Recognition of plane-to-plane map-germs and …fj_singularities/FJV2014/poster/... · and...
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Recognition of plane-to-plane map-germs and applications to generic differential geometry Yutaro KABATA, D1, Department of Mathematics, Hokkaido University , e-mail address: [email protected] Introduction The aims of this poster are 1. to introduce useful criteria for map-germs from plane to plane of corank one with A-codimension≤ 6, 2. to show applications of our criteria to the study of singularities arising in the projection of surfaces. Classification of map germs • Objects: smooth map germs R 2 , 0 → R 2 , 0 of corank 1. • Classification: f,g : R n , 0 → R p , 0 are A-equivalent, if there exist diffeomorphism germs ϕ and ψ of R n , 0 and R m , 0 so that f = ψ ◦ g ◦ ϕ -1 . f and g are called Topologi- cally A-equivalent if ϕ and ψ are homeomorphisms. J. H. Rieger (1986, 1988) gave Topologically A- and A-classification of corank 1 germs (A-cod≤ 6) [4, 5] as in the following: C 0 -A-specified jet A-type Criteria [7, 3] regular :(x, y ) 1 λ(0) ̸=0 （we may assume λ(0) = 0 in the following） fold :(x, y 2 ) 2 ηλ(0) ̸=0 cusp :(x, xy + y 3 ) 3 dλ(0) ̸=0,ηλ(0) = 0,η 2 λ(0) ̸=0 I 2 :(x, y 3 ± x 2 y ) 4 ± 2 dλ(0) = 0, det H λ (0) ̸=0,η 2 λ(0) ̸=0 I * :(x, y 3 ) 4 ± k (k ≥ 3) dλ(0) = 0, rankH λ (0) = 1,η 2 λ(0) ̸=0 II 4 :(x, xy + y 4 ) 5 dλ(0) ̸=0,ηλ(0) = η 2 λ(0) = 0,η 3 λ(0) ̸=0 II 5 :(x, xy + y 5 ) 6, 7 dλ(0) ̸=0,ηλ(0) = η 2 λ(0) = η 3 λ(0) = 0,η 4 λ(0) ̸=0 II 6 :(x, xy + y 6 ) 8, 9 dλ(0) ̸=0,ηλ(0) = ··· = η 4 λ(0) = 0,η 5 λ(0) ̸=0 II 7 :(x, xy + y 7 ) 10 dλ(0) ̸=0,ηλ(0) = ··· = η 4 λ(0) = η 5 λ(0) = 0,η 6 λ(0) ̸=0 III * :(x, xy 2 + y 4 ) 11 2k +1 dλ(0) = 0, det H λ (0) < 0,η 2 λ(0) = 0,η 3 λ(0) ̸=0 IV 5 :(x, xy 2 + y 5 ) 12, 13, (14) dλ(0) = 0, det H λ (0) < 0,η 2 λ(0) = η 3 λ(0) = 0,η 4 λ(0) ̸=0 IV 6 :(x, xy 2 + y 6 ) 15 dλ(0) = 0, det H λ (0) < 0,η 2 λ(0) = ··· = η 4 λ(0) = 0,η 5 λ(0) ̸=0 V 1 :(x, x 2 y + y 4 ) 16, 17 dλ(0) = 0, rankH λ (0) = 1,η 2 λ(0) = 0,η 3 λ(0) ̸=0 V 2 :(x, x 2 y + Axy 3 ) 18 dλ(0) = 0, rankH λ (0) = 1,η 2 λ(0) = η 3 λ(0) = 0 VI :(x, y 4 + αx 2 y 2 + Ax 3 y ) 19 dλ(0) = 0, rankH λ (0) = 0,η 3 λ(0) ̸=0 Table 1: Classification and criteria Remark that numbers in the second column mean normal forms of each A-types; for example 6 ··· (x, xy + y 5 ± y 7 ), 7 ··· (x, xy + y 5 ), 8 ··· (x, xy + y 6 ± y 8 + αy 9 ). ✓ ✏ Recognition problem ··· How should we detect the type of given map-germs? ✒ ✑ Recognition A geometric characterization is given to each specified jets of top. A-type as in the third column of the Table1 [7, 3] in terms of • the Jacobian λ(x, y ) := ∂ (f 1 ,f 2 ) ∂ (x,y ) • a vector field η so that η | λ=0 spans ker df •H λ : Hess matrix of the function λ. A-type is determined by additional condition to the above geometric condition [3]. For example, ✓ ✏ Proposition 1 (K [3]). If we write f =(x, xy + y 5 + ∑ i+j ≥6 c ij x i y j ), then c 07 - 5 8 c 2 06 ̸=0 ⇐⇒ f ∼ A (x, xy + y 5 ± y 7 ) Butterfly 6 c 07 - 5 8 c 2 06 =0 ⇐⇒ f ∼ A (x, xy + y 5 ) Elder butterfly 7 ✒ ✑ Application I (Central projection of smooth surface) Let ι : M, → R 3 be an embedding of smooth surface and p =(a, b, c) ∈ R 3 not lying on M . We define π p : R 3 -{p}→ RP 2 , x 7→ the line generated by x - p. The central projection of M from p is defined as φ p := π p ◦ ι : M → RP 2 . ✓ ✏ Theorem 2 (Arnold (1979), Platonova (1986) [1]). For a generic surface M , and for ar- bitrary p ∈ R 3 - M , the germ of the central projection φ p : M → RP 2 at any point is A-equivalent to one of A-types in Rieger’s list up to codim. 5 except for types 8, 12, 16. ✒ ✑ Using our criteria and based on the method of J. W. Bruce [2], we got a transparent re-proof of Theorem 2 and the following extension: ✓ ✏ Theorem 3 (K [3]). For a generic one-parameter family of embeddings M × I → R 3 , (x, t) 7→ ι t (x), the central projection π p ◦ ι t : M → RP 2 for some t and some viewpoint p may admit the singularity types 12, 16, 8, 4 5 , 9, 11 9 , 13, 17, 19. That is, types 10, 15, 18 (codim. 6) are excluded. ✒ ✑ Application II (Projection of crosscap) • Central projection of crosscaps admits corank 2 singularities. •A-simple corank 2 germs are of type in the following [6]: I ℓ,m 2,2 :(x 2 + y 2ℓ+1 ,y 2 + x 2m+1 ), II n 2,2 :(x 2 - y 2 + y 2n+1 , xy ) ✓ ✏ Theorem 4 (Yoshida, Ohmoto, K [9]). For a generic crosscap, the corank 2 singularities arising in central projection are of type I 1,1 2,2 (Sharksfin) and II 1 2,2 (Deltoid), i.e., I 2,1 2,2 (Odd sharksfin) and I 2,3 are not generic. ✒ ✑ We look at generic elliptic crosscap (i.e., the projection is of type Sharksfin) especially. There are two characteristic curves as invariants in affine differential geometry: 1.The parabolic curve (the loci of beaks), 2.The frecnodal curve (the loci of swallowtail). ✓ ✏ Theorem 5 (Yoshida, Ohmoto, K). The two curves have 3-point contact at the crosscap point in the source plane if and only if the elliptic crosscap is generic. ✒ ✑ We also studied the bifurcation of degenerate crosscaps [9, 10]. c.f. Bifurcation diagram of Odd-sharksfin in abc-space of the A-versal unfolding F (x, y, a, b, c)=(x 2 + y 5 + cy 3 + ay, y 2 + x 3 + bx) [9]: beaks gulls tacnode sharksfin swallowtail b c a Generic section of the bifurcation diagram References [1] V. I. Arnold, Singularities of caustics and wavefronts, Kluwer Acad. Publ. (1991). [2] J. W. Bruce, Projections and reflections of generic surfaces in R 3 . Math. Scand. 54 no.2 (1984), 262-278. [3] Y. Kabata, Recognition of plane-to-plane map-germs, preprint. [4] J. H. Rieger, Families of maps from the plane to the plane. J. London Math. Soc. (2) 36 (1987), no. 2, 351-369. [5] J. H. Rieger, Versal topological stratification and the bifurcation geometry of map-germs of the plane. Math. Proc. Cambridge Philos. Soc. 107, no. 1, (1990), 127-147. [6] J. H. Rieger and M. A. S. Ruas, Classification of A-simple germs from k n to k 2 , Compositio Math. 79 no. 1, (1991), 99-108. [7] K. Saji, Criteria for singularities of smooth maps from the plane into the plane and their applications. Hiroshima Math. J. 40, (2010), 229-239. [8] J. West, The Differential Geometry of the Cross-Cap, Dissertation, University of Liverpool (1995). [9] T. Yoshida, Y. Kabata and T. Ohmoto, Bifurcation of plane-to-plane map-germs of corank 2, to appear in Quarterly Jour. Math. (2014) [10] T. Yoshida, Y. Kabata and T. Ohmoto, Bifurcation of plane-to-plane map-germs with corank 2 of parabolic type, preprint (2014)
Transcript of Recognition of plane-to-plane map-germs and …fj_singularities/FJV2014/poster/... · and...
Recognition of plane-to-plane map-germsand applications to generic differential geometry
Yutaro KABATA, D1, Department of Mathematics, Hokkaido University , e-mail address: [email protected]
IntroductionThe aims of this poster are1. to introduce useful criteria for map-germs from plane to plane of corank one with
A-codimension≤ 6,2. to show applications of our criteria to the study of singularities arising in the
projection of surfaces.
Classification of map germs• Objects: smooth map germs R2, 0 → R2, 0 of corank 1.• Classification: f, g : Rn, 0 → Rp, 0 are A-equivalent, if there exist diffeomorphism
germs ϕ and ψ of Rn, 0 and Rm, 0 so that f = ψ ◦ g ◦ ϕ−1. f and g are called Topologi-cally A-equivalent if ϕ and ψ are homeomorphisms.
J. H. Rieger (1986, 1988) gave Topologically A- and A-classification of corank 1 germs(A-cod≤ 6) [4, 5] as in the following:
C0-A-specified jet A-type Criteria [7, 3]regular : (x, y) 1 λ(0) ̸= 0（we may assume λ(0) = 0 in the following）fold : (x, y2) 2 ηλ(0) ̸= 0cusp : (x, xy + y3) 3 dλ(0) ̸= 0, ηλ(0) = 0, η2λ(0) ̸= 0I2 : (x, y
3 ± x2y) 4±2 dλ(0) = 0, detHλ(0) ̸= 0, η2λ(0) ̸= 0
I∗ : (x, y3) 4±k (k ≥ 3) dλ(0) = 0, rankHλ(0) = 1, η2λ(0) ̸= 0
II4 : (x, xy + y4) 5 dλ(0) ̸= 0, ηλ(0) = η2λ(0) = 0, η3λ(0) ̸= 0II5 : (x, xy + y5) 6, 7 dλ(0) ̸= 0, ηλ(0) = η2λ(0) = η3λ(0) = 0, η4λ(0) ̸= 0II6 : (x, xy + y6) 8, 9 dλ(0) ̸= 0, ηλ(0) = · · · = η4λ(0) = 0, η5λ(0) ̸= 0II7 : (x, xy + y7) 10 dλ(0) ̸= 0, ηλ(0) = · · · = η4λ(0) = η5λ(0) = 0, η6λ(0) ̸= 0III∗ : (x, xy2 + y4) 112k+1 dλ(0) = 0, detHλ(0) < 0, η2λ(0) = 0, η3λ(0) ̸= 0IV5 : (x, xy
2 + y5) 12, 13, (14) dλ(0) = 0, detHλ(0) < 0, η2λ(0) = η3λ(0) = 0, η4λ(0) ̸= 0IV6 : (x, xy
2 + y6) 15 dλ(0) = 0, detHλ(0) < 0, η2λ(0) = · · · = η4λ(0) = 0, η5λ(0) ̸= 0V1 : (x, x
2y + y4) 16, 17 dλ(0) = 0, rankHλ(0) = 1, η2λ(0) = 0, η3λ(0) ̸= 0V2 : (x, x
2y + Axy3) 18 dλ(0) = 0, rankHλ(0) = 1, η2λ(0) = η3λ(0) = 0V I : (x, y4 + αx2y2 + Ax3y) 19 dλ(0) = 0, rankHλ(0) = 0, η3λ(0) ̸= 0
Table 1: Classification and criteria
Remark that numbers in the second column mean normal forms of each A-types;for example 6 · · · (x, xy + y5 ± y7), 7 · · · (x, xy + y5), 8 · · · (x, xy + y6 ± y8 + αy9).
� �Recognition problem · · · How should we detect the type of given map-germs?� �
RecognitionA geometric characterization is given to each specified jets of top. A-type as in thethird column of the Table1 [7, 3] in terms of• the Jacobian λ(x, y) := ∂(f1,f2)
∂(x,y)• a vector field η so that η|λ=0 spans ker df
•Hλ: Hess matrix of the function λ.
A-type is determined by additional condition to the above geometric condition [3]. Forexample,� �Proposition 1 (K [3]). If we write f = (x, xy + y5 +
∑i+j≥6 cijx
iyj), then
c07 − 58c
206 ̸= 0 ⇐⇒ f ∼A (x, xy + y5 ± y7) Butterfly 6
c07 − 58c
206 = 0 ⇐⇒ f ∼A (x, xy + y5) Elder butterfly 7� �
Application I (Central projection of smooth surface)
Let ι : M ↪→ R3 be an embedding of smooth surface and p = (a, b, c) ∈ R3 not lying on M .We define
πp : R3 − {p} → RP 2, x 7→ the line generated by x− p.
The central projection of M from p is defined as
φp := πp ◦ ι :M → RP 2.� �Theorem 2 (Arnold (1979), Platonova (1986) [1]). For a generic surface M , and for ar-bitrary p ∈ R3 −M , the germ of the central projection φp : M → RP 2 at any point isA-equivalent to one of A-types in Rieger’s list up to codim. 5 except for types 8, 12, 16.� �
Using our criteria and based on the method of J. W. Bruce [2], we got a transparentre-proof of Theorem 2 and the following extension:� �Theorem 3 (K [3]). For a generic one-parameter family of embeddings
M × I → R3, (x, t) 7→ ιt(x),
the central projection πp ◦ ιt : M → RP 2 for some t and some viewpoint p may admitthe singularity types 12, 16, 8, 45, 9, 119, 13, 17, 19. That is, types 10, 15, 18 (codim. 6) areexcluded.� �
Application II (Projection of crosscap)• Central projection of crosscaps admits corank 2 singularities.• A-simple corank 2 germs are of type in the following [6]:
Iℓ,m2,2 : (x2 + y2ℓ+1, y2 + x2m+1), IIn2,2 : (x
2 − y2 + y2n+1, xy)
� �Theorem 4 (Yoshida, Ohmoto, K [9]). For a generic crosscap, the corank 2 singularitiesarising in central projection are of type I1,12,2 (Sharksfin) and II12,2 (Deltoid), i.e., I2,12,2 (Oddsharksfin) and I2,3 are not generic.� �
We look at generic elliptic crosscap (i.e., the projection is of type Sharksfin) especially.There are two characteristic curves as invariants in affine differential geometry:1.The parabolic curve (the loci of beaks), 2.The frecnodal curve (the loci of swallowtail).
� �Theorem 5 (Yoshida, Ohmoto, K). The two curves have 3-point contact at the crosscappoint in the source plane if and only if the elliptic crosscap is generic.� �
We also studied the bifurcation of degenerate crosscaps [9, 10].
c.f. Bifurcation diagram of Odd-sharksfin in abc-space of the A-versal unfoldingF (x, y, a, b, c) = (x2 + y5 + cy3 + ay, y2 + x3 + bx) [9]:
beaks gulls tacnode
sharksfin swallowtail
b
ca
Generic section of the bifurcation diagram
References
[1] V. I. Arnold, Singularities of caustics and wavefronts, Kluwer Acad. Publ. (1991).
[2] J. W. Bruce, Projections and reflections of generic surfaces in R3. Math. Scand. 54 no.2 (1984), 262-278.
[3] Y. Kabata, Recognition of plane-to-plane map-germs, preprint.
[4] J. H. Rieger, Families of maps from the plane to the plane. J. London Math. Soc. (2) 36 (1987), no. 2, 351-369.
[5] J. H. Rieger, Versal topological stratification and the bifurcation geometry of map-germs of the plane. Math. Proc. Cambridge Philos. Soc. 107, no. 1,(1990), 127-147.
[6] J. H. Rieger and M. A. S. Ruas, Classification of A-simple germs from kn to k2, Compositio Math. 79 no. 1, (1991), 99-108.
[7] K. Saji, Criteria for singularities of smooth maps from the plane into the plane and their applications. Hiroshima Math. J. 40, (2010), 229-239.
[8] J. West, The Differential Geometry of the Cross-Cap, Dissertation, University of Liverpool (1995).
[9] T. Yoshida, Y. Kabata and T. Ohmoto, Bifurcation of plane-to-plane map-germs of corank 2, to appear in Quarterly Jour. Math. (2014)
[10] T. Yoshida, Y. Kabata and T. Ohmoto, Bifurcation of plane-to-plane map-germs with corank 2 of parabolic type, preprint (2014)
