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

0.18 MB·

by  Shunsuke Ichiki

GEOMETRIC INTERPRETATION OF GENERALIZED
DISTANCE-SQUARED MAPPINGS OF R2 INTO Rℓ (ℓ≥3)
7
1
SHUNSUKEICHIKI
0
2

n Abstract. Generalized distance-squared mappings are quadratic mappings
a of Rm into Rℓ of special type. In the case that matrices A constructed by
J coefficientsofgeneralizeddistance-squaredmappingsofR2intoRℓ(ℓ≥3)are
7 fullrank, thegeneralized distance-squared mappings having ageneric central
2 point have the followingproperties. In the case of ℓ=3, they have only one
singularpoint. Ontheotherhand,inthecaseofℓ>3,theyhavenosingular
] points. Hence,inthispaper,thereasonwhyinthecaseofℓ=3(resp.,inthe
G caseofℓ>3), theyhaveonlyonesingularpoint(resp.,nosingularpoints)is
explainedbygivingageometricinterpretationtothesephenomena.
D
.
h
t
a
m 1. 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
1
v f : Rm → Rℓ and g : Rm → Rℓ are said to be A-equivalent if there exist two
8 diffeomorphisms h : Rm → Rm and H : Rℓ → Rℓ satisfying f = H ◦g◦h−1. Let
3 p = (p ,p ,...,p ) (1 ≤ i ≤ ℓ) (resp., A = (a ) ) be a point of Rm
i i1 i2 im ij 1≤i≤ℓ,1≤j≤m
9 (resp., an ℓ×m matrix with non-zero entries). Set p = (p ,p ,...,p ) ∈ (Rm)ℓ.
1 2 ℓ
7 Let G :Rm →Rℓ be the mapping defined by
0 (p,A)
. m m m
1
0 G(p,A)(x)= a1j(xj−p1j)2, a2j(xj−p2j)2,..., aℓj(xj−pℓj)2 ,
!
7 Xj=1 Xj=1 Xj=1
1 where x = (x ,x ,...,x ) ∈ Rm. The mapping G is called a generalized
1 2 m (p,A)
v: distance-squared mapping, and the ℓ-tuple of points p = (p1,...,pℓ) ∈ (Rm)ℓ is
i called the central point of the generalized distance-squared mapping G . For a
X (p,A)
givenmatrixA,apropertyofgeneralizeddistance-squaredmappingswillbesaidto
r
a be true for a generalized distance-squared mapping having a generic central point
if there exists a subset Σ with Lebesgue measure zero of (Rm)ℓ such that for any
p∈(Rm)ℓ−Σ, the mapping G has the property. A distance-squared mapping
(p,A)
D (resp.,Lorentziandistance-squaredmappingL )isthemappingG satisfying
p p (p,A)
that each entry of A is 1 (resp., a =−1 and a =1 (j 6=1)). In [3] (resp., [4]), a
i1 ij
classification result on distance-squared mappings D (resp., Lorentzian distance-
p
squared mappings L ) is given. Moreover, in [6] (resp., [5]), a classification result
p
on generalized distance-squared mappings G of R2 into R2 (resp., Rm+1 into
(p,A)
Rℓ (ℓ≥2m+1)) is given.
2010 Mathematics Subject Classification. 53A04,57R45,57R50.
Key words and phrases. generalized distance-squared mapping, geometric interpretation,
singularity,A-equivalence.
Theauthor isResearchFellowDC1ofJapanSocietyforthePromotionofScience.
1
2 SHUNSUKEICHIKI
Theimportantmotivationfortheseinvestigationsisasfollows. Heightfunctions
anddistance-squaredfunctionshavebeeninvestigatedindetailsofar,andtheyare
well known as a useful tool in the applications of singularity theory to differential
geometry(forexample, see [1]). The mapping inwhicheachcomponentis a height
function is nothing but a projection. Projections as well as height functions or
distance-squaredfunctions have been investigated so far.
Onthe other hand, the mapping in whicheachcomponentis a distance-squared
functionisadistance-squaredmapping. Besides,thenotionofgeneralizeddistance-
squaredmappingsisanextensionofthedistance-squaredmappings. Therefore,itis
naturaltoinvestigategeneralizeddistance-squaredmappingsaswellasprojections.
In[6],aclassificationresultongeneralizeddistance-squaredmappingsofR2 into
R2 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
pointexceptonecusppoint. Thesingularsetisarectangularhyperbola. Moreover,
in [6], a geometric interpretation of a singular set of generalized distance-squared
mappings of R2 into R2 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 [5], a classification result on generalizeddistance-squared
mappings of Rm+1 into Rℓ (ℓ≥2m+1) is given. As the special case of m=1, we
have the following.
Theorem 1 ([5]). Let ℓ be an integer satisfying ℓ≥3. Let A=(a ) be
ij 1≤i≤ℓ,1≤j≤2
an ℓ×2 matrix with non-zero entries satisfying rank A = 2. Then, the following
hold:
(1) In the case of ℓ=3, there exists a proper algebraic subset Σ ⊂(R2)3 such
A
that for any p ∈ (R2)3 −Σ , the mapping G is A-equivalent to the
A (p,A)
normal form of Whitney umbrella (x ,x )7→(x2,x x ,x ).
1 2 1 1 2 2
(2) In the case of ℓ>3, there exists a proper algebraic subset Σ ⊂(R2)ℓ such
A
that for any p ∈ (R2)ℓ −Σ , the mapping G is A-equivalent to the
A (p,A)
inclusion (x ,x )7→(x ,x ,0,...,0).
1 2 1 2
As described above, in [6], a geometric interpretation of generalized distance-
squared mappings of R2 into R2 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
interpretationof generalizeddistance-squaredmappings of R2 into Rℓ (ℓ≥3) hav-
ing 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 Re-
mark 1.1). Hence, by [6] 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 mainpurposeofthispaper istogiveageometricinterpretationofTheorem
1. Namely, the main purpose of this paper is to answer the following question.
Question 1.1. Let ℓ be an integer satisfying ℓ ≥ 3. Let A = (a ) be
ij 1≤i≤ℓ,1≤j≤2
an ℓ×2 matrix with non-zero entries satisfying rank A=2.
(1) In the case of ℓ=3, why do generalized distance-squared mappings G :
(p,A)
R2 →R3 having a generic central point have only one singular point ?
GEOMETRIC INTERPRETATION OF GENERALIZED DISTANCE-SQUARED MAPPINGS 3
(2) Ontheother hand, inthecaseofℓ>3,whydogeneralizeddistance-squared
mappings G :R2 →Rℓ having a generic central point have no singular
(p,A)
points ?
1.1. Remark. In the casethat the matrix Ais notfull rank(rank A=1), forany
ℓ ≥ 3, the generalized distance-squared mapping of R2 into Rℓ having a generic
central point is A-equivalent to only the inclusion (x ,x )7→(x ,x ,0,...,0) (see
1 2 1 2
Theorem 3 in [5]). On the other hand, in the case that the matrix A is full rank
(rank A=2), the phenomenonin the case ofℓ=3 is completely different fromthe
phenomenon in the case of ℓ > 3 (see Theorem 1). Hence, we concentrate on the
case that the matrix A is full rank .
InSection2,someassertionsanddefinitionsarepreparedforansweringQuestion
1.1, and the answer to Question 1.1 is stated. In Section 3, the proof of a lemma
of Section 2 is given.
2. The answer to Question 1.1
Firstly, in order to answer Question 1.1, some assertions and definitions are
prepared. By Theorem 1, it is clearly seen that the following assertion holds. The
assertion is important for giving an geometric interpretation.
Corollary 1. Let ℓ be an integer satisfying ℓ ≥ 3. Let A = (a )
ij 1≤i≤ℓ,1≤j≤2
(resp.,B = (b ) ) be an ℓ×2 matrix with non-zero entries satisfying
ij 1≤i≤ℓ,1≤j≤2
rank A=2 (resp., rank B =2). Then, there exist proper algebraic subsets Σ and
A
Σ of (R2)ℓ such that for any p ∈ (R2)ℓ −Σ and for any q ∈ (R2)ℓ −Σ , the
B A B
mapping G is A-equivalent to the mapping G .
(p,A) (q,B)
Let F :R2 →Rℓ (ℓ≥3) be the mapping defined by
p
F (x ,x )
p 1 2
= a(x −p )2+b(x −p )2,(x −p )2+(x −p )2,...,
1 11 2 12 1 21 2 22
(cid:0)
(x −p )2+(x −p )2 ,
1 ℓ1 2 ℓ2
(cid:1)
where 0 < a < b and p = (p ,p ,...,p ,p ). Remark that the mapping F is
11 12 ℓ1 ℓ2 p
the generalized distance-squared mapping G , where
(p,B)
a b
 
1 1
B = .. .. .
 . . 
 
 1 1 
SincetherankofthematrixB istwo,byCorollary1,inordertoanswerQuestion
1.1, it is sufficient to answer the following question.
Question 2.1. Let ℓ be an integer satisfying ℓ≥3.
(1) In the case of ℓ = 3, why do the mappings F : R2 → R3 having a generic
p
central point have only one singular point ?
(2) On the other hand, in the case of ℓ>3, why do the mappings F :R2 →Rℓ
p
having a generic central point have no singular points ?
4 SHUNSUKEICHIKI
ThemappingF =(F ,...,F )determinesℓ-foliationsC (c ),...,C (c )inthe
p 1 ℓ p1 1 pℓ ℓ
plane defined by
C (c )={(x ,x )∈R2 |F (x ,x )=c },
pi i 1 2 i 1 2 i
where c ≥ 0 (1 ≤ i ≤ ℓ) and p = (p ,...,p ) ∈ (R2)ℓ. For a given central point
i 1 ℓ
p = (p ,...,p ) ∈ (R2)ℓ, a point q ∈ R2 is a singular point of the mapping F if
1 ℓ p
andonly if the ℓ-foliations C (c ) (1≤i≤ℓ) defined by the point p are tangent at
pi i
the point q, where (c ,...,c )=F (q).
1 ℓ p
Foragivencentralpointp=(p ,...,p )∈(R2)ℓ,inthecasethatapointq ∈R2
1 ℓ
is a singular point of the mapping F , there may exist an integer i such that the
p
foliation C (c ) is merely a point, where c = F (q). However, by the following
pi i i i
lemma,we see that the trivialphenomenonseldomoccurs (for the proofof Lemma
2.1, see Section 3).
Lemma 2.1. Let ℓ be an integer satisfying ℓ ≥ 3. Then, there exists a proper
algebraic subsetΣ⊂(R2)ℓ suchthat for anycentralpointp=(p ,...,p )∈(R2)ℓ−
1 ℓ
Σ, if a point q ∈ R2 is a singular point of the mapping F , then the ℓ-foliations
p
C (c ) and C (c ) (2≤i≤ℓ) are an ellipse and (ℓ−1)-circles, respectively, where
p1 1 pi i
(c ,...,c )=F (q).
1 ℓ p
2.1. Answer to Question 1.1. As described above, in order to answer Question
1.1, it is sufficient to answer Question 2.1.
(1) We will answer (1) of Question 2.1. The phenomenon that the mapping
F :R2 →R3 havingagenericcentralpointhasonlyonesingularpointcan
p
be explained by the following geometric interpretation. Namely, constants
c ≥ 0 (i = 1,2,3) such that three foliations C (c ),C (c ) and C (c )
i p1 1 p2 2 p3 3
definedbythecentralpointp=(p ,p ,p )∈(R2)3aretangentareuniquely
1 2 3
determined, and the tangent point is also unique. Moreover, in the case,
remarkthatbyLemma2.1, the threefoliationsC (c ),C (c )andC (c )
p1 1 p2 2 p3 3
definedbyalmostall(inthesenseofLebesguemeasure)(p ,p ,p )∈(R2)3
1 2 3
are an ellipse and two circles, respectively.
Furthermore, by the geometric interpretation, we can also see the loca-
tion of the singular point of the mapping F having a generic centralpoint
p
(for example, see Figure 1).
(2) We will answer (2) of Question 2.1. The phenomenon that the mapping
F :R2 →Rℓ (ℓ>3) having a generic central point has no singular points
p
can be explained by the following geometric interpretation. Namely, for
any constants c ≥0 (1≤i≤ℓ), ℓ-foliationsC (c ),...,C (c ) defined by
i p1 1 pℓ ℓ
thecentralpointp=(p ,...,p )∈(R2)ℓ arenottangentatanypoints(for
1 ℓ
example, see Figure 2).
GEOMETRIC INTERPRETATION OF GENERALIZED DISTANCE-SQUARED MAPPINGS 5
Figure 1. A figure for (1) of Subsection 2.1
Figure 2. A figure for (2) of Subsection 2.1
2.2. Remark. The geometric interpretation (the answer to Question 1.1) has one
more advantage. By the interpretation, we get the following assertion from the
viewpoint of the contacts amongst one ellipse and some circles.
Corollary 2. Let a,b be real numbers satisfying 0<a<b.
(1) ThereexistsaproperalgebraicsubsetΣof(R2)3 suchthatforany(p ,p ,p )∈
1 2 3
(R2)3−Σ, constants c >0 (i=1,2,3)such that one ellipse a(x −p )2+
i 1 11
b(x −p )2 = c and two circles (x −p )2+(x −p )2 = c (i = 2,3)
2 12 1 1 i1 2 i2 i
are tangent are uniquely determined, where p = (p ,p ). Moreover, the
i i1 i2
tangent point is also unique.
(2) On the other hand, in the case of ℓ > 3, there exists a proper algebraic
subset Σ of (R2)ℓ such that for any (p ,...,p )∈(R2)ℓ−Σ, for any c >0
1 ℓ i
6 SHUNSUKEICHIKI
(i = 1,...,ℓ), the one ellipse a(x − p )2 + b(x − p )2 = c and the
1 11 2 12 1
(ℓ−1)-circles (x −p )2+(x −p )2 = c (i = 2,...,ℓ) are not tangent
1 i1 2 i2 i
at any points, where p =(p ,p ).
i i1 i2
3. Proof of Lemma 2.1
The Jacobian matrix of the mapping F at (x ,x ) is the following.
p 1 2
a(x −p ) b(x −p )
1 11 2 12
 
. .
JF =2 . . .
p(x1,x2) . .
 
 x1−pℓ1 x2−pℓ2 
LetΣ be a subset of(R2)ℓ consisting ofp=(p ,...,p )∈(R2)ℓ satisfying p ∈R2
i 1 ℓ i
isasingularpointofF (1≤i≤ℓ). Namely,forexample,Σ isthe subsetof(R2)ℓ
p 1
consisting of p=(p ,...,p )∈(R2)ℓ satisfying
1 ℓ
0 0
 
p −p p −p
11 21 12 22
rank  .. .. <2.
 . . 
 
 p11−pℓ1 p12−pℓ2 
By ℓ≥3, it is clearly seen that Σ is a proper algebraic subset of (R2)ℓ. Similarly,
1
for any i (2≤i≤ℓ), we see that Σ is also a proper algebraic subset of (R2)ℓ. Set
i
Σ=∪ℓ Σ . Then, Σ is also a proper algebraic subset of (R2)ℓ.
i=1 i
Let p=(p ,...,p )∈(R2)ℓ−Σ be a centralpoint, andlet q be a singularpoint
1 ℓ
ofthe mapping F definedbythe centralpoint. Then, supposethatthereexists an
p
integerisuchthatthefoliationC (c )isnotanellipseoracircle,wherec =F (q)
pi i i i
(F =(F ,...,F )). Then, we get c =0. Hence, we have q =p . This contradicts
p 1 ℓ i i
the assumption p∈(R2)ℓ−Σ.
✷
Acknowledgements
The author is grateful to Takashi Nishimura for his kind advices. The author is
supported by JSPS KAKENHI Grant Number 16J06911.
References
[1] J.W.BruceandP.J.Giblin,CurvesandSingularities(secondedition),CambridgeUniversity
Press,Cambridge,1992.
[2] M.GolubitskyandV.Guillemin,Stablemappingsandtheirsingularities,GraduateTextsin
Mathematics14,Springer,NewYork,1973.
[3] S.Ichiki andT.Nishimura,Distance-squared mappings, TopologyAppl.,160(2013), 1005–
1016.
[4] S. Ichiki and T. Nishimura, Recognizable classification of Lorentzian distance-squared map-
pings,J.Geom.Phys.,81(2014), 62–71.
[5] S. Ichiki and T. Nishimura, Generalized distance-squared mappings of Rn+1 into R2n+1,
ContemporaryMathematics, Amer.Math.Soc.,ProvidenceRI,675(2016), 121-132.
[6] S. 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.
GraduateSchoolofEnvironmentandInformationSciences,YokohamaNationalUni-
versity,Yokohama240-8501,Japan
E-mail address: [email protected]