gmain}. 
Let $H:\mathbb{R}^{\ell} \to \mathbb{R}^{\ell}$ 
be \<a \> diffeomorphism of the target for deleting constant terms. 
The composition $H\circ G_{(p,A)}:\mathbb{R}^{m}\to \mathbb{R}^{\ell}$ 
is \<given \> as follows:
\begin{eqnarray*}
H\circ G_{(p, A)}(x)&=&\left(
\sum_{j=1}^m a_{1j}x_j^2-2\sum_{j=1}^m a_{1j}p_{1j}x_j, 
\<
\sum_{j=1}^m a_{2j}x_j^2-2\sum_{j=1}^m a_{2j}p_{2j}x_j, \right. 
\>
\\&&
\left. 
\ldots, 
\sum_{j=1}^m a_{\ell j}x_j^2-2\sum_{j=1}^m a_{\ell j}p_{\ell j}x_j
\right), 
\end{eqnarray*}
where $x=(x_1,\<x_2,\>\ldots ,x_m)$. 

Let $\psi :(\mathbb{R}^m)^\ell \to \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ 
be the mapping defined by 
\begin{eqnarray*}
\psi (p_{11},p_{12},\ldots ,p_{\ell m})=-2(a_{11}p_{11},a_{12}p_{12},\ldots ,
a_{\ell m}p_{\ell m}). 
\end{eqnarray*}
Remark that \<we have \> the natural identification 
$\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})=(\mathbb{R}^m)^\ell$. 
Since $a_{ij}\not =0$ for any $i$, $j$ $(1\leq i \leq \ell$, $1\leq j \leq m)$, 
it is clearly seen that $\psi$ is a $C^\infty$ diffeomorphism. 

Set $F_{i}(x)=\sum_{j=1}^m a_{ij}x_j^2$ $(1\leq i \leq \ell)$ and 
$F=(F_1,\<F_2,\>\ldots ,F_\ell)$.  
By Theorem \ref{main2}, 
there exists a subset $\Sigma$ of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ with Lebesgue measure zero  
such that 
for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\Sigma$, 
\<and \>
for any $s$ $(2\leq s\leq s_f)$, 
the mapping $(F_{\pi}\circ f)^{(s)}:N^{(s)}\to (\mathbb{R}^\ell)^s$ 
is transverse to $\Delta_s$. 
Moreover, if the mapping $F_\pi$ 
satisfies that $| F_\pi^{-1}(y) | \leq s_f$ for any $y\in\mathbb{R}^\ell$, 
then the mapping $F_\pi \circ f:N\to \mathbb{R}^\ell$ is a mapping with normal crossings. 
Since $\psi^{-1}:\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell}) \to 
(\mathbb{R}^m)^\ell$ is a $C^\infty$ mapping, 
$\psi^{-1}(\Sigma)$ is a subset of $(\mathbb{R}^m)^\ell$
with Lebesgue measure zero. 
For any $p\in (\mathbb{R}^m)^\ell-\psi^{-1}(\Sigma)$, 
we have $\psi(p)\in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\Sigma$. 
Hence, for any $p\in (\mathbb{R}^m)^\ell-\psi^{-1}(\Sigma)$, 
and    
for any $s$ $(2\leq s\leq s_f)$, 
the mapping 
$(H\circ G_{(p, A)}\circ f)^{(s)}:N^{(s)}\to (\mathbb{R}^\ell)^s$ 
is transverse to $\Delta_s$. 
Moreover, if the mapping $H\circ G_{(p,A)}$ satisfies that $|(H\circ  G_{(p,A)})^{-1}(y) | \leq s_f$ for any $y\in\mathbb{R}^\ell$, 
then the mapping $H\circ G_{(p,A)} \circ f:N\to \mathbb{R}^\ell$ is a mapping with normal crossings. 
Then, since $H:\mathbb{R}^{\ell} \to \mathbb{R}^{\ell}$ 
is a diffeomorphism, the proposition holds. 
%%%%%%%%%%%%%%%%%

\hfill 
$\Box$
 
%%%%%%%%%%%%%%%%%
\subsubsection{Remark}\label{remarkGmain2}
As applications of Proposition \ref{Gmain2}, regarding generalized distance-squared mappings, 
we get analogies of Proposition \ref{normal}, 
Corollaries \ref{injective}, \ref{injective immersion} and \ref{embedding}. 
\bigskip

As the special case of the classification result of 
distance squared mappings $D_p$ 
(resp., Lorentzian distance-squared mappings $L_p$) in \cite{D} (resp., \cite{L}), 
we have Lemma \ref{Dlemma} (resp., Lemma \ref{Llemma}). 
\begin{lemma}[\cite{D}] \label{Dlemma}
Let $D_p:\mathbb{R}^m\to \mathbb{R}^\ell$ be a distance-squared mapping. 
Then, the following holds: 
\begin{enumerate}
\item
In the case of $m=\ell=1$, 
for any $p \in \mathbb{R}$, 
the mapping $D_p: \mathbb{R}\to \mathbb{R}$ 
is $\mathcal{A}$-equivalent to $x\mapsto x^2$. 
\item
In the case of $2\leq m=\ell$, 
there exists a subset $\Sigma$ of $(\mathbb{R}^m)^m$ with Lebesgue measure zero such that 
for any $p \in (\mathbb{R}^m)^m-\Sigma $, 
the mapping $D_p: \mathbb{R}^m\to \mathbb{R}^m$ 
is $\mathcal{A}$-equivalent to the normal form of definite fold mappings 
\<$(x_1, x_2, \ldots ,x_m)\mapsto (x_1, x_2, \ldots ,x_{m-1},x_m^2)\>$. 
\>
\item
In the case of $1\leq m< \ell$, 
there exists a subset $\Sigma$ of $(\mathbb{R}^m)^\ell$ with Lebesgue measure zero such that 
for any $p \in (\mathbb{R}^m)^\ell-\Sigma $, 
the mapping $D_p: \mathbb{R}^m\to \mathbb{R}^\ell$ 
is $\mathcal{A}$-equivalent to the inclusion 
$\<(x_1,x_2, \ldots ,x_m)\mapsto (x_1,x_2, \ldots ,x_{m},0,\ldots ,0)\>$. 
\end{enumerate}
\end{lemma}
\>
\begin{lemma}[\cite{L}] \label{Llemma}
Let $L_p:\mathbb{R}^m\to \mathbb{R}^\ell$ be a Lorentzian distance-squared mapping. 
Then, the following holds: 
\begin{enumerate}
\item
In the case of $m=\ell=1$, 
for any $p \in \mathbb{R} $, 
the mapping $L_p: \mathbb{R}\to \mathbb{R}$ 
is $\mathcal{A}$-equivalent to $x\mapsto x^2$. 
\item
In the case of $2\leq m=\ell$, 
there exists a subset $\Sigma$ of $(\mathbb{R}^m)^m$ with Lebesgue measure zero such that 
for any $p \in (\mathbb{R}^m)^m-\Sigma $, 
the mapping $L_p: \mathbb{R}^m\to \mathbb{R}^m$ 
is $\mathcal{A}$-equivalent to the normal form of definite fold mappings 
\<$(x_1,x_2, \ldots ,x_m)\mapsto (x_1,x_2,\ldots ,x_{m-1},x_m^2)$. \>
\item
In the case of $1\leq m< \ell$, 
there exists a subset $\Sigma$ of $(\mathbb{R}^m)^\ell$ with Lebesgue measure zero such that 
for any $p \in (\mathbb{R}^m)^\ell-\Sigma $, 
the mapping $L_p: \mathbb{R}^m\to \mathbb{R}^\ell$ 
is $\mathcal{A}$-equivalent to the inclusion 
\<$(x_1,x_2,\ldots ,x_m)\mapsto (x_1,x_2,\ldots ,x_{m},0,\ldots ,0)$.\> 
\end{enumerate}
\end{lemma}
\>
Notice that in the case of $m<\ell$, 
since $D_p:\mathbb{R}^m\to \mathbb{R}^\ell$ and 
$L_p:\mathbb{R}^m\to \mathbb{R}^\ell$ 
having a generic central point are $\mathcal{A}$-equivalent to the inclusion, 
\<the conclusions trivially hold\>. 
Thus, we will consider the case of $m=\ell$. 
\begin{proposition}\label{DL}. 
Let $N$ be a manifold of dimension $n$. 
Let $f  : N\to \mathbb{R}^m$ be an injection. 
Let $D_p:\mathbb{R}^m\to \mathbb{R}^m$ 
$($resp., $L_p:\mathbb{R}^m\to \mathbb{R}^m$$)$ be 
a distance-squared mapping $($resp., Lorentzian distance-squared mapping$)$. 
Then, the following holds:
\begin{enumerate}
\item
There exists a subset $\Sigma$ of $(\mathbb{R}^m)^m$ with Lebesgue measure zero such that 
for any $p \in (\mathbb{R}^m)^m-\Sigma $, 
the mapping $D_p \circ f:N\to \mathbb{R}^m$ 
is a mapping with normal crossings. 
\item
There exists a subset $\Sigma$ of $(\mathbb{R}^m)^m$ with Lebesgue measure zero such that 
for any $p \in (\mathbb{R}^m)^m-\Sigma $, 
the mapping $L_p \circ f:N\to \mathbb{R}^m$ 
is a mapping with normal crossings. 
\end{enumerate}
\end{proposition}
{\it Proof.}\qquad 
The proofs of 
the assertions 1 and 2 of this proposition are almost the same. 
Thus, it is sufficient to show the assertion 1.  
By the assertions 1 and 2 of Lemma \ref{Dlemma}, 
there exists a subset $\Sigma_1$ of $(\mathbb{R}^m)^m$ with Lebesgue measure zero such that 
for any $p \in (\mathbb{R}^m)^m-\Sigma _1$, 
the mapping $D_p: \mathbb{R}^m\to \mathbb{R}^m$ 
satisfies that $|D_p(y)| \leq 2$ 
for any $y\in \mathbb{R}^m$. 

On the other hand, by Proposition \ref{Gmain2}, 
there exists a subset $\Sigma_2$ of $(\mathbb{R}^m)^m$ with Lebesgue measure zero such that 
for any $p \in (\mathbb{R}^m)^m-\Sigma_2 $, 
if the mapping $D_p$ satisfies that $| D_p^{-1}(y) | \leq s_f$ for any $y\in\mathbb{R}^m$, 
then the mapping $D_p \circ f:N\to \mathbb{R}^m$ 
is a mapping with normal crossings. 
Set $\Sigma=\Sigma_1\cup \Sigma_2$. 
It is clearly seen that $\Sigma$ is a subset of 
 $(\mathbb{R}^m)^m$ with Lebesgue measure zero. 
Then, 
 for any $p \in (\mathbb{R}^m)^m-\Sigma$, 
 the mapping $D_p \circ f:N\to \mathbb{R}^m$ 
is a mapping with normal crossings. 
%%%%%%%%%%%%%%%%%

\hfill 
$\Box$
 
%%%%%%%%%%%%%%%%%
By combining Proposition \ref{DL} and the analogy of 
Corollary \ref{immersion} in Remark \ref{remarkGmain}, we have the 
following. 
\begin{corollary}\label{immersion with}
Let $N$ be a manifold of dimension $n$. 
Let $f : N\to \mathbb{R}^m$ be an injective immersion $(2n\leq m)$.  Let $D_p:\mathbb{R}^{m}\to \mathbb{R}^{m}$ 
$($resp., $L_p:\mathbb{R}^{m}\to \mathbb{R}^{m}$$)$ be 
a distance-squared mapping $($resp., Lorentzian distance-squared mapping$)$. 
Then, the following holds:
\begin{enumerate}
\item
There exists a subset $\Sigma$ of $(\mathbb{R}^{m})^{m}$ 
with Lebesgue measure zero such that 
for any $p \in (\mathbb{R}^{m})^{m}-\Sigma $, 
the mapping $D_p \circ f:N\to \mathbb{R}^{m}$ 
is an immersion with normal crossings. 
\item
There exists a subset $\Sigma$ of $(\mathbb{R}^{m})^{m}$ 
with Lebesgue measure zero such that 
for any $p \in (\mathbb{R}^{m})^{m}-\Sigma $, 
the mapping $L_p \circ f:N\to \mathbb{R}^{m}$ 
is an immersion with normal crossings. 
\end{enumerate}
\end{corollary}
\subsubsection{Remark}
In Corollary \ref{immersion with}, if the mapping 
$D_p \circ f:N\to \mathbb{R}^{2n}$ (resp., $L_p \circ f:N\to \mathbb{R}^{2n}$) 
is proper, 
then the immersion with normal crossings $D_p \circ f:N\to \mathbb{R}^{2n}$ 
(resp., $L_p \circ f:N\to \mathbb{R}^{2n}$) 
is necessarily stable 
(see \cite{GG}, p.\,86).   
Thus, we get the following. 
\begin{corollary}\label{stable}
Let $N$ be a compact manifold of dimension $n$. 
Let $f : N\to \mathbb{R}^{2n}$ be an embedding. 
Let $D_p:\mathbb{R}^{2n}\to \mathbb{R}^{2n}$ 
$($resp., $L_p:\mathbb{R}^{2n}\to \mathbb{R}^{2n}$$)$ be 
a distance-squared mapping $($resp., Lorentzian distance-squared mapping$)$. 
Then, the following holds:
\begin{enumerate}
\item 
There exists a subset $\Sigma$ of $(\mathbb{R}^{2n})^{2n}$ with Lebesgue measure zero such that 
for any $p \in (\mathbb{R}^{2n})^{2n}-\Sigma $, 
the mapping $D_p \circ f:N\to \mathbb{R}^{2n}$ 
is stable. 
\item
There exists a subset $\Sigma$ of $(\mathbb{R}^{2n})^{2n}$ with Lebesgue measure zero such that 
for any $p \in (\mathbb{R}^{2n})^{2n}-\Sigma $, 
the mapping $L_p \circ f:N\to \mathbb{R}^{2n}$ 
is stable. 
\end{enumerate}
\end{corollary}
We will compare Corollaries \ref{embedding} and \ref{stable} from 
the viewpoint of stability. 
Remark that the dimension of the target space in Corollary \ref{stable} is 
smaller than that in Corollary \ref{embedding}. 

\section{Appendix}\label{section 7}
In this section, 
the main theorems in \cite{F1} and \cite{GP} are stated. 
For this, 
we prepare some notions. 

Let $N$ and $P$ be manifolds. 
Let ${}_s J^r(N,P)$ be the space consisting of 
elements $(j^r g(q_1),j^r g(q_2), \ldots ,j^r g(q_s))\in J^r(N,P)^s$ 
satisfying $(q_1,q_2,\ldots ,q_s)\in N^{(s)}$. 
Since $N^{(s)}$ is an open submanifold of $N^s$, 
the space ${}_s J^r(N,P)$ is also an open submanifold of $J^r(N,P)^s$.
For a given mapping $g:N\to P$, 
the mapping ${}_s j^rg:N^{(s)}\to {}_s J^r(N,P)$ 
is defined by 
\< 
$(q_1,q_2,\ldots ,q_s) \mapsto (j^rg(q_1),j^rg(q_2),\ldots ,j^rg(q_s))\>$.  

Let $W$ be a submanifold of ${}_s J^r(N,P)$. 
A mapping $g:N\to P$ will be said to be {\it transverse with respect to} $W$ 
if ${}_s j^r g:N^{(s)}\to {}_s J^r(N,P)$ is transverse to $W$. 

Following Mather (\cite{GP}), we can partition $P^s$ as follows. 
Given any partition $\pi$ of \<$\{1,2, \ldots ,s\}\>$,  
let $P^{\pi}$ denote the set of $s$-tuples 
$(y_1,y_2,\ldots ,y_s)\in P^s$ 
such that $y_i=y_j$ 
if and only if the 
two positive integers 
$i$ and $j$ are in the same member of the partition $\pi$. 

Let Diff $N$ denote 
the group of diffeomorphisms of $N$. 
We have the natural action of Diff $N$ $\times $ Diff $P$ 
on ${}_s J^r(N,P)$ 
such that for a mapping $g:N\to P$, 
the equality 
\<$(h,H)\cdot {}_sj^rg(q)={}_sj^r(H\circ g\circ h^{-1})(q')$ 
holds, where $q=(q_1,q_2, \ldots ,q_s)$ and $q'=(h(q_1),h(q_2), \ldots ,h(q_s))\>$. 
A subset $W$ of ${}_s J^r(N,P)$ is called {\it invariant} 
if it is invariant under this action. 

We recall the following identification $(\ast\ast\ast)$ from \cite{GP}. 
Let \<$q=(q_1,q_2, \ldots ,q_s)\in N^{(s)}\>$, 
let $g:U\to P$ be a mapping defined 
in a neighborhood $U$ of \<$\{q_1,q_2, \ldots ,q_s\}\>$ in $N$, and 
let $z={}_s j^r g(q)$, \<$q'=(g(q_1),g(q_2), \ldots ,g(q_s))\>$. 
Let ${}_s J^r(N,P)_q$ and ${}_s J^r(N,P)_{q,q'}$ denote 
the fibers of ${}_s J^r(N,P)$ over $q$ and over $(q,q')$ respectively. 
Let $J^r(N)_q$ denote the $\mathbb{R}$-algebra of 
$r$-jets at $q$ of functions on $N$. 
Namely, 
\begin{eqnarray*}
J^r(N)_q={}_s J^r(N,\mathbb{R})_q.
\end{eqnarray*} 
Set $g^*TP=\bigcup_{q\in N}T_{g(q)}P$, 
where $TP$ is the tangent bundle of $P$.    
Let $J^r(g^*TP)_q$ denote 
the $J^r(N)_q$-module of $r$-jets at $q$ of 
sections of the bundle $g^*TP$. 
Let $\mathfrak{m}_q$ be the ideal in $J^r(N)_q$ 
consisting of jets of functions which vanish at $q$. 
Namely,   
\begin{eqnarray*}
\mathfrak{m}_q=\{{}_s j^r h(q)\in {}_s J^r(N,\mathbb{R})_q \mid 
h(q_1)=h(q_2)=\cdots =h(q_s)=0\}. 
\end{eqnarray*}
Let $\mathfrak{m}_q J^r(g^*TP)_q$ be the set consisting of finite sums 
of the product of 
an element of $\mathfrak{m}_q$ and an element of $J^r(g^*TP)_q$. 
Namely, we get 
\begin{eqnarray*}
\mathfrak{m}_q J^r(g^*TP)_q
=J^r(g^*TP)_q\cap \{{}_sj^r \xi(q)\in {}_s J^r(N,TP)_q \mid \xi(q_1)=
\xi(q_2)=
\cdots =\xi(q_s)=0\}.
\end{eqnarray*}
Then, it is easily seen that the following canonical identification 
of $\mathbb{R}$ vector spaces $(\ast\ast\ast)$ holds.
\begin{align}
T({}_s J^r(N,P)_{q,q'})_z=\mathfrak{m}_q J^r(g^*TP)_q. \tag{$\ast\ast\ast $}
\end{align}

Let $W$ be a non-empty submanifold of 
${}_s J^r(N,P)$. 
Choose \<$q=(q_1,q_2, \ldots ,q_s)\in N^{(s)}\>$ and 
$g:N\to P$, and 
let $z={}_s j^r g(q)$ and \<$q'=(g(q_1),g(q_2),\ldots ,g(q_s))\>$. 
Suppose that the choice is made so that $z\in W$. 
Let $W_{q,q'}$ denote the fiber of $W$ over $(q,q')$. 

Then, under the identification $(\ast\ast\ast)$, the tangent space 
$T(W_{q,q'})_z$ can be identified with a vector subspace of 
$\mathfrak{m}_q$$J^r(g^*TP)_q$. 
We denote this vector subspace by $E(g,q,W)$. 
\>
\begin{definition}\label{modular}
{\rm 
The submanifold $W$ is called {\it modular} 
if conditions $(\alpha)$ and $(\beta)$ below are satisfied:
\begin{enumerate}
\item[$(\alpha)$]
The set $W$ is an invariant submanifold of ${}_s J^r(N,P)$, 
and lies over $P^{\pi}$ 
for some partition $\pi$ of \<$\{1,2, \ldots ,s\}\>$.
\>
\item[$(\beta)$] 
For any $q\in N^{(s)}$ and any mapping $g:N \to P$ 
such that ${}_s j^r g(q)\in W$, 
the subspace $E(g,q,W)$ is a $J^r(N)_q$-submodule.
\end{enumerate}
}
\end{definition}

Now, suppose that $P=\mathbb{R}^\ell$. 
The main theorem in \cite{GP} is the following. 
\begin{theorem}[\cite{GP}] \label{@m}
Let $N$ be a manifold of dimension $n$. 
Let $f$ be an embedding 
of $N$ into $\mathbb{R}^m$. 
If $W$ is a modular submanifold of ${}_{s}J^{r}(N,\mathbb{R}^\ell)$ and $m>\ell$, 
then there exists a subset $\Sigma$ with Lebesgue measure zero 
of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ 
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\Sigma$, 
$\pi \circ f:
N\to \mathbb{R}^\ell$ is transverse with respect to $W$.
\end{theorem}
Then, the main theorem in \cite{F1} is the following. 
\begin{theorem}[\cite{F1}]\label{@f}
Let $N$ be a manifold of dimension $n$. 
Let $f$ be an embedding 
of $N$ into an open subset $U$ of $\mathbb{R}^m$. 
Let $F:U\to \mathbb{R}^\ell$ be a mapping. 
If $W$ is a modular submanifold of ${}_{s}J^{r}(N,\mathbb{R}^\ell)$, 
then there exists a subset $\Sigma$ with Lebesgue measure zero 
of $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$ 
such that for any $\pi \in \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})-\Sigma$, 
$F_\pi \circ f:
N\to \mathbb{R}^\ell$ is transverse with respect to $W$.
\end{theorem}
The assertion (6) in Section \ref{section 1}, 
Corollary \ref{embedding} in Section \ref{section 5} and Corollary \ref{stable} in Section \ref{section 6} of the present paper are obtained as corollaries of 
Theorems \ref{main} and \ref{main2} in this paper. 
On the other hand, they are also corollaries of 
Theorem \ref{@f}. 
%\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    
\section*{Acknowledgements}
The author is most grateful to the anonymous reviewers for carefully reading 
the first manuscript of this paper and for giving invaluable suggestions. 
He is also grateful to Takashi Nishimura for his kind advices, and 
to Atsufumi Honda, Satoshi Koike, 
Osamu Saeki, 
Hajime Sato, Masatomo Takahashi, Takahiro Yamamoto for their valuable comments.
The author is supported by JSPS KAKENHI Grant Number 16J06911. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\>
\begin{thebibliography}{99}
\bibitem{abra}R.~Abraham,  
\textit{Transversality in manifolds of mappings}, 
Bull. Amer. Math. Soc. 
\textbf{69} (1963), 470--474.   

%\bibitem{Arnold}V.~I.~Arnol'd,  
%\textit{Singularities of smooth mappings}, 
%(Russian) 
%Uspehi Mat. Nauk, \textbf{23} (1968), 3--44. 

%\bibitem{Boardman}J.~M.~Boardman,  
%\textit{Singularities of differentiable maps}, 
%Inst. Hautes \'Etudes Sci. Publ. Math., \textbf{33} (1967), 21--57.

\bibitem{CS}J.~W.~Bruce and P.~J.~Giblin, 
\textit{Curves and singularities (second edition)}, 
Cambridge University Press, Cambridge, 1992. 

%\bibitem{bruce}J. W. Bruce and N. P. Kirk, 
%\textit{Generic projections of stable mappings}, 
%Bull. London Math. Soc. \textbf{32}, (2000), no. 6, 718-728.

\bibitem{GG}M.~Golubitsky and V.~Guillemin, 
\textit{Stable mappings and their singularities}, 
Graduate Texts in Mathematics \textbf{14}, Springer, New York, 1973.

%\bibitem{arnoldetall}V.~I.~Arnol'd, S.~M.~Gusein-Zade and A.~N.~Varchenko, 
%{\rm Singularities of Differentiable Maps I.} {Monographs in Mathematics}, 
%\textbf{82}, Birkh\"auser, Boston Basel Stuttgart, 1985. 

%\bibitem{duplessisgaffneywilson}A.~du Plessis, T.~Gaffney and L.~Wilson, 
%\textit{Map-germs determined by their discriminant}, 
%in \lq\lq Stratifications, singularities and differential equations, I (marseille 1990; Honolulu 1990)\rq\rq, 
%1--40, Travaux en Cours, \textbf{54}, Hermann, Paris, 1997.   

\bibitem{F1}S.~Ichiki, 
\textit{Generic linear perturbations}, 
preprint (available
from arXiv:1607.03220 [math.MG]).


\bibitem{D}S.~Ichiki and T.~Nishimura, 
\textit{Distance-squared mappings}, 
Topology Appl., 
\textbf{160} (2013), 1005--1016.   

\bibitem{L}S.~Ichiki and T.~Nishimura, 
\textit{Recognizable classification of Lorentzian distance-squared 
mappings}, J.~Geom.~Phys., 
\textbf{81} (2014), 62--71.   

\bibitem{G2}S.~Ichiki and T.~Nishimura, 
\textit{Generalized distance-squared mappings of 
$\mathbb{R}^{n+1}$ into $\mathbb{R}^{2n+1}$}, 
Contemporary Mathematics, Amer. Math. Soc., Providence RI, \textbf{675}  (2016),
 121--132.

\bibitem{G3}S.~Ichiki, T.~Nishimura, 
\textit{Preservation of immersed or injective properties by composing generic 
generalized distance-squared mappings}, accepted for publication in Proceedings of BRAZIL-MEXICO-2015 Singularity meetings at Salvador, to be published in a volume of Springer Proceedings in Mathematics and Statistics.

\bibitem{G1}S.~Ichiki, T.~Nishimura, R.~Oset Sinha and M.~A.~S.~Ruas, 
\textit{Generalized distance-squared mappings of the plane into the plane}, 
Adv. Geom., \textbf{16} (2016), 189--198. 


%\bibitem{mather2}J.~N.~Mather, 
%\textit{Stability of $C^\infty$ mappings II. Infinitesimal stability implies stability}, 
%{Ann. of Math.}, 
%\textbf{89} (2) (1969), 254--291. 
%\bibitem{mather4}J.~Mather, 
%\textit{Stability of}  
%$C^\infty$  
%\textit{mappings, \textit{IV},{ }Classification of stable 
%map-germs by} $\mathbb{R}$-\textit{algebras}, 
%{\rm Publ. Math. Inst. Hautes \'Etudes Sci}. 
%\textbf{37}, (1970), 223--248.

%\bibitem{mather5}J.~N.~Mather, 
%\textit{Stability of $C^\infty$ mappings V. Transversality}, 
%{Adv.~in~Math.}, 
%\textbf{4} (1970), 301--336. 

%\bibitem{mather6}J.~N.~Mather, 
%\textit{Stability of $C^\infty$ mappings V\!I. The nice dimensions}, 
%{Lecture
%Notes in Math.}, 
%\textbf{192} (1971), 207--253. 

%\bibitem{montesinos}A.~M.~Montesinos, 
%{\rm Maps $\mathbb{R}^2$ to $\mathbb{R}^2$, Computer program available by anonymous ftp at http://www.uv.es/montesin/}. 
%\bibitem{wall}C.~T.~C.~Wall, 
%\textit{Finite determinacy of smooth map-germs}, 
%{Bull. London Math. Soc.} 
%\textbf{13}, (1981), 481--539.

%\bibitem{TB}J. N. Mather, 
%\textit{On Thom-Boardman singularities}, 
%Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), 
%Academic Press, New York, 1973, 233--248. 

\bibitem{GP}J. N. Mather, \textit{Generic projections}, 
Ann. of Math., (2), \textbf{98} (1973), 226--245.

%\bibitem{Thom}R.~Thom, 
%\textit{Les singularit\'es des applications diff\'erentiables}, 
%Ann. Inst. Fourier, Grenoble, \textbf{6} (1955--1956), 43--87.

%\bibitem{whitney umbrella}H.~Whitney,
%\textit{The singularities of a smooth $n$-manifold in $(2n-1)$-space},
%{Ann. of Math.}, \textbf{45} (1944),  247--293.

%\bibitem{spanier}E. H. Spanier, 
%\textit{Algebraic Topology }, 
%McGraw-Hill Book Company, 1966. 

%\bibitem{plane to plane}H.~Whitney, 
%\textit{On singularities of mappings of euclidean spaces. I. 
%Mappings of the plane into the plane}, 
%Ann. of Math., (2), 
%\textbf{62} (1955), 374--410.   
%  
\end{thebibliography}
\end{document}


1612.01100.extracted_bib
{abraR.~Abraham, Transversality in manifolds of mappings, Bull. Amer. Math. Soc. 69 (1963), 470--474. %
1612.01100.extracted_bib
{ArnoldV.~I.~Arnol'd, %Singularities of smooth mappings, %(Russian) %Uspehi Mat. Nauk, 23 (1968), 3--44. %
1612.01100.extracted_bib
{BoardmanJ.~M.~Boardman, %Singularities of differentiable maps, %Inst. Hautes \'Etudes Sci. Publ. Math., 33 (1967), 21--57.
1612.01100.extracted_bib
{CSJ.~W.~Bruce and P.~J.~Giblin, Curves and singularities (second edition), Cambridge University Press, Cambridge, 1992. %
1612.01100.extracted_bib
{bruceJ. W. Bruce and N. P. Kirk, %Generic projections of stable mappings, %Bull. London Math. Soc. 32, (2000), no. 6, 718-728.
1612.01100.extracted_bib
{GGM.~Golubitsky and V.~Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics 14, Springer, New York, 1973. %
1612.01100.extracted_bib
{arnoldetallV.~I.~Arnol'd, S.~M.~Gusein-Zade and A.~N.~Varchenko, %{\rm Singularities of Differentiable Maps I. {Monographs in Mathematics, %82, Birkh\"auser, Boston Basel Stuttgart, 1985. %
1612.01100.extracted_bib
{duplessisgaffneywilsonA.~du Plessis, T.~Gaffney and L.~Wilson, %Map-germs determined by their discriminant, %in \lq\lq Stratifications, singularities and differential equations, I (marseille 1990; Honolulu 1990)\rq\rq, %1--40, Travaux en Cours, 54, Hermann, Paris, 1997.
1612.01100.extracted_bib
{F1S.~Ichiki, Generic linear perturbations, preprint (available from arXiv:1607.03220 [math.MG]).
1612.01100.extracted_bib
{DS.~Ichiki and T.~Nishimura, Distance-squared mappings, Topology Appl., 160 (2013), 1005--1016.
1612.01100.extracted_bib
{LS.~Ichiki and T.~Nishimura, Recognizable classification of Lorentzian distance-squared mappings, J.~Geom.~Phys., 81 (2014), 62--71.
1612.01100.extracted_bib
{G2S.~Ichiki and T.~Nishimura, Generalized distance-squared mappings of $\mathbb{R^{n+1$ into $\mathbb{R^{2n+1$, Contemporary Mathematics, Amer. Math. Soc., Providence RI, 675 (2016), 121--132.
1612.01100.extracted_bib
{G3S.~Ichiki, T.~Nishimura, Preservation of immersed or injective properties by composing generic generalized distance-squared mappings, accepted for publication in Proceedings of BRAZIL-MEXICO-2015 Singularity meetings at Salvador, to be published in a volume of Springer Proceedings in Mathematics and Statistics.
1612.01100.extracted_bib
{G1S.~Ichiki, T.~Nishimura, R.~Oset Sinha and M.~A.~S.~Ruas, Generalized distance-squared mappings of the plane into the plane, Adv. Geom., 16 (2016), 189--198. %
1612.01100.extracted_bib
{mather2J.~N.~Mather, %Stability of $C^\infty$ mappings II. Infinitesimal stability implies stability, %{Ann. of Math., %89 (2) (1969), 254--291. %
1612.01100.extracted_bib
{mather4J.~Mather, %Stability of %$C^\infty$ %mappings, IV,{ Classification of stable %map-germs by $\mathbb{R$-algebras, %{\rm Publ. Math. Inst. Hautes \'Etudes Sci. %37, (1970), 223--248. %
1612.01100.extracted_bib
{mather5J.~N.~Mather, %Stability of $C^\infty$ mappings V. Transversality, %{Adv.~in~Math., %4 (1970), 301--336. %
1612.01100.extracted_bib
{mather6J.~N.~Mather, %Stability of $C^\infty$ mappings V\!I. The nice dimensions, %{Lecture %Notes in Math., %192 (1971), 207--253. %
1612.01100.extracted_bib
{montesinosA.~M.~Montesinos, %{\rm Maps $\mathbb{R^2$ to $\mathbb{R^2$, Computer program available by anonymous ftp at http://www.uv.es/montesin/. %
1612.01100.extracted_bib
{wallC.~T.~C.~Wall, %Finite determinacy of smooth map-germs, %{Bull. London Math. Soc. %13, (1981), 481--539. %
1612.01100.extracted_bib
{TBJ. N. Mather, %On Thom-Boardman singularities, %Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), %Academic Press, New York, 1973, 233--248.
1612.01100.extracted_bib
{GPJ. N. Mather, Generic projections, Ann. of Math., (2), 98 (1973), 226--245. %
1612.01100.extracted_bib
{ThomR.~Thom, %Les singularit\'es des applications diff\'erentiables, %Ann. Inst. Fourier, Grenoble, 6 (1955--1956), 43--87. %
1612.01100.extracted_bib
{whitney umbrellaH.~Whitney, %The singularities of a smooth $n$-manifold in $(2n-1)$-space, %{Ann. of Math., 45 (1944), 247--293. %
1612.01100.extracted_bib
{spanierE. H. Spanier, %Algebraic Topology , %McGraw-Hill Book Company, 1966. %
1612.01100.extracted_bib
{plane to planeH.~Whitney, %On singularities of mappings of euclidean spaces. I. %Mappings of the plane into the plane, %Ann. of Math., (2), %62 (1955), 374--410. %