Projections of hypersurfaces with boundary in R4 to planespjgiblin/B-W2012/talks/...Projections of...

Projections of hypersurfaces with boundary in R4

to planes

Ana Claudia Nabarro

ICMC - USP - Sao Carlos - Brazil

Liverpool- June 2012

Dedicated to Bruce and Wall.

Joint work with Luciana F. Martins - UNESP - Brazil

Ana Claudia Nabarro Projections of hypersurfaces with boundary in R4 to planes

Projections of hypersurfaces with boundary in R4 to planes

M hypersurface with boundary in R4.

We study singularities of these projections, that measure thecontact of M with planes.

At interior points we classify it by using the Mather’s group A(2003).

For boundary points we need to use the subgroup B (Damon’sgeometric group) of A which preserves a plane in the source.

J. W. Bruce and P. J. Giblin investigated the singularities ofprojections of surfaces with boundary in R

3 to a plane.


The projection

The family of planes in R4 close to the plane u generated by

a = (1, 0, 0, 0) and b = (0, 0, 0, 1) may be given taking (1, β1, γ1, 0)and (0, β2, γ2, 1) as generators of those planes, for β1, β2, γ1, γ2close to 0. So, we get that the projection is locally the map

Πβ,γ(X ,Y ,Z ,W ) = (X + β1Y + γ1Z ,W + β2Y + γ2Z ) ,

where β and γ denote the pairs (β1, β2) and (γ1, γ2). Note thatΠ0,0 = Πu .

Let i(x , y , z) = (X ,Y ,Z ,W ) a germ of immersion, then weclassify simple germs of maps R3, 0 → R

2, 0 of corank 1 andBe-codimension ≤ 4.


The method of classification of map germs is well known for thegroup A, but also works for the geometric group B.

We used the Transversal package that was written by Neil Kirk(see http://old.lms.ac.uk/jcm//3/lms1999-002/). The programwas written in Maple and we adapted such package to work in thecase of hypersurface with boundary that is for the group B.


Theorem

(Bruce, du Plessis, Wall, 1987)A germ f : Rn, 0 → R

p, 0 is (2r + 1)-A1-determined if

mr+1n .E(n, p) ⊂ LA1.f +m2r+2

n .E(n, p).


Proposition

(Complete transversal, Bruce, Kirk, du Plessis, 1997) Let g be ak-jet in Jk(n, p), and let T be a vector subspace of the setHk+1(n, p) of homogeneous jets of degree k + 1, such that

Hk+1(n, p) ⊂ T + L(Jk+1A1)(g) .

Then any (k + 1)-jet jk+1f with jkg = jk f is Jk+1A1-equivalentto g + t for some t ∈ T. (The vector subspace T is called thecomplete (k + 1)-transversal of g.)

Lemma

(Mather’s Lemma, 1969) Let G be a Lie group acting smoothly ona finite dimensional manifold X . Let V be a connected submanifoldof X . Then V is contained in a single orbit of G if and only if1. for each x ∈ V , TxV ⊂ TxG(x) = LG(x);2. dimTxG(x) is constant for all x ∈ V .


The codimension of the orbit of f is given by

dimR(m3.E(3, 2)/LA(f ))

and the codimension of the extended orbit (Ae-codimension) is

dimR(E(3, 2)/LeA(f )) .


The method

The classification of germs is carried out inductively on the jetlevel until a sufficient jet is found. To do this we have three mainresults: finite determinacy (FD); Complete Transversal Theorem(CT) and Mather’s Lemma (ML).

• jk f is f.d? If yes then stop. (FD result by using the unipotentgroup A1. )

• If not we add some k + 1-degree monomials given by CT (thatalso uses A1 ).

• Finally Mather’s Lemma says it these monomials are reallynecessary for the A group.

After that we return to the FD result to see if the jk+1f is f.d. andso on.


The Transversal Program

Then to work with the Transversal package we use twosimultaneous windows of maple:• one to work with A1 (to apply FD and CT results)• the other one to work with the group A (to apply ML).

This method also works for the group B then we adapted theTransversal Program to use the group B.


The group B is the subgroup of A consisting of pairs of germs ofdiffeomorphisms (h, k) in Diff (R3)× Diff (R3) with h preservingthe manifold as well as its boundary (R2, 0) (that is, h takes thevariety V = {(x , y , z) : z ≥ 0} into itself).

Therefore, if (h, k) ∈ B we can writeh(x , y , z) = (h1(x , y , z), h2(x , y , z), zh3(x , y , z)) withh3(0, 0, 0) > 0, for germs of smooth functions hi , i = 1, 2, 3.


The B (resp. B1, i.e. the subgroup of B whose elements have with1-jets at 0 the identity) tangent space of f ∈ m(x , y , z).E(3, 2) isgiven by

LB.f = m(x , y , z).{fx , fy}+ E3{zfz}+ f ∗m(u, v).{e1, e2},LB1.f = m2(x , y , z).{fx , fy}+m(x , y , z).{zfz}+ f ∗m2(u, v).{e1, e2}

where fxi denotes the partial derivative with respect to x , y ,{e1, e2} is the standard basis vectors of R2 considered as elementsof E(3, 2), and f ∗(m2) is the pull-back of the maximal ideal in E2.

The extended tangent space to the B-orbit of f at the germ f isgiven by

LeB(f ) = E3.{fx , fy}+ E3{zfz}+ f ∗m(u, v).{e1, e2},


Classification

The 1-jets

j1f = (a1x + a2y + a3z , b1x + b2y + b3z). Cases a1b2 − a2b1 6= 0and a1b2 − a2b1 = 0, give us the orbits in J1(3, 2):

(x , y), (x, z), (x,0), (z,0)ns, (0, 0).

(0, 0) leads to germs of corank 2(x , y) is 1-B-determined and is stable (Be-cod 0).


Higher jets1. j1f ∼ (x , z) then f ∼ (x , z + g(x , y)), deg(g) ≥ 2.

Proposition

The map germ f (x , y , z) = (x , z + g(x , y)) is r -B-determined(resp. simple) if and only if the map-germ h(x , y) = (x , g(x , y)) isr -A-determined (resp. simple). We also haveBe-cod(f ) = Ae-cod(h).

Proof.

Observe that

LB.f = E3.{(z , 0), (0, z)} + LA.h.


Table 1: A-simple germs of map-germs R2, 0 → R2, 0, ε = 1, given

by Rieger.

Type Normal form Ae-codimension

1 (x , y) 02 (x , εy2) 03 (x , xy + y3) 04k (x , y3 + (±1)k−1xky), k ≥ 2 k − 15 (x , xy + εy4) 16 (x , xy + y5 ± y7) 27 (x , xy + y5) 39 (x , xy + εy6 + y9) 4112k+1 (x , xy2 + εy4 + y2k+1), k ≥ 2 k12 (x , xy2 + y5 + εy6) 313 (x , xy2 + y5 ± y9) 416 (x , x2y + εy4 ± y5) 317 (x , x2y + εy4) 4


2. j1f ∼ (x , 0). A complete 2-transversal is given by

(x , b1xy + b2xz + b3yz + b4y2 + b5z

2)

for some b1, b2, b3, b4, b5 ∈ R. We obtain the following orbits inJ2(3, 2):

Case (a)

b4 6= 0 ⇒ (x , y2)(ns), (x , y2 + xz ± z2), (x , y2 + xz), (x , y2 ± z2).Case (b)

b4 = 0, b1 6= 0 ⇒ (x , xy)(ns), (x , xy + yz), (x , xy + z2).Case (c)

b4 = 0, b1 = 0 ⇒ (x , yz)(ns), (x , xz + z2)(ns), (x , xz)(ns),

(x , z2)(ns), (x , 0)(ns).


Case (a): j2f = (x , y2) + j2(0, g(x , y , z)).

Proposition

The map-germ f (x , y , z) = (x , y2 + g(x , z)) is r -B-determined(resp. simple) if and only if the map-germ h(x , z) = (x , g(x , z)) isr -B-determined (resp. simple). We have Be-cod(f ) = Be-cod(h).

Proof.

Observe that

LB.f = E3.{(y , 0), (0, xy), (0, y2), (0, yz)} + LB.h.


Table 2: Corank 1 B-simple singularities of map-germsg : R2, 0 → R

2, 0, ε = 1, from Bruce and Giblin.

Normal form Be-codimension

(x , εz2 + (±1)kxk−1z), k ≥ 2 k − 2(x , xz + εz3) 1(x , xz + εz4 ± z6), 2(x , xz + εz4) 3

Then for us the germs are (x , y2 + g(x , y)) with ε = ±1.We follow the other cases with 2 < cod ≤ 4. We show that theyare all no simple germs.


Case (b): j2f ∼ (x , xy + yz) or j2f ∼ (x , xy + z2)

The simple cases we summarize in the next table.

Table 3: Corank 1 B-singularities of map-germs f : R3, 0 → R2, 0

of Be-codimension ≤ 4

Normal form Be-codimension

(x , yz + xy + εy3) 1(x , yz + xy + y4 + εy6) 2(x , yz + xy + y4) 3(x , xy + z2 + y3 + εykz), k ≥ 2 k


As a consequence of the analysis, we have the following results.

Theorem

The map-germ f : R3, 0 → R2, 0 with 1-jet equivalent to (z , 0) or

2-jet equivalent to (x , y2), (x , xy), (x , yz), (x , xz + z2), (x , xz),(x , z2), (x , 0) or 3-jet (x , xy + z2 + εy2z), (x , xy + z2) or 4-jet(x , y2 + xz), (x , xy + yz) are no simple germs.

Theorem

The map-germ f : R3, 0 → R2, 0 of corank at most 1 and

Be-codimension ≤ 1 are simple germs.

We study geometrically the germs given by the last theorem.


The geometry of codimension ≤ 1 singularities

Table 4: B-simple map germs (R3, 0) → (R2, 0) of codimension ≤ 1(ε and ε are independently ±1).

No. Normal form Be-cod Versal unfolding

I (x , y) 0II (x , z + εy2) 0III (x , z + xy + y3) 0IV (x , z + εx2y + y3) 1 (x , z + y3 + εx2y + λy)V (x , z + xy + εy4) 1 (x , z + xy + εy4 + λy2)VI (x , y2 + εz2 + xz) 0VII (x , y2 + εz2 + εx2z) 1 (x , y2 + εz2 + εx2z + λz)VIII (x , y2 + xz + εz3) 1 (x , y2 + xz + εz3 + λ(z + z2)) (IX (x , yz + xy + εy3) 1 (x , yz + xy + εy3 + λ(y2 + z)) (


Πβ,γ(X ,Y ,Z ,W ) = (X + β1Y + γ1Z ,W + β2Y + γ2Z ) ,

where β and γ denote the pairs (β1, β2) and (γ1, γ2). Π0,0 = Πu.

To realize a versal unfolding with 1-parameter we takeβ1 = γ1 = β2 = 0 and call γ2 = λ, wherei : (x , y , z) → (X ,Y ,Z ,W ) is an imersion at (0, 0, 0).

I. i(x , y , z) = (x , 0, z , y)II. i(x , y , z) = (x , y , 0, z + εy2)III. i(x , y , z) = (x , y , 0, z + xy + y3)IV. i(x , y , z) = (x , y , 0, z + y3 + εx2y)V. i(x , y , z) = (x , y2, y , z + xy + εy4)VI. i(x , y , z) = (x , y , z , y2 + εz2 + xz)VII. i(x , y , z) = (x , y , z , y2 + εz2 + εx2z)VIII. i(x , y , z) = (x , y , z + z2, y2 + xz + εz3)IX. i(x , y , z) = (x , y , y2 + z , yz + xy + εy3)


Geometrical information in Cases I-IX.

We consider the following geometric sets to recognize the cases:• the fibre f −1(0);• kernel K of df (0);• Σ, the critical set of the map f ;• singularities of the restriction of f to the boundary• critical loci (image of Σ) and the image of the boundary, for fand for members of the versal family unfolding f .

Parts of Σ or its image that is ‘virtual’, in the sense ofcorresponding to the part z < 0 in the source R

3, appear dashed inthe figures. The boundary (plane-xy) and its image are drawn withthe gray color.


The following geometric facts recognize the cases.Cases I to V are submersions.

I. The unique case where K is a line transverse to the boundarythat is, on M, the direction of projection is transverse to ∂M.

II. The fibre f −1(0) is a curve tangent to the boundary, for ε = −1and f −1(0) = 0, for ε = 1. The singularity of the restriction of fto the boundary is the fold (x , y2).

III. The set f −1(0) is a half curve tangent to the boundary. Thesingularity of the restriction of f to the boundary is the cusp(x , xy + y3).


x

y

z

K

K KK

f−1(0)f−1(0)

(a) (b) (c) (d)

Figure: Submersions. (a) Case I. (b) Cases II and V, for ε = −1. (c) Cases IIand V, for ε = 1. (d) Cases III and IV.


The cod 1 submersions IV an V have a cod 1 singularity for therestriction of f to the boundary.

IV. The singularity of the restriction of f to the boundary is thelips/beaks (x , εx2y + y3), which is a codimension 1 singularity.

V. The singularity of the restriction of f to the boundary is theswallowtail (x , xy + y4), which is a codimension 1 singularity.Besides the fibre f −1(0) is a curve tangent to the boundary, forε = −1 and f −1(0) = 0, for ε = 1.

1 1

22 5

42

λ

λ

ε = +1 ε = −1

Figure: Left: The image of the boundary of Case V. Right: Bifurcationdiagrams of the singularities for the restriction of f to the boundary, cases IVand V. Ana Claudia Nabarro Projections of hypersurfaces with boundary in R

4 to planes

VI-IX. These are those germs with K = ker df (0) being theplane-yz .

For each no submersion germ f of codimension 1 we give abifurcation diagram to show, for germs close to f in the Be-versalunfolding, which types of boundary singularities occur.

VI. The image do Σ and the fibre f −1(0) distinguish cases ε = 1and ε = −1: f −1(0) = 0 or it is a set transverse to the boundary,respectively.

K K

Σ

Σ f−1(0)

(a) (b)

Figure: (a) Case VI for ε = 1. (b) Case VI for ε = −1.Ana Claudia Nabarro Projections of hypersurfaces with boundary in R

4 to planes

VII. The singular set Σ (or the image of Σ) distinguishes this caseof all other cases. f −1(0) distinguishes the cases ε.

K

K

λ

λ

Σ

Σ

λ = 0 λ > 0λ < 0

ε = 1

f−1(0) ε = −1

VII

VII(VI+)2

(VI−)2

Figure: Case VII for εε = 1.


K

K

λ

λ

Σ

Σ

λ = 0 λ > 0λ < 0

ε = 1

f−1(0) ε = −1

VII

VII (VI+)2

(VI−)2

Figure: Case VII for εε = −1.


VIII. f |Σ, unlike all other cases, has a cusp singularity.

K

K

λ

λ

Σ

Σ

λ = 0 λ > 0λ < 0

ε = 1

ε = −1f−1(0)

VI−

VI−

VI+

VI+

VIII

VIII


IX. This is the unique no submersion whose image of the boundaryis the hole plane. f −1(0) distinguishes the cases ε.

y

K

K

λ

λ

Σ

Σ

λ = 0 λ > 0λ < 0

ε = 1

ε = −1

f−1(0)

f−1(0)

VI−

VI−

VI−

VI−

IX

IX


Some references:

• J. W. Bruce and P. J. Giblin, Projections of surfaces withboundary. Proc. London Math. Soc. (3) 60 (1990), 392–416.

• J. W. Bruce, N. P. Kirk and A. A. du Plessis, Completetransversals and the classification of singularities, Nonlinearity 10(1997), 253-275.

• J. W. Bruce, A. A. du Plessis and C. T. C. Wall, Determinacyand unipotency, Invent. Math. 88 (1987), 521–554.

• A. C. Nabarro, Projections of hypersurfaces in R4 to planes,

Lecture Notes in Pure and Applied Maths, Marcel Dekker - NewYork, vol. 232 (2003), 283–300.

• J. H. Rieger, Families of maps from the plane to the plane, J.London Math. Soc. (2) 36 (1987), 351369.


Thank you very much for your attention!

Happy birthday, Bruce and Wall!!


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