Geometric interpretation of generalized distance-squared mappings of $\mathbb{R}^2$ into $\mathbb{R}^\ell$ $(\ell \geq 3)$
Shunsuke Ichiki









Generalized distance-squared mappings  
are quadratic mappings of ℝ^m into ℝ^ℓ of special type. 
In the case that matrices A 
constructed by coefficients of generalized distance-squared mappings 
of ℝ^2 into ℝ^ℓ  (ℓ≥3) are full rank, 
the generalized distance-squared mappings having a generic central point have the following properties. 
In the case of ℓ=3, they have only one singular point. 
On the other hand, in the case of ℓ>3, they have no singular points. 
Hence, in this paper, the reason why in the case of ℓ=3 
(resp., in the case of ℓ>3), 
they have only one singular point (resp., no singular points) is explained 
by giving a geometric interpretation to these phenomena.  

[2010]53A04, 57R45, 57R50 	
 

A unified description of 
collective magnetic excitations
    Michael Farle
    
=========================================================



§ INTRODUCTION



Throughout this paper, positive integers are expressed by i, j, ℓ and m. 
In this paper, unless otherwise stated, all mappings belong to class C^∞. 
Two mappings f : ℝ^m→ℝ^ℓ and 
g:ℝ^m→ℝ^ℓ are said  
to be  𝒜-equivalent if 
there exist two diffeomorphisms h : ℝ^m→ℝ^m and 
H : ℝ^ℓ→ℝ^ℓ satisfying 
f=H∘ g ∘ h^-1. 
Let p_i=(p_i1, p_i2, …, p_im)  (1≤ i≤ℓ) 
(resp., A=(a_ij)_1≤ i≤ℓ, 1≤ j≤ m) 
be a point of ℝ^m 
(resp., an ℓ× m matrix with non-zero entries).
Set p=(p_1,p_2,…,p_ℓ)∈ (ℝ^m)^ℓ. 
Let G_(p, A):ℝ^m →ℝ^ℓ be the mapping 
defined by 

    G_(p, A)(x)=(
    ∑_j=1^m a_1j(x_j-p_1j)^2, 
    ∑_j=1^m a_2j(x_j-p_2j)^2, 
    …, 
    ∑_j=1^m a_ℓ j(x_j-p_ℓ j)^2
    ),
 where x=(x_1, x_2, …, x_m)∈ℝ^m. 
The mapping G_(p, A) is called a generalized distance-squared mapping, 
and the ℓ-tuple of points p=(p_1,… ,p_ℓ)∈ (ℝ^m)^ℓ 
is called the central point 
of the generalized distance-squared mapping G_(p,A). 
For a given matrix A, a property of 
generalized distance-squared mappings will be said to be 
true for a generalized distance-squared mapping 
having a generic central point 
if there exists a subset Σ with Lebesgue measure zero 
of (ℝ^m)^ℓ such that for any p ∈ (ℝ^m)^ℓ-Σ, 
the mapping G_(p,A) has the property. 


A distance-squared mapping D_p 
(resp., Lorentzian distance-squared mapping
L_p) is the mapping G_(p,A) 
satisfying that each entry of A is 1 
(resp., a_i1=-1 and a_ij=1 (j 1)). 
In <cit.> (resp., <cit.>), 
a classification result on distance-squared mappings D_p 
(resp., Lorentzian distance-squared mappings
L_p) is given. Moreover, in <cit.> (resp., <cit.>), a classification result on generalized distance-squared mappings G_(p,A) of ℝ^2 into  ℝ^2 
(resp., ℝ^m+1 into  ℝ^ℓ (ℓ≥ 2m+1)) is given. 

The important motivation for these investigations is as follows. 
Height functions and distance-squared functions have been investigated 
in detail so far, 
and they are well known as a useful tool 
in the applications of singularity theory to differential geometry 
(for example, see <cit.>). 
The mapping in which each component is a height function is 
nothing but a projection. 
Projections as well as height functions or distance-squared functions 
have been investigated so far. 

On the other hand, 
the mapping in which each component 
is a distance-squared function is a distance-squared mapping. 
Besides, the notion of generalized distance-squared mappings is 
an extension of the distance-squared mappings. 
Therefore, it is natural to investigate generalized distance-squared mappings 
as well as projections. 

In <cit.>, 
a classification result on 
generalized distance-squared mappings of ℝ^2 into ℝ^2 
is given. 
If the rank of A is two,  
the generalized distance-squared mapping 
having a generic central point 
is a mapping of which any singular point is 
a fold point 
except one cusp point.     

The singular set is a rectangular hyperbola. 
Moreover, in <cit.>, a 
geometric interpretation of a singular set of 
generalized distance-squared mappings of ℝ^2 into ℝ^2 having a generic central point is also given in the case of rank A=2.  
By the geometric interpretation,  
the reason why the mappings have only one cusp point is explained.  








On the other hand, in <cit.>, 
a classification result on 
generalized distance-squared mappings 
of ℝ^m+1 into ℝ^ℓ (ℓ≥ 2m+1) 
is given. 



















As the special case of m=1, we have the following. 

Let ℓ be an integer satisfying ℓ≥ 3. 
Let A=(a_ij)_1≤ i ≤ℓ, 1≤ j ≤ 2 be an ℓ× 2 matrix with 
non-zero entries satisfying rank A=2. 
Then, the following hold: 


  * In the case of ℓ=3, 
there exists a proper algebraic subset Σ _A⊂ (ℝ^2)^3 
such that for any p∈  (ℝ^2)^3-Σ _A, 
the mapping G_(p,A) is 𝒜-equivalent to 
the normal form of Whitney umbrella
(x_1,x_2)↦ (x_1^2, x_1x_2, x_2). 

  * In the case of ℓ>3, 
there exists a proper algebraic subset Σ _A⊂ (ℝ^2)^ℓ 
such that for any p∈  (ℝ^2)^ℓ-Σ _A, 
the mapping G_(p,A) is 𝒜-equivalent to the inclusion 
(x_1,x_2)↦ (x_1, x_2, 0,… ,0). 



As described above, in <cit.>, a geometric interpretation of 
generalized distance-squared mappings of ℝ^2 into ℝ^2 
having a generic central point is given in the case that the matrix A is full rank.   
On the other hand, in this paper, 
a geometric interpretation of 
generalized distance-squared mappings of ℝ^2 into ℝ^ℓ 
(ℓ≥3)  
having a generic central point is given in the case that the matrix A is full rank 
(for the reason why we concentrate on the case that the matrix A is full rank, 
see Remark <ref>). 
Hence, by <cit.> and this paper, 
geometric interpretations of generalized distance-squared mappings of the plane 
having a generic central point in the case that the matrix A is full rank 
are completed. 

The main purpose of this paper is to give a geometric interpretation of  
Theorem <ref>. 
Namely, the main purpose of this paper is to answer the following question. 

Let ℓ be an integer satisfying ℓ≥ 3. 
Let A=(a_ij)_1≤ i ≤ℓ, 1≤ j ≤ 2 be an ℓ× 2 matrix with 
non-zero entries satisfying rank A=2. 


  * In the case of ℓ=3, 
why do generalized distance-squared mappings 
G_(p,A):ℝ^2→ℝ^3 having a generic central point have only one singular point ?

  * On the other hand, 
in the case of ℓ>3, 
why do generalized distance-squared mappings 
G_(p,A):ℝ^2→ℝ^ℓ having a generic central point have 
no singular points ?
























 §.§ Remark

In the case that the matrix A is not full rank ( rank A=1), 
for any ℓ≥ 3, 
the generalized distance-squared mapping 
of  ℝ^2 into ℝ^ℓ having a generic central point  
is 𝒜-equivalent to only the inclusion (x_1,x_2)↦ (x_1, x_2, 0, … , 0) (see Theorem 3 in <cit.>). 
On the other hand, in the case that the matrix A is full rank ( rank A=2), 
the phenomenon in the case of ℓ=3 is completely different from the phenomenon  in 
the case of ℓ>3 (see Theorem <ref>). 
Hence, we concentrate on the case that the matrix A is full rank . 

In Section <ref>, 
some assertions and definitions are prepared for 
answering Question <ref>, and 
the answer to Question <ref> is stated. 
In Section <ref>, the proof of a lemma of Section <ref> is given.    




§ THE ANSWER TO QUESTION <REF>



Firstly, in order to answer Question <ref>, 
some assertions and definitions 
 are prepared. 
By Theorem <ref>, it is clearly seen that the following assertion holds. 
The assertion is important for giving an geometric interpretation. 

Let ℓ be an integer satisfying ℓ≥ 3. 
Let A=(a_ij)_1≤ i ≤ℓ, 1≤ j ≤ 2 
(resp., B=(b_ij)_1≤ i ≤ℓ, 1≤ j ≤ 2) be an ℓ× 2 matrix with 
non-zero entries 
satisfying rank A=2 (resp., rank B=2).  
Then, there exist  proper algebraic subsets Σ_A and Σ_B of 
(ℝ^2)^ℓ such that for any p ∈ (ℝ^2)^ℓ-Σ_A 
and for any q ∈ (ℝ^2)^ℓ-Σ_B, the mapping 
G_(p,A) is 𝒜-equivalent to the mapping G_(q, B). 

Let F_p:ℝ^2→ℝ^ℓ (ℓ≥3) be the mapping 
defined by 

    F_p(x_1,x_2)
       =   (
    a(x_1-p_11)^2+b(x_2-p_12)^2, (x_1-p_21)^2+(x_2-p_22)^2, … , 
    . 
    
          . 
    (x_1-p_ℓ 1)^2+(x_2-p_ℓ 2)^2
    ),

where 0<a<b and p=(p_11,p_12, … ,p_ℓ 1,p_ℓ 2).  
Remark that the mapping F_p is the generalized distance-squared mapping G_(p,B),  
where 

    B=
    (
        [ a b; 1 1; ⋮ ⋮; 1 1;   ]).

 
Since the rank of the matrix B is two, 
by Corollary <ref>, in order to answer 
Question <ref>, it is sufficient to answer the following question. 


Let ℓ be an integer satisfying ℓ≥ 3. 


  * In the case of ℓ=3, 
why do the mappings F_p : ℝ^2→ℝ^3 having a generic central point have only one singular point ?

  * On the other hand, 
in the case of ℓ>3, 
why do the mappings 
F_p : ℝ^2→ℝ^ℓ having a generic central point have 
no singular points ?



The mapping F_p=(F_1,… ,F_ℓ) determines ℓ-foliations 
𝒞_p_1(c_1), … ,𝒞_p_ℓ(c_ℓ) 
in the plane defined by 

    𝒞_p_i(c_i)={(x_1, x_2)∈ℝ^2 | F_i(x_1,x_2)=c_i},

where c_i≥ 0 (1≤ i ≤ℓ) and 
p=(p_1,… ,p_ℓ)∈ (ℝ^2)^ℓ. 

For a given central point p=(p_1,… ,p_ℓ)∈ (ℝ^2)^ℓ, 
a point q∈ℝ^2 is a singular point of the mapping F_p 
if and only if the ℓ-foliations 𝒞_p_i(c_i) (1≤ i ≤ℓ) 
defined by the point p are tangent at the point q, 
where (c_1,… ,c_ℓ)=F_p(q). 

For a given central point p=(p_1,… ,p_ℓ)∈ (ℝ^2)^ℓ, in the case that a point q∈ℝ^2 is a singular point of 
the mapping F_p, there may exist an integer i such that the foliation 𝒞_p_i(c_i) is merely a point, where c_i=F_i(q). 
However, by the following lemma, 
we see that the trivial phenomenon seldom occurs 
(for the proof of Lemma <ref>, see Section <ref>). 






























Let ℓ be an integer satisfying ℓ≥ 3. 
Then, there exists a proper algebraic subset Σ⊂ (ℝ^2)^ℓ such that 
for any central point p=(p_1,… ,p_ℓ)∈ (ℝ^2)^ℓ-Σ, 
if a point q∈ℝ^2 is a singular point of the mapping F_p,  then 
the ℓ-foliations 𝒞_p_1(c_1) and 𝒞_p_i(c_i) (2≤ i ≤ℓ) are an ellipse and (ℓ-1)-circles, respectively, where (c_1,… ,c_ℓ)=F_p(q).   


















 §.§ Answer to Question <ref>

As described above, in order to answer Question <ref>, 
it is sufficient to answer Question <ref>. 


  * We will answer (1) of Question <ref>. 
The phenomenon that the mapping F_p:ℝ^2→ℝ^3 having 
a generic central point has only one singular point  
can be explained by the following geometric interpretation. Namely,  
constants c_i ≥0 (i=1,2,3) such that three foliations 
 𝒞_p_1(c_1), 𝒞_p_2(c_2)  and 𝒞_p_3(c_3) defined by the central point p=(p_1, p_2, p_3)∈ (ℝ^2)^3 are tangent are uniquely determined, 
 and the tangent point is also unique. 
Moreover, in the case, remark that by Lemma <ref>, 
the three foliations 𝒞_p_1(c_1), 𝒞_p_2(c_2)  and 𝒞_p_3(c_3) defined by almost all (in the sense of Lebesgue measure) (p_1,p_2,p_3)∈ (ℝ^2)^3 are an ellipse and two circles, respectively.  

Furthermore, by the geometric interpretation, 
we can also see the location of the singular point of the mapping F_p 
having a generic central point
(for example, see Figure <ref>). 

  * We will answer (2) of Question <ref>. 
The phenomenon that the mapping F_p:ℝ^2→ℝ^ℓ (ℓ >3) having 
a generic central point has no singular points 
can be explained by the following geometric interpretation. Namely, 
for any constants c_i≥0 (1≤ i ≤ℓ), 
ℓ-foliations 𝒞_p_1(c_1), … , 𝒞_p_ℓ(c_ℓ) defined by the central point p=(p_1, … , p_ℓ)∈ (ℝ^2)^ℓ are not tangent at any points 
(for example, see Figure <ref>).  



  
  




 §.§ Remark

The geometric interpretation (the answer to Question <ref>) has one more advantage. 
By the interpretation, we get the following assertion from the viewpoint of the contacts amongst one ellipse and some circles. 

Let a, b be real numbers satisfying 0<a<b.  


  * There exists a proper algebraic subset Σ of (ℝ^2)^3 
such that for any (p_1,p_2,p_3)∈ (ℝ^2)^3-Σ, 
constants c_i>0 (i=1,2,3) such that one ellipse a(x_1-p_11)^2+b(x_2-p_12)^2=c_1 and two circles (x_1-p_i1)^2+(x_2-p_i2)^2=c_i (i=2, 3) are 
tangent are uniquely determined, where p_i=(p_i1, p_i2). Moreover, the tangent point 
is also unique. 

  * On the other hand, in the case of ℓ>3, 
there exists a proper algebraic subset Σ of (ℝ^2)^ℓ 
such that for any (p_1,…, p_ℓ)∈ (ℝ^2)^ℓ-Σ, 
for any c_i>0 (i=1,… ,ℓ), 
 the one ellipse a(x_1-p_11)^2+b(x_2-p_12)^2=c_1 and the (ℓ-1)-circles   
 (x_1-p_i1)^2+(x_2-p_i2)^2=c_i (i=2,… ,ℓ) are not tangent 
 at any points, where p_i=(p_i1, p_i2).  




























































































































































































































































































































































§ PROOF OF LEMMA <REF>
 


The Jacobian matrix of the mapping F_p at (x_1,x_2) is the following.  

    JF_p(x_1,x_2) =2(
        [ a(x_1-p_11) b(x_2-p_12);           ⋮           ⋮;   x_1-p_ℓ 1   x_2-p_ℓ 2 ]).

Let Σ_i be a subset of (ℝ^2)^ℓ consisting of 
p=(p_1,… ,p_ℓ)∈ (ℝ^2)^ℓ satisfying    
p_i∈ℝ^2 is a singular point of F_p (1≤ i ≤ℓ). 
Namely, 
for example, Σ_1 is the subset of (ℝ^2)^ℓ consisting of 
p=(p_1,… ,p_ℓ)∈ (ℝ^2)^ℓ satisfying 

    rank (
        [          0          0;  p_11-p_21  p_12-p_22;          ⋮          ⋮; p_11-p_ℓ 1 p_12-p_ℓ 2 ])<2.

By ℓ≥ 3, it is clearly seen that Σ_1 is a proper algebraic subset of 
(ℝ^2)^ℓ. 
Similarly, for any i (2≤ i ≤ℓ), we see that  Σ_i is also a proper algebraic subset 
of 
(ℝ^2)^ℓ. 
Set Σ=∪_i=1^ℓΣ_i. Then, Σ is also a proper algebraic subset 
of (ℝ^2)^ℓ. 

Let p=(p_1,… ,p_ℓ)∈ (ℝ^2)^ℓ-Σ be a central point, and let q be 
a singular point of the mapping F_p defined by the central point. 
Then, suppose that there exists an integer i such that 
the foliation 𝒞_p_i(c_i) is not an ellipse or a circle, 
where c_i=F_i(q) (F_p=(F_1, … ,F_ℓ)). 
Then, we get c_i=0. Hence, we have q=p_i. 
This contradicts the assumption p∈ (ℝ^2)^ℓ-Σ. 

























































































































































§ ACKNOWLEDGEMENTS

The author is grateful to Takashi Nishimura for his kind advices. 
The author is supported by JSPS KAKENHI Grant Number 16J06911.


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