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: kabata@math.sci.hokudai.ac.jp

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)