Equivalences of 5 -Dimensional CR Manifolds [Joint Work with Wei-Guo - - PowerPoint PPT Presentation

Equivalences of 5-Dimensional CR Manifolds

[Joint Work with Wei-Guo Foo and The-Anh Ta]

Einstein-Weyl Structures

[Joint Work with Paweł Nurowski]

Differential Invariants of Parabolic Surfaces

[Joint Work with Zhangchi Chen]

JOËL MERKER

www.math.u-psud.fr/∼merker/ Département de Mathématiques d’Orsay Bibliothèque mathématique Jacques Hadamard 2019 Taipei Conference on Complex Geometry Institute of Mathematics, Academia Sinica Wednesday 18 December 2019

2

Cartan’s Method of Equivalence Cartan devised a quite sophisticated and proteiform method of equiva-

• lence. Given a manifold M equipped with a certain class of geometric, say

CR here, structures, Cartan’s method of equivalence consists in constructing a bundle π: P − → M together with an absolute (co)parallelism on P, namely a coframe of everywhere linearly independent 1-forms θ1, . . . , θdim P on P such that: P

Π

• π
• P ′

π′

• M

Φ

M′

• every local CR diffeomorphism Φ: M −

→ M′ between two CR manifolds lifts uniquely as a diffeomorphism Π: P − → P ′ satisfying Π∗θ′i = θi for 1 i dim P, with P ′ and the θ′i similarly constructed;

• conversely, every diffeomorphism Π: P −

→ P ′ commuting with projec- tions π, π′ whose horizontal part is a diffeomorphims M − → M′ and which

satisfies Π∗θ′i = θi for 1 i dim P, has a horizontal part which is Cauchy- Riemann diffeomorphism (or, more generally, a diffeomorphism respecting the considered geometric structure). [Beyond, there can exist Cartan connections associated to (modifications

• f) P −

→ M, but we will not need this concept.] Rexpressing the exterior differentials dθi and dθ′i from both sides in terms

• f the basic 2-forms provided by the two ambient coframes:

dθi =

• j<k

T i

j,k(p) θj∧θk

and dθ′i =

• j<k

T ′i

j,k(p′) θ′j∧θ′k,

certain structure functions appear, defined for p ∈ P and for p′ ∈ P ′, and the exact pullback relations Π∗θ′i = θi force individual invariancy of all them: T ′i

j,k

• Φ(p)
• = T i

j,k(p) (∀ p ∈ P).

As is known, Cartan’s method is computationally extremely intensive, es- pecially in CR geometry, where several normalizations and prolongations are

• required. Explicit expressions of intermediate torsion coefficients which con-

duct to the final T i

j,k(p) grow dramatically in complexity.

4

One reason for such a complexity is the presence of large isotropy groups for the CR automorphisms groups of (standard) models, which imposes a great number of steps. Another reason is the nonlinear character of differential al- gebraic polynomial expressions that must be handled progressively. The last reason is that Cartan’s method studies geometric structures at every point of the base manifold, and there is a price to pay for this generality. In most existing references (cf. the bibliography), the trick that Cartan him- self devised to avoid nonlinear complications while retaining anyway some es- sential information, is the so-called Cartan Lemma. It is explicit only at the level of linear algebra. Even admitting to only deal with linear algebra com- putations, as Chern always did, Cartan’s method is often long and demanding.

2-Nondegenerate Levi Rank 1 Hypersurfaces M5 ⊂ C3

[Joint Work with Wei-Guo Foo and The-Anh Ta]

• Coordinates:
• z, ζ, w
• ∈ C3.

The right graphed equation for the model light cone MLC ⊂ C3 in C2,1 was discovered by Gaussier-M. in 2003: MLC: u = zz + 1

2 z2ζ + 1 2 z2ζ

1 − ζζ =: ♠

• z, ζ, z, ζ
• ,

Start with M5 ⊂ C3, with 0 ∈ M, rigid, graphed as: u = F(z, ζ, z, ζ). Constant Levi rank 1 means, possibly after a linear transformation in C2

z,ζ,

that: Fzz = 0 ≡

• Fzz Fζz

Fzζ Fζζ

• =: Levi(F),

(0.1)

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while 2-nondegeneracy means that: 0 =

• Fzz

Fzz Fzzζ Fzzζ

• .

(0.2) At the origin, MLC of equation: u = zz + 1

2 z2ζ + 1 2 z2ζ + Oz,ζ,z,ζ(4),

is obviously 2-nondegenerate, thanks to the cubic monomial 1

2 z2ζ which gives

that (0.2) at (z, ζ) = (0, 0) becomes

• 1 0

∗ 1

• = 1. As for constant Levi rank 1,
• rder two terms u = zz + · · · show that this condition is true at the origin, and

simple computations show that (0.1) is identically zero:

• 1

1−ζζ z+zζ (1−ζζ)2 z+zζ (1−ζζ)2 (z+zζ)(z+zζ) (1−ζζ)3

• ≡ 0

(– indeed!).

Consider as before a rigid M5 ⊂ C3 with 0 ∈ M, which is 2-nondegenerate and has Levi form of constant rank 1, i.e. belongs to the class C2,1, and which is graphed as: u = F

• z1, z2, z1, z2
• .

The letter ζ is protected, hence not used instead of z2, since ζ will denote a 1-form. The two natural generators of T 1,0M and T 0,1M are: L1 := ∂z1 − i Fz1 ∂v and L2 := ∂z2 − i Fz2 ∂v, in the intrinsic coordinates (z1, z2, z1, z2, v) on M. The Levi kernel bundle K1,0M ⊂ T 1,0M is generated by: K := ❦ L1 + L2, where ❦ := − Fz2z1 Fz1z1 , is the slant function. The hypothesis of 2-nondegeneracy is equivalent to the nonvanishing: 0 = L 1(❦). Also, the conjugate K generates the conjugate Levi kernel bundle K0,1 ⊂ T 0,1M. There is a second fundamental function, and no more: P := Fz1z1z1 Fz1z1 . In the rigid case, it looks so simple! But in the nonrigid case, P has a numerator involving 69 differential monomials!

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Foo-Merker-Ta produced reduction to an {e}-structure for the equivalence problem, under rigid (local) biholomorphic transformations, of such rigid M5 ∈ C2,1.

• Theorem. [Foo-M.-Ta 2019] There exists an invariant 7-dimensional bundle

P 7 − → M5 equipped with coordinates:

• z1, z2, z1, z2, v, c, c
• ,

with c ∈ C, together with a collection of seven complex-valued 1-form which make a frame for TP 7, denoted:

• ρ, κ, ζ, κ, ζ, α, α
• (ρ = ρ),

which satisfy 7 invariant structure equations of the form: dρ =

• α + α
• ∧ ρ + i κ ∧ κ,

dκ = α ∧ κ + ζ ∧ κ, dζ =

• α − α
• ∧ ζ + 1

c ■0 κ ∧ ζ + 1 cc ❱0 κ ∧ κ, dα = ζ ∧ ζ − 1 c ■0 ζ ∧ κ + 1 cc ◗0 κ ∧ κ + 1 c ■0 ζ ∧ κ, conjugate structure equations for dκ, dζ, dα being easily deduced.

Here, as in Pocchiola’s Ph.D., there are exactly two primary Cartan- curvature invariants: ■0 := − 1 3 K

• L 1
• L 1(❦)
• L 1(❦)2

+ 1 3 K

• L 1(❦)
• L 1
• L 1(❦)
• L 1(❦)3

+ + 2 3 L1

• L1(❦)
• L1(❦)

+ 2 3 L1

• L 1(❦)
• L 1(❦)

, ❱0 := − 1 3 L 1

• L 1
• L 1(❦)
• L 1(❦)

+ 5 9 L 1

• L 1(❦)
• L 1(❦)

2 − − 1 9 L 1

• L 1(❦)
• P

L 1(❦) + 1 3 L 1(P) − 1 9 P P. One can check that Pocchiola’s ❲0 which occurs under general biholomorphic transformations of C3 (not necessarily rigid!), when written for a rigid M5 ⊂ C3, identifies with: ■0

• F(z1, z2, z1, z2)
• ≡ ❲0
• F(z1, z2, z1, z2)
• .

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Furthermore, there is one secondary invariant whose unpolished expression is:

◗0 := 1 2 L 1

• ■0
• − 1

3

• P − L1
• L1(❦)
• L1(❦)
• ■0 − 1

6

• P − L 1
• L 1(❦)
• L 1(❦)
• ■0 − 1

2 K (❱0) L 1(❦) .

Visibly indeed, the vanishing of ■0 and ❱0 implies the vanishing of ◗0. In fact, a consequence of Cartan’s general theory is: 0 ≡ ■0 ≡ ❱0 ⇐ ⇒ M is rigidly equivalent to the Gaussier-Merker model. By deducing new relations from the structure equations above, it was proved that ◗0 is real-valued, but a finalized expression was missing there. A clean finalized expression of ◗0, in terms of only the two fundamental

functions ❦, P (and their conjugates), from which one immediately sees real- valuedness, is: ◗0 := 2 Re 1 9 K

• L 1(❦)
• L 1
• L 1(❦)

2 L 1(❦)4 − − 1 9 K

• L 1
• L 1(❦)
• L 1
• L 1(❦)
• L 1(❦)3

− 1 9 K

• L 1(❦)
• L 1
• L 1(❦)
• P

L 1(❦)3 − − 1 9 L1

• L 1(❦)
• L 1
• L 1(❦)
• L 1(❦)2

+ 1 9 K

• L 1
• L 1(❦)
• P

L 1(❦)2 − − 2 9 L1

• L 1(❦)
• P

L 1(❦) − 1 9 L 1

• L 1(❦)
• P

L 1(❦) + 1 3 L1

• L 1
• L 1(❦)
• L 1(❦)

+ 1 6 L 1(P − 1 9

• P
• 2 + 1

3

• L 1
• L 1(❦)
• L 1(❦)
• 2

.

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Rigid Normal Forms for Rigid CR Manifolds

In his works, Moser usually searched for wisdom rather than simply knowledge, and thus he strongly emphasized developments

• f methods and insights over pushing a specific result to the limit.

Accordingly, he sometimes described the outcome of his own work as methods rather than theorems. Katok-Hasselblatt 2002

Moser’s method is more ‘down to Earth’, computationally speaking, since it usually proceeds at only one point, often the origin, of a manifold, manip- ulating power series expanded at that point. Hence it needs geometric objects

• f class C ω, while adaptations to the C ∞ or C K≫1 classes can concern only

formal Taylor expansions at the point.

• Main goal: Construct a bridge:

Cartan’s method Moser’s method, and exhibit how differential invariants pass from one side of the river to the

• ther side, computationally.

Two Invariant Determinants for Hypersurfaces M5 ⊂ C3 Consider a rigid biholomorphism: H : (z, ζ, w) − →

• f(z, ζ), g(z, ζ), ρ w + h(z, ζ)
• =:
• z′, ζ′, w′

(ρ ∈ R∗),

hence with Jacobian fzgζ − fζgz = 0, between two rigid C ω hypersurfaces: w = − w+2 F

• z, ζ, z, ζ
• =: Q

and w′ = − w′+2 F ′ z′, ζ′, z′, ζ′ =: Q′. Plugging the three components of H in the target equation: ρ w + h(z, ζ) + ρ w + h(z, ζ) = 2 F ′ f(z, ζ), g(z, ζ), f(z, ζ), g(z, ζ)

• ,

and replacing w + w = 2 F, one receives the fundamental equation expressing H(M) ⊂ M′: 2 ρ F

• z, ζ, z, ζ
• +h(z, ζ)+h(z, ζ) ≡ 2 F ′

f(z, ζ), g(z, ζ), f(z, ζ), g(z, ζ)

• .

By differentiating it (exercise! use a computer!), one expresses as follows the invariancy of the Levi determinant defined for general biholomorphisms

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as:

• Qz

Qζ Qw Qzz Qzζ Qzw Qζz Qζζ Qζw

• = 22
• Fz

Fζ −1 Fzz Fzζ Fζz Fζζ

• .
• Proposition. Through any rigid biholomorphism:
• F ′

z′z′ F ′ z′ζ′

F ′

ζ′z′ F ′ ζ′ζ′

• =

ρ2

• fz fζ

gz gζ

• fz fζ

gz gζ

• Fzz Fzζ

Fζz Fζζ

• .
• Consequently, the property that the Levi form is of constant rank 1 is bi-

holomorphically invariant. The 2-nondegeneracy property then expresses as the nonvanishing of:

• Qz

Qζ Qw Qzz Qzζ Qzw Qzzz Qzzζ Qzzw

• = 22
• Fz

Fζ −1 Fzz Fzζ Fzzz Fzzζ

• .

• Proposition. When the Levi form is of constant rank 1, through any rigid

biholomorphism:

• F ′

z′z′

F ′

z′ζ′

F ′

z′z′z′ F ′ z′z′ζ′

• = ρ2

gζ Fzz − gz Fζz 3

• fz fζ

gz gζ

• 3
• fz fζ

gz gζ

• Fzz

Fzζ Fzzz Fzzζ

• .
• Recall that we denote the class of (local) hypersurfaces M5 ⊂ C3 pass-

ing by the origin 0 ∈ M that are 2-nondegenerate and whose Levi form has constant rank 1 as: C2,1.

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Rigid Infinitesimal CR Automorphisms of the Model appropriate model MLC is rigid and was set up by Gaussier-Merker and Fels-Kaup 2007: MLC: u = zz + 1

2 z2ζ + 1 2z2ζ

1 − ζζ =: ♠

• z, ζ, z, ζ
• .

locally graphed representation of the tube in C3 over the future light

• R3. The 10-dimensional simple Lie algebra of its infinitesimal CR

automorphisms: g := autCR

• MLC

∼ = so2,3(R), natural generators X1, . . . , X10, which are (1, 0) vector fields having holomorphic coefficients with Xσ +Xσ tangent to MLC. Assigning weights to ariables, to vector fields, and the same weights to their conjugates: 1 [ζ] := 0, [w] := 2

• ∂z
• := − 1
• ∂ζ
• := 0
• ∂w
• := − 2,

algebra of vector fields isomorphic to so2,3(R) can be graded as: g = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2,

where: g−2 := Span

• i ∂w
• ,

g−1 := Span

• (ζ − 1) ∂z − 2z ∂w,

(i + iζ) ∂z − 2iz ∂w

• ,

where g0 = gtrans ⊕ giso

0 :

gtrans := Span

• zζ ∂z + (ζ2 − 1) ∂ζ − z2 ∂w,

izζ ∂z + (i + iζ2) ∂ζ − iz2 ∂w

• ,

giso := Span

• z ∂z + 2w ∂w,

iz ∂z + 2iζ ∂ζ

• ,

while: g1 := Span

• z2 − ζw − w) ∂z +
• 2zζ + 2z
• ∂ζ + 2zw ∂w,
• − iz2 + iζw − iw
• ∂z +
• − 2izζ + 2iz
• ∂ζ − 2izw ∂w
• ,

g2 := Span

• izw ∂z − iz2 ∂ζ + iw2 ∂w
• .

Calling these X1, . . . , X10 in order of appearance, the five Xσ+Xσ for σ = 1, 2, 3, 4, 5 span TM5 while those for σ = 6, 7, 8, 9, 10 generate the isotropy subgroup of the origin. Theorem.

[Chen-Foo-Merker-Ta 2019] Every hypersurface M5 ∈ C2,1 is

equivalent, through a local rigid biholomorphism, to a rigid C ω hyper- surface M′5 ⊂ C′3 which, dropping primes for target coordinates, is a

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perturbation of the Gaussier-Merker model: u = zz + 1

2 z2ζ + 1 2 z2ζ

1 − ζζ +

• a,b,c,d∈N

a+c3

Ga,b,c,d zaζbzcζd, with a simplified remainder G which: (1) is normalized to be an Oz,z(3); (2) satisfies the prenormalization conditions G = Oz(3)+Oζ(1) = Oz(3)+ Oζ(1): Ga,b,0,0 = 0 = G0,0,c,d, Ga,b,1,0 = 0 = G1,0,c,d, Ga,b,2,0 = 0 = G2,0,c,d; (3) satisfies in addition the sporadic normalization conditions: G3,0,0,1 = 0 = G0,1,3,0, Im G3,0,1,1 = 0 = Im G1,1,3,0. Furthermore, two such rigid C ω hypersurfaces M5 ⊂ C3 and M′5 ⊂ C′3, both brought into such a normal form, are rigidly biholomorphically equivalent if and only if there exist two constants ρ ∈ R∗

+, ϕ ∈ R, such

that for all a, b, c, d: Ga,b,c,d = G′

a,b,c,d ρ

a+c−2 2

eiϕ(a+2b−c−2d).

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Einstein-Weyl Structures

[Joint Work with Paweł Nurowski]

• 3-dimensional manifold:

M.

• Weyl geometry: Consists of pairs:

g := Riemannian metric, A := 1-form, modulo gauge transformations:

• g = e2ϕg,
• A = A + dϕ.
• Theorem. [Élie Cartan 1943] Every solution to the Einstein-Weyl equations:

R(µν) − 1 3 R gµν = 0

comes from an appropriate 3D leaf space quotient of a 7D connection bundle associated with a 3rd order ODE: y′′′ = H(x, y, y′, y′′), modulo point transformations . . . provided 2 among 3 primary invari- ants vanish:

Wünschmann(H) ≡ 0 ≡ Cartan(H).

• Obstacle: These two invariants are very difficult to integrate.
• For instance: Wünschmann’s nonlinear equation incorporates 25 differen-

tial monomials:

0 ≡ ❲(H) := − 18 qHqHpq + 9 pHyHqq + 18 qHHpqq + 9 q HpHqq − 18 pHqHyq + 18 pHHyqq − 9 HHqHqq + 18 pqHypq + 18 pHxyq + 18 q Hxpq + 9 HxHqq + 18 HHxqq − 18 HqHxq + 18 HpHq + 9 Hxxq − 27 Hxp + 4 H3

q + 9 p2Hyyq − 27 pHyp + 9 qHyq + 9 q2Hppq − 27 qHpp − 18 HHpq + 9 H2Hqqq

+ 54 Hy.

• Application of Cartan-Kähler theorem:

Corollary.

[Élie Cartan 1943] The general solution to the Einstein-Weyl

equations depends on 4 functions of 2 variables.

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• Open problem: Construct somewhat explicit solutions to the Einstein-Weyl
• equations. [Dunajski, Eastwood, Godlinski, Hitchin, Jones, LeBrun, Makhmali, Ma-

son, Nurowski, Tod].

• Examples: Some particular solutions are known, e.g.:

H = 3 q2 2 q , H = 3 q2p p2 + 1, H = q3/2, H =

• 2 qy − p23/2

y2 ,

• r the ‘horrible’:

H = pq

• − 12 + 3 pq − 8 √1 − pq
• + 8
• 1 + √1 − pq
• p3

. They were all obtained by rather ad hoc methods.

• Recent unexpected new approach: [Merker-Nurowski 2019] Study instead

point equivalence classes of a single partial differential equation: zy = F(x, y, z, zx), with para-CR integrability condition:

DF := Fx + zxFz ≡ 0.

• Observation. Cartan’s method of equivalence leads to a very similar 7D

Cartan bundle and connection. Magical simplification: The (complicated) Wünschmann equation above:

0 ≡ ❲(H) := − 18 qHqHpq + 9 pHyHqq + 18 qHHpqq + 9 q HpHqq − 18 pHqHyq + 18 pHHyqq − 9 HHqHqq + 18 pqHypq + 18 pHxyq + 18 q Hxpq + 9 HxHqq + 18 HHxqq − 18 HqHxq + 18 HpHq + 9 Hxxq − 27 Hxp + 4 H3

q + 9 p2Hyyq − 27 pHyp + 9 qHyq + 9 q2Hppq − 27 qHpp − 18 HHpq + 9 H2Hqqq

+ 54 Hy.

becomes: 0 ≡ Monge(F) := 9 F 2

pp Fppppp − 45 Fpp Fppp Fpppp + 40 F 3 ppp (p := zx).

• Question: What is this Monge invariant?

Very Old Theorem. [Explain more later] The solutions of: 0 ≡ Monge(F), are just conics in the {p, F}-plane.

24

• As an Ansatz, take: [Joint with Wei Guo Foo]

F(x, y, z, p) := α(y)(z−xp)2+β(y)(z−xp)p+γ(y)(z−xp)+δ(y)p2+ε(y)p+ζ(y)

λ(y) (z−xp)+µ(y) p+ν(y)

, with 9 arbitrary functions α, . . . , ν of y.

• Immediate: This function F satisfies both:

0 ≡ DF, 0 ≡ Monge(F).

• Expectably: The last condition from the ODE side:

0 ≡ Cartan(H) = 18 Hqq DHq − 12 Hqq H2

q − 54 Hqq Hp + 36 Hpq Hq − 108 Hyq + 54 Hpp,

translates to the PDE side as a certain condition: 0 ≡ ❑(F).

• Serendipitous Fact: This condition ❑(F) ≡ 0 holds identically for any

choice of α(y), . . . , ν(y).

• Conclusion: [Explain more later] As Élie Cartan did, descending to the leaf

space quotient, we gain ∞-dimensional functionally parametrized and explicit

families of Einstein-Weyl structures

• (g, A)
• in 3D. These structures are non-

trivial in the sense that dA ≡ 0 and Cotton([g]) ≡ 0. [978 pages Mathematica

export pdf]

Theorem.

[Merker-Nurowski 2019 [Foo participated]] All pairs

• g, A
• with g

Lorentzian of signature (2, 1): g := τ1 ⊗ τ2 + τ2 ⊗ τ1 + τ3 ⊗ τ3, A := τ3 1

2

1 Π

• γλx − γµ + xλν′ + βλz + λµ′z − 2αµz − λ′µz

− µν′ − xλ′ν − 2xαν + βν + µ′ν

• ,

with the coframe:

τ 1 := dx + dy xλ − µ

• xβ − γ − x2α
• ,

τ 2 := 2 dy xλ − µ Π, τ 3 :=

• − λz − ν
• dx +

1 xλ − µ dz

• − εµ + 2x2αν + xγµ − 2xβν − βµz + 2δλz

+ 2xαµz + xελ + 2xδ − x2γλ − xβλz

• ,

and the function:

26

Π := x2ζλ2 + αµ2z2 + 2xαµνz + x2αν2 − βλµz2 − xβλνz + δλ2z2 + xελ2z − 2xζλµ − βµνz − xβν2 + 2δλνz − ελµz + xελν − xγλµz − x2γλν + ζµ2 + δν2 − εµν + γµ2z + xγµν, satisfy the Einstein-Weyl equations, hence define a Lorentzian Einstein- Weyl structure on R3. Moreover, all such examples are generically conformally non-flat, namely dA ≡ 0, and each of the 5 independent components of the Cot- ton tensor of the underlying conformal structure (M, [g]) is not identically zero.

Elementary Monge Invariant In R2 ∋ (x, u), consider a graphed curve passing through the origin nor- malized to order 1: 0 = − u + F(x) = − u + F2 x2

2! + F3 x3 3! + · · · ,

which satisfies F2 = 0. At the beginning, assign these Taylor coefficients to be initial functional jets: Fi := ui = uxi(x), corresponding to ith derivatives uxi(x) at arbitrary points

• x, u(x)
• f the

curve. The special affine linear group SL2(R) consists of matrices a b

c d

• having

determinant 1 = ad − bc. Target coordinates will be denoted (y, v).

28

u R2 x R2 v y

• v = G(y)
• u = F(x)
• special affine

inverse

Any analytic graph

• u = F(x)
• is sent to a similar graphed curve
• v =

G(y)

• .
• Goal: Simplify the target power series G(y) =

j2 Gj yj j!.

• Work with: The inverse special affine transformation:

x = a y + b v, u = c y + d v. Then the graphing function G(y) is uniquely determined, by a fundamental equation: 0 ≡ −

• c y + d v
• + F
• a y + b v
• replace v=G(y)

(in R{y}).

• Rule of the game: Use the group parameters freedom a, b, c, d in order to

‘annihilate’ as much as possible coefficients Gj. After an affine transformation, we may of course assume that our target graph enjoys a similar first-order normalization v = Oy(2), namely: 0 = − v + G(y) = − v + G2 y2

2! + G2 y3 3! + · · · .

Then performing the plain replacement above: 0 ≡ − c y − d G(y) + F

• a y + b G(y)
• ,

we glean first-order terms which must vanish: 0 ≡ − c y + Oy(2)

(in R{y}).

• Lemma. The subgroup of SL2(R) sending v = Oy(2) to u = Ox(2) is

2-dimensional and consists of matrices: G(1)

stab:

a b 0 1

a

• (a = 0).
• Next:

SL2(R) ⊃ G(1)

stab ⊃ G(2) stab ⊃ G(3) stab ⊃ · · · ⊃ G(τ) stab = {e}.

30

• Theorem. (1) Given a real analytic curve
• u = F(x)
• in R2 passing

through the origin which satisfies: Fxx(0) = 0, there always exists an SL2(R) transformation which puts it into the form: u = x2

2! + 0 + F4 x4 4! + F5 x5 5! +

• i6

Fi xi

i! .

(2) Any other such real analytic curve

• v = G(y)
• similarly put into the

form: v = y2

2! + 0 + G4 y4 4! + G5 y5 5! +

• j6

Gj

yj j!,

is SL2(R)-equivalent to

• u = F(x)
• above if and only if all Taylor coeffi-

cients match: G4 = F4, G5 = F5, Gi = Fi

(∀ i 6).

• Explicit expressions: [Chen-Merker 2019] The power series coefficients so
• btained F4, F5, are differential invariants at any point
• x, u(x)
• f the curve,

and beyond, the explicit expressions of the next two are: F6 := 1 9 9 u3

2u6 − 63 u2 2u3u5 + 105 u2u2 3u4 − 35 u4 3

u16/3

2

, F7 := 1 9 9 u4

2u7 − 84 u3 2u3u6 + 210 u2 2u2 3u5 − 105 u2 2u3u2 4 + 210 u2u3 3u4 − 280 u5 3

u20/3

2

. Furthermore, in terms of the total differentiation operator: ❉x := ∂ ∂x + u1 ∂ ∂u +

∞

• i=1

ui+1 ∂ ∂ui the affine-invariant differentiation operator: Dx :=

1 u1/3

2

❉x enables to produce higher order invariants, for instance: Dx 1 3 −5 u2

3 + 3 u2u4

u8/3

2

• = 1

9 9 u2

2u5 − 45 u2u3u4 + 40 u3 3

u4

2

.

32

• Lemma. [Quite ancient!] (1) A curve u = u(x) with uxx = 0 is affinely

equivalent to a parabola v = y2 if and only if: 0 ≡ P(u) := 1 3 3 uxx uxxxx − 5 u2

xxx

u8/3

xx

. (2) A curve u = u(x) with uxx = 0 is affinely equivalent to a nonde- generate conic in the plane if and only if: 0 ≡ ❈(u) := 1 9 9 u2

xx uxxxxx − 45 uxx uxxx uxxxx + 40 u3 xxx

u4

xx

. (3) P(u) ≡ 0 implies ❈(u) ≡ 0.

• Proof. (1) The target being v = y2, we have after an affine transformation

whose linear part a b

k l

• ∼

1 0

0 1

• may be assumed close to the identity:

k x + l u + m =

• a x + b u + c

2, hence we may solve for u thanks to the positivity of the discriminant: u = l − 2 bc − 2 ab x ±

• 4 b2k − 4 abl
• x + l2 − 4 bcl + 4 b2m

2 b2 .

Thus with different constants the general equation of parabolas is: u = d x + e +

• 2 g x + h.

At first, in order to eliminate d and e, we just differentiate two times: uxx = − g2

• 2 g x + h

3/2, and next, to eliminate the remaining constants, we upside-down: 1 u2/3

xx

= 2 g1/3 x + h g4/3, and we again differentiate twice: 1 u2/3

xx

• xx = −2

9 3 uxx uxxxx − 5 u2

xxx

u8/3

xx

≡ 0.

(2) Now, the target is a general conic in the R2 y,v-plane, hence an x2 mono-

mial must be present under the square root: u = d x + e +

• f x2 + 2 g x + h.

34

Quite similarly: uxx = f h − g2

• f x2 + 2 g x + h

3/2, whence: 1 u2/3

xx

= f (f h − g2)2/3 x2 + 2 g (f h − g2)2/3 x + h (f h − g2)2/3, and lastly we have to differentiate three times to get rid of all remaining con- stants: 1 u2/3

xx

• xxx = − 2

27 9 u2

xx uxxxxx − 45 uxx uxxx uxxxx + 40 u3 xxx

u11/3

xx

≡ 0. The converse is also left as an exercise.

(3) follows from a direct differentiation of 0 ≡ 3 uxx(x) uxxxx(x) −

5 uxxx(x)2.

• Conclusion: The Monge equation:

0 ≡ Monge(F) characterizes {u = F(x)

• to be a conic in R2.

• In Cauchy-Riemann geometry: Denote the class of (local) hypersurfaces

M5 ⊂ C3 passing by the origin 0 ∈ M that are 2-nondegenerate and whose Levi form has constant rank 1 as: C2,1. Consider therefore a not necessarily rigid hypersurface M5 ⊂ C3 which be- longs to this class C2,1, and which is graphed as: u = F

• z1, z2, z1, z2, v
• = z1z1 + 1

2 z2 1z2 + 1 2 z2 1z2 + O(4).

The two natural generators of T 1,0M and T 0,1M are: L1 := ∂ ∂z1 −i Fz1 1 + i Fv ∂ ∂v and L2 := ∂ ∂z2 −i Fz2 1 + i Fv ∂ ∂v, in the intrinsic coordinates (z1, z2, z1, z2, v) on M. We will use the abbrevia- tions: ❆1 := − i ❋z1 1 + i ❋v and ❆2 := − i ❋z2 1 + i ❋v . Clearly, the real differential 1-form: ̺0 := dv − ❆1 dz1 − ❆2 dz2 − ❆1 dz1 − ❆2 dz2

36

has kernel:

• ̺0 = 0
• = T 1,0M ⊕ T 0,1M.

At various points: p =

• z1, z2, z1, z2, v
• ∈ M,

and in terms of ̺0, the hypothesis that M has everywhere degenerate Levi form writes as: 0 ≡ =

• ̺0
• i [L1, L 1]
• ̺0
• i [L2, L 1]
• ̺0
• i [L1, L 2]
• ̺0
• i [L2, L 2]
• (p).

The hypothesis that the Levi form has constant rank equal to 1 — not to 0! — expresses as the fact that the real CR-transversal vector field: T := i

• L1, L 1
• = i
• L1
• ❆1

− L 1

• ❆1 ∂

∂v =: ℓ ∂ ∂v, has nowhere vanishing real coefficient: ℓ := i

• ❆1

z1 + ❆1 ❆1 v − ❆1 z1 − ❆1 ❆1 v

• = 0.

The Levi kernel bundle K1,0M ⊂ T 1,0M is then generated by: K := ❦ L1 + L2,

where: ❦ := − L2

• ❆1

− L 1

• ❆2

L1

• ❆1

− L 1

• ❆1

is the fundamental slant function. As is known, the hypothesis of 2- nondegeneracy is then equivalent to the nonvanishing: 0 = L 1(❦). Also, the conjugate field K generates the conjugate Levi kernel bundle K0,1M ⊂ T 0,1M. There also is a second fundamental function: P := ℓz1 + ❆1 ℓv − ℓ ❆1

v

ℓ .

• Pocchiola, Ph.D., Orsay 2014: Pocchiola conducted the Cartan equivalence

method for such M5 ∈ C2,1 under general (local) biholomorphic transforma-

• tions. Reduction to an explicit {e}-structure was later done [Foo-Merker-2019].

38

Introducing the five 1-forms: ρ0 = dv − ❆1dz1 − ❆2dz2 − ❆1dz1 − ❆2dz2 ℓ , κ0 = dz1 − ❦ dz2, ζ0 = dz2, κ0 = dz1 − ❦ dz2, ζ0 = dz2, after very, very intensive computations, redone manually by Foo-Merker all along ∼ 50 pages, Pocchiola obtained modifications

• ρ, κ, ζ, κ, ζ
• f these 1-

forms

• ρ0, κ0, ζ0, κ0, ζ0
• , together with four complicated 1-forms π1, π2, π1,

π2 which satisfy structure equations of the specific concise shape: dρ =

• π1 + π1

∧ ρ + i κ ∧ κ, dκ = π2 ∧ ρ + π1 ∧ κ + ζ ∧ κ, dζ =

• π1 − π1

∧ ζ + i π2 ∧ κ + + ❘ ρ ∧ ζ + i 1 c3 ❏0 ρ ∧ κ + 1 c ❲0 κ ∧ ζ,

in which ❘ is a secondary invariant: ❘ := Re

• i e

cc ❲0 + 1 cc

• − i

2 L 1

• ❲0
• + i

2

• − 1

3 L 1

• L 1(❦)
• L 1(❦)

+ 1 3 P

• ❲0
• ,

expressed in terms of Pocchiola’s two primary invariants whose explicit ex- pressions have been confirmed by Alexander Isaev assuming M is rigid: ❲0 := − 1 3 K

• L 1
• L 1(❦)
• L 1(❦)2

+ 1 3 K

• L 1(❦)
• L 1
• L 1(❦)
• L 1(❦)3

+ + 2 3 L1

• L1(❦)
• L1(❦)

+ 2 3 L1

• L 1(❦)
• L 1(❦)

+ i 3 T (❦) L 1(❦), ❏0 := 1 6 L 1

• L 1
• L 1
• L 1(❦)
• L 1(❦)

− 5 6 L 1

• L 1
• L 1(❦)
• L 1
• L 1(❦)
• L 1(❦)2

− 1 6 L 1

• L

❦ L ❦ P + 20 27 L 1

• L 1(❦)

3 L 1(❦)3 + 5 18 L 1

• L 1(❦)

2 L 1(❦)2 P + 1 6 L 1

• L 1(❦)
• L 1
• P
• L 1(❦)

− 1 9 L 1

• ❦

L ❦ P P − 1 6 L 1

• L 1
• P
• + 1

3 L 1

• P
• P − 2

27 P P P. When M is assumed to be rigid for simplicity, the numerator of ❲0 contains 52 differential monomials. When M is not assumed rigid, it contains hundreds

40

• f thousands of differential monomials instead! Furthermore, the numerator of

❏0 is even huger! Thus, as is known, the complexity increases spectacularly from rigid to nonrigid CR manifolds. This justifies, in a way, to devote some mathematical works to rigid or tube CR manifolds, as Alexander Isaev did.

• Theorem. [Isaev 2018] When M is tube, namely when F = F(Re z1, Re z2)

then in terms of: S := ∂ ∂x1 Fx1x2 Fx1

• ,

Pocchiola’s two invariants express as: J = 5 18 (Sx1)2 S2 Fx1x1x1 Fx1x1 + · · · , W = 4 3 Sx1 S + · · · .

• Corollary. Furthermore:

0 ≡ J ≡ W implies: 0 ≡ 9 Fx1x1x1x1x1 Fx1x1 − 45 Fx1x1 Fx1x1x1 Fx1x1x1x1 + 40 F 3

x1x1x1.

Theorem.

[Isaev 2018] More generally, when M is rigid, namely when

F = F(z1, z2, z1, z2) is independent of v = Im w, with: S := ∂ ∂z1 Fz1z2 Fz1z1

• ,

the vanishing: 0 ≡ J ≡ 5 18 (Sz1)2 S2 Sz1z1z1 Sz1z1 + · · · , 0 ≡ W ≡ 2 3 Sz1 S + · · · , implies: 0 ≡ 9 Fz1z1z1z1z1

• Fz1z1

2 − 45 Fz1z1z1z1 Fz1z1z1 Fz1z1 + 40

• Fz1z1z1

3. Lastly, we recall that Cartan adopted Lie’s principle of thought, as we do too, which admits that either a given differential invariant, call it P, is identi- cally zero, or is assumed to be nowhere zero, after restriction to an appropriate

42

• pen subset:

P ≡ 0, P

• P ≡ 0.

Mixed cases where some invariant is nonzero on some nonempty open subset and vanishes on a nonempty closed subset are excluded from exploration. Therefore there is essentially no necessity to set up an {e}-structure when ❲0 ≡ 0 ≡ ❏0, because when either ❲0 ≡ 0, hence ❲0 = 0 after restriction, or ❏0 ≡ 0, hence ❏0 = 0 after restriction, Cartan’s method commands to continue the group parameter normalizations! Pocchiola indeed listened to captain Cartan, and was able to prove the

• Theorem. [Pocchiola 2013] Only two primary invariants, ❲0 and ❏0, occur

for biholomorphic equivalences of C2,1 real analytic hypersurfaces M5 ⊂ C3, and: 0 ≡ ❲0 ≡ ❏0 ⇐ ⇒ M is equivalent to the Gaussier-Merker model. Furthermore, when either ❲0 = 0 or ❏0 = 0, the equivalence problem reduces to a 5-dimensional {e}-structure on M5.

As a corollary known from general Cartan theory, every non-flat M5 ∈ C2,1 has CR automorphisms group of dimension 5. This confirmed the same dimensional gap estimate 10 ↓ 5 obtained by Fels-Kaup [Acta Math.

2008,

• pp. 1–82], who assumed M to be homogeneous from the beginning.

44

PDE System Now we consider a system of two PDEs on the plane associated to Pocchi-

• la’s CR structures:

zy = F(x, y, z, zx, zxx), zxxx = H(x, y, z, zx, zxx). We introduce the standard notation p := zx, q := zy, r := zxx, i.e. we have: zy = F(x, y, z, p, r), zxxx = H(x, y, z, p, r). For this system of equations to be equivalent to a 2-nondegenerate para-CR manifold we have to assume: Fr ≡ 0 and: Fpp = 0.

In addition, this system is of finite type, or, what is the same, its general solu- tion can be written as z = z(x, y; ¯ x, ¯ y, ¯ z), if and only if D3F = ∆H, with: D := ∂x + p∂z + r∂p + H∂r, ∆ := ∂y + F∂z + DF∂p + D2F∂r. Introduce contact 1-forms: λ′ := dz − pdx − Fdy µ1′ := dp − rdx − DFdy µ2′ := dr − Hdx − D2Fdy ν1′ := dx ν2′ := dy. These forms live on a manifold M parameterized by (x, y, z, p, r), which is the 5-dimensional manifold of second jets for functions z = z(x, y). we first use

46

the allowed transformations, in a special form:       λ µ1 µ2 ν1 ν2       :=         −1 1

1 18(2H2 r + 9Hp − 3DHr) 1 3Hr −1

1 Fp

3FppFpppp−5F 2

ppp

18F 2

pp

Fppp 3Fpp FpFppp−3F 2

pp

3Fpp

              λ′ µ1′ µ2′ ν1′ ν2′       , to bring the initial forms (λ′, µ1′, µ2′, ν1′, ν2′) to forms (λ, µ1, µ2, ν1, ν2) satis- fying, in particular, the following normalizations: dλ ∧ λ = µ1 ∧ ν1 ∧ λ dµ1 ∧ λ ∧ µ1 = µ2 ∧ ν1 ∧ λ ∧ µ1 dν1 ∧ λ ∧ ν1 = ν2 ∧ µ1 ∧ λ ∧ ν1.

Then we use the most general forms:        θ1 θ2 θ3 θ4 θ5        =        f1 f2 ρeφ f4 f5 f6 f7 ¯ f2 0 ρe−φ ¯ f4 ¯ f5 ¯ f6 ¯ f7              λ µ1 µ2 ν1 ν2       ,

• n the bundle M × G0 → M, and force them to satisfy nonzero curvature
• equations. In particular we want the forms (θ1, . . . , θ5) to satsify the first five

48

• f these equations:

0 = dθ1 −

• θ2 ∧ θ4 − θ1 ∧ Ω1
• −
• 1≤i<j≤5

t1ijθi ∧ θj 0 = dθ2 −

• θ3 ∧ θ4 − 1

2θ2 ∧ (Ω1 + Ω2) − θ1 ∧ Ω3

• −
• 1≤i<j≤5

t2ijθi ∧ θj 0 = dθ3 −

• − θ3 ∧ Ω2 − θ2 ∧ Ω3
• −
• 1≤i<j≤5

t3ijθi ∧ θj 0 = dθ4 −

• − θ2 ∧ θ5 − 1

2θ4 ∧ (Ω1 − Ω2) − θ1 ∧ Ω4

• −
• 1≤i<j≤5

t4ijθi ∧ θj 0 = dθ5 −

• θ5 ∧ Ω2 + θ4 ∧ Ω4
• −
• 1≤i<j≤5

t5ijθi ∧ θj,

with ‘torsions’ tijk as minimal as possible. More precisely, natural normaliza- tions conduct to require: 0 = E1 = dθ1 −

• θ2 ∧ θ4 − θ1 ∧ Ω1
• 0 = E2 = dθ2 −
• θ3 ∧ θ4 − 1

2θ2 ∧ (Ω1 + Ω2) − θ1 ∧ Ω3

• 0 = E3 = dθ3 −
• − θ3 ∧ Ω2 − θ2 ∧ Ω3
• − t313θ1 ∧ θ3 − t314θ1 ∧ θ4 − t323θ2 ∧ θ3

0 = E4 = dθ4 −

• − θ2 ∧ θ5 − 1

2θ4 ∧ (Ω1 − Ω2) − θ1 ∧ Ω4

• 0 = E5 = dθ5 −
• θ5 ∧ Ω2 + θ4 ∧ Ω4
• − t512θ1 ∧ θ2 − t313θ1 ∧ θ5 − t545θ4 ∧ θ5.

Note that we require equality of the coefficients at θ1 ∧ θ3 in dθ3 and at θ1 ∧ θ5 in dθ5.

50

Theorem [Merker-Nurowksi 2018, soon on arxiv] The torsion normalizations equations above define the forms (θ1, . . . , θ5) as:        θ1 θ2 θ3 θ4 θ5        =          ρ2 f ρeφ

f2 2ρ2 feφ ρ

e2φ ¯ f ρe−φ −

¯ f2 2ρ2

−

¯ fe−φ ρ

e−2φ          ·       λ µ1 µ2 ν1 ν2       . The nonvanishing torsions read: t314 = − e3φ

54ρ3I1,

t512 = e−3φ

54ρ3I2,

t323 = e−φ

3ρ I3,

t545 = eφ

3ρI4,

where: I1 = 9D2Hr − 27DHp − 18DHrHr + 18HpHr + 4H3

r + 54Hz,

I2 = 40F 3

ppp − 45FppFpppFpppp + 9F 2 ppFppppp

F 3

pp

, I3 = 2Fppp + FppHrr Fpp , I4 =

• ∆FpppFpp − DFpppFppFp + 3DFppF 2

pp − ∆FppFppp + DFppFpppFp+

DFpFppFppp + 3F 2

ppFpz + FpppFppFz + 2F 3 ppHr

• F −3

pp .

The vanishing or not of each of the quantities I1, I2, I3, I4 is an invariant property of the corresponding para-CR structure. The forms Ω1, Ω2, Ω3,Ω4 are given explicitly in terms of the defining functions and of the para-CR structure, its derivatives, fiber variables (ρ, φ, f2, ¯ f2), and one new real variable, which we call u1.

52

Einstein-Wey Geometry: A Summary In Einstein’s theory, gravity is described in terms of a (pseudo-)riemannian metric g called the gravitational potential. In Maxwell’s theory, the electro- magnetic field is described in terms of a 1-form A called the Maxwell poten- tial. In his attempt Raum, Zeit, Materie (1919) of unifying gravitation and elec- tromagnetism, Weyl was inspired to introduce the synthetic geometric struc- ture on any n-dimensional manifold Mn which consists of classes of such pairs

• (g, A)
• under the equivalence relation:
• g, A
• ∼
• g,

A

• holding by definition if and only if there exists a function ϕ: M −

→ R such that: (1) g = e2 ϕ g; (2) A = A + dϕ. Clearly, the electromagnetic field strength F := dA depends only on the

• class. The signature (k, n − k) of g can be arbitrary. Conformally Einstein

structures from ordinary conformal geometry are a special class of Weyl struc- tures, corresponding to the choice of a closed — hence locally exact — 1- form A. Inspired by Levi-Civita, Weyl established that to such a Weyl structure:

• M, [(g, A)]
• is associated a unique connection ❉ on TM satisfying:

(A) ❉ has no torsion; (B) ❉g = 2 A g for any representative (g, A) of the class [(g, A)]. In any (local) coframe ωµ, µ = 1, . . . , n, for the cotangent bundle T ∗M in which g = gµν ωµ ⊗ ων, the connection 1-forms Γµν of ❉, or equivalently the Γµν := gµρ Γρν, are indeed uniquely defined from the more explict conditions: (A’) dωµ + Γµν ∧ ων = 0; (B’) ❉gµν := dgµν − Γµν − Γνµ = 2 A gµν. Then the curvature of this Weyl connection identifies with the collection of n2 curvature 2-forms: Ωµν := dΓµν + Γµρ ∧ Γρν,

54

which produce the curvature tensor Rµνρσ by expanding in the given coframe ωµ: Ωµν = 1

2 Rµνρσ ωρ ∧ ωσ.

It turns out that Rµνρσ is a tensor density, which means in particular that its vanishing is independent of the choice of a representative (g, A), and hence as such, serves as a starting point for all invariants of a Weyl geometry

• M, [(g, A)]
• , produced by covariant differentiation.

Other invariant objects are:

• the (Weyl-)Ricci tensor Rµν := Rρµρν;
• its symmetric part R(µν) := 1

2

• Rµν + Rνµ
• ;
• its antisymmetric part R[µν] := 1

2

• Rµν − Rνµ
• .

In particular, an appropriately contracted Bianchi identity shows that in 3-dimensions: R[µν] = − 3

2 Fµν,

where F = dA =: 1

2 Fµν ωµ ∧ ων.

In 1943, Élie Cartan proposed dynamical Einstein equations for a Weyl geometry

• M, [(g, A)]
• postulating that the trace-free part of the symmetric

Ricci tensor vanishes: R(µν) − 1

n R gµν = 0,

where R := gµν Rµν, with gµρ gρν = δµν and n = dim M. These equations are called Einstein-Weyl equations, and a Weyl geometry satisfying them is called an Einstein-Weyl structure. The reason for this name is as follows. Since a Weyl structure

• M, [g, A]
• with vanishing F = dA ≡ 0 is equiva-

lent to a plain (pseudo-)conformal structure (M, [g]) and since the Weyl con- nection ❉ then reduces to the Levi-Civita connection, these equations are a natural generalization of Einstein’s field equations. According to Weyl’s ap- proach, a gravity potential g is thereby coupled with an electromagnetic field F = dA.

56

Third-Order ODEs Modulo Point Transformations of Variables It was Cartan who solved the equivalence problem for 3rd order ODEs considered modulo point transformations. Nowadays, the result may be stated more elegantly in terms of a certain Cartan connection, as follows. To any 3rd order ODE: y′′′ = H

• x, y, y′, y′′

, (0.4)

• ne associates a contact-like coframe on the space J4 ∋ (x, y, p, q) of 2-jets of

graphs x − → y(x):            ω1 := dy − p dx, ω2 := dx, ω3 := dp − q dx, ω4 := dq − H(x, y, p, q) dx. (0.5) It follows that if a 3rd order ODE undergoes a point transformation of vari- ables: (x, y) − →

• x, y
• =
• x(x, y), y(x, y)
• ,

then the 1-forms

• ω1, ω2, ω3, ω4

transform as:      ω1 ω2 ω3 ω4      − →     u1 0 u2 u3 0 u4 0 u5 0 u6 0 u7 u8          ω1 ω2 ω3 ω4      =:      θ1 θ2 θ3 θ4      , (0.6) where the ui are certain functions on J4. Actually, Cartan assures us that the entire equivalence problem for 3rd order ODEs considered modulo point transformations of variables is the same as the equivalence problem for 1-forms, considered modulo transformations. There is a unique way of reducing these eight group parameters ui to only three u3, u5, u7, the other ones being expressed in terms of them. This is achieved by forcing the exterior differentials of the θµ’s to satisfy the EDS below. Theorem.

[Cartan 1941, Godlinski-Nurowski-2009] A 3rd order ODE y′′′ =

H(x, y, y′, y′′) with its associated 1-forms: ω1 = dy − p dx, ω2 = dx, ω3 = dp − q dx, ω4 = dq − H(x, y, p, q) dx,

58

uniquely defines a 7-dimensional fiber bundle P7 − → J4 over the space of second jets J4 ∋ (x, y, p, q) and a unique coframe

• θ1, θ2, θ3, θ4, Ω1, Ω2, Ω3
• n P7 enjoying structure equations of the

shape:                      dθ1 = Ω1 ∧ θ1 − θ2 ∧ θ3, dθ2 = (Ω1 − Ω3) ∧ θ2 + ❇1 θ1 ∧ θ3 − ❇2 θ1 ∧ θ4, dθ3 = Ω2 ∧ θ1 + Ω3 ∧ θ3 + θ2 ∧ θ4, dθ4 = (2Ω3 − Ω1) ∧ θ4 − Ω2 ∧ θ3 − ❆1 θ1 ∧ θ2, dΩ1 = Ω2 ∧ θ2 + (❆2 − 2❈1) θ1 ∧ θ2 + (3❇3 + ❊1) θ1 ∧ θ3 + (2❇1 − 3❇4) θ1 ∧ θ4 + ❇2 θ3 ∧ θ4, dΩ2 = Ω2 ∧ (Ω1 − Ω3) − ❆3 θ1 ∧ θ2 + ❊2 θ1 ∧ θ3 − (❇3 + ❊1) θ1 ∧ θ4 + ❈1 θ2 ∧ θ3 + (❇1 − 2❇4) θ3 ∧ θ4, dΩ3 = − ❈1 θ1 ∧ θ2 + (2❇3 + ❊1) θ1 ∧ θ3 + 2 (❇1 − ❇4) θ1 ∧ θ4 + 2❇2 θ3 ∧ θ4.

Moreover, two equations y′′′ = H(x, y, y′, y′′) and y′′′ = H

• x, y, y′, y′

are locally point equivalent if and only if there exists a local bundle iso- morphism Φ: P7

∼

− → P 7 between the corresponding bundles P7 − → J4 and P 7 − → J4 satisfying: Φ∗ θµ = θµ and Φ∗ Ωi = Ωi

(µ = 1, 2, 3, 4; i = 1, 2, 3).

Exactly 3 (boxed) invariants are primary: ❆1, ❇1, ❈1, while others express in terms of them and their covariant derivatives. Point equiva- lence to y′′′ = 0 is characterized by 0 ≡ ❆1 ≡ ❇1 ≡ ❈1. Two relevant explicit expressions are: ❆1 =

1 54 u3

3

u3

1

❲, ❈1 =

1 54 u3 u2

1

• ❈ + 1

27 ❲q

• .

60

The seven 1-forms

• θ1, θ2, θ3, θ4, Ω1, Ω2, Ω3
• set up a Cartan connec-

tion ω on P7 via:

• ω :=

          

1 2Ω1 1 2Ω2

−θ2 Ω3 − 1

2Ω1

θ3 −θ4

1 2Ω1 − Ω3 −1 2Ω2

2θ1 θ3 θ2 −1

2Ω1

           , and the structure equations are just the equations for the curvature K

• f this connection:

d ω + ω ∧ ω =: K.

• Now, the structure equations guarantee that the bundle P7 is foliated by a

4-dimensional distribution annihilating the three 1-forms

• θ1, θ3, θ4

, and that the leaf space M3 of this foliation is equipped with a natural Weyl geometry, if and only if two among three primary invariants vanish identically: 0 ≡ ❆1(H) ≡ ❈1(H).

A representative (g, A) of the concerned Weyl class

• (g, A)
• n M3 has

then the signature (2, 1) symmetric bilinear form: g := θ3 ⊗ θ3 + θ1 ⊗ θ4 + θ4 ⊗ θ1, which is obtained as the determinant of the lower-left 2 × 2 submatrix of the connection matrix ω, while the 1-form is defined as: A := Ω3. It is thanks to the hypothesis ❆1 ≡ 0 ≡ ❈1 that g and A, originally defined on P7, descend on M3. Furthermore, it is the result of Cartan that any such Weyl geometry

• (g, A)
• defined on such a leaf space M3 is automatically Einstein-Weyl!

We stress that given H = H(x, y, p, q) satisfying ❆1 ≡ 0 ≡ ❈1, or equiva- lently: ❲(H) ≡ 0 ≡ ❈(H),

• ne can in principle set up explicit formulae for the corresponding forms θ1,

θ3, θ4, Ω3 on P7, and this in turn can provide explicit formulae for (g, A) on

• M3. However, one substantial obstacle is the:
• Question: How to solve ❲(H) ≡ 0 ≡ ❈(H)?

62

PDE on the Plane zy = F(x, y, z, zx) Modulo Point Transformations Hill-Nurowski (2010) showed that the equivalence problem for 3rd-order ODEs considered modulo point transformations of variables is in one-to-

• ne correspondence with the equivalence problem for 4-dimensional para-CR

structures of type (1, 1, 2). This thus suggests a new approach for constructing Lorentzian Einstein-Weyl structures via para-CR structures of type (1, 1, 2). Instead of working with general para-CR structures of type (1, 1, 2), we will concentrate on a subclass determined in the following way. We start with a class of PDES of the form: zy = F(x, y, z, zx), considered modulo point transformations, for an unknown function z = z(x, y). We then ask when this class defines a para-CR structure of type (1, 1, 2). To answer this, we need a little preparation. Using the abbreviation zx =: p, we indeed consider such PDEs modulo point transformations of variables: (x, y, z) − → (x, y, z) =

• x(x, y, z), y(x, y, z), z(x, y, z)
• .

This conducts to an equivalence problem for the four 1-forms:            ω1

0 := dz − p dx − F(x, y, z, p) dy,

ω2

0 := dp,

ω3

0 := dx,

ω4

0 := dy,

given up to transformations:      ω1 ω2 ω3 ω4      − →     u1 0 u2 u3 0 u4 0 u5 u6 u7 0 u8 u9          ω1 ω2 ω3 ω4      . (0.7) Within this coframe

• ω1

0, ω2 0, ω3 0, ω4

• , in terms of the two operators:

D := ∂x + p ∂z and ∆ := ∂y + F ∂z, the exterior differential of any function F = F(x, y, z, p) rewrites as: dF = Fz ω1

0 + Fp ω2 0 + DF ω3 0 + ∆F ω4 0.

• Proposition. The coframe of 1-forms {ω1

0, ω2 0, ω3 0, ω4 0} modulo transfor-

mations (0.7) defines a para-CR structure of type (1, 1, 2) if and only

64

if: 0 ≡ DF = Fx + p Fz.

• Proof. The only nontrivial integrability condition required to constitute a true

para-CR structure comes from: 0 = dω1

0 ∧ ω1 0 ∧ ω2 0 = − DF ω1 0 ∧ ω2 0 ∧ ω3 0 ∧ ω4 0.

• We will now show that for this class of para-CR structures there is an amaz-

ing coincidence between its main invariant, which will happen to be the Monge invariant with respect to p, and the classical Wünschmann invariant of the cor- responding class of 3rd order ODEs modulo point transformations. From now on, we will only consider PDEs zy = F(x, y, z, zx) satisfying DF ≡ 0. Furthermore, we will also assume that another point-invariant con- dition holds: 0 = Fpp

(everywhere).

Cartan’s process conducts to choose more convenient representatives of these forms:

ω1 := ω1

0,

ω2 := ω2

0 − ∆FpppFpp − ∆FppFppp + 3 Fp Fpp Fzpp − 3 F 2 pp Fzp − 2 Fp Fppp Fzp

6 F 3

pp

ω1

0,

ω3 := ω3

0 + Fp ω4 0 − 1

3 Fppp Fpp ω1

0,

ω4 := Fpp ω4

0 + 4 F 2 ppp − 3 Fpp Fpppp

18 F 2

pp

ω1

0,

and we will use this choice in the sequel. Using Cartan’s method, it is then straightforward to solve the equivalence problem for point equivalence classes of such PDEs zy = F(x, y, z, zx). The solution is summarized in the following

• Theorem. A PDE system zy = F(x, y, z, zx) satisfying the two point-

invariant conditions: DF ≡ 0 = Fzxzx, with its associated 1-forms ω1, ω2, ω3, ω4 as above, uniquely defines a 7-dimensional principal H3-bundle H3 − → P7 − → J4 over the space of first jets J4 ∋ (x, y, z, p) with the (reduced) structure group H3 consisting

66

• f matrices:

     u3u5 u3 0 −u3u8 0 u5 0 −u3u2

8

2 u5

0 u8 u5

u3

    

(u3 ∈ R∗, u5 ∈ R∗, u8 ∈ R),

together with a unique coframe

• θ1, θ2, θ3, θ4, Ω1, Ω2, Ω3
• n P7 where:

     θ1 θ2 θ3 θ4      :=      u3u5 u3 0 −u3u8 0 u5 0 −u3u2

8

2 u5

0 u8 u5

u3

          ω1 ω2 ω3 ω4      , such that the coframe enjoys precisely the structure equations of 3rd

• rder ODES. This time however, the curvature invariants ❆1, ❆2, ❆3,

❇1, ❇2, ❇3, ❇4, ❈1, ❈2, ❈3, ❊1, ❊2 depend on F = F(x, y, z, p) and its derivatives up to order 6.

Two relevant explicit expressions are: ❆1 = − 1

54 1 u3

3

▼ F 3

pp

, ❈1 = 1

3 1 u2

3 u5

❑ F 5

pp

, where:

▼ := 9 Fppppp F 2

pp − 45 Fpppp Fppp Fpp + 40 F 3 ppp,

❑ := ∆Fppppp F 3

pp − 5 ∆Fpppp F 2 pp Fppp + 12 ∆FpppFppF 2 ppp − 12 ∆FppF 3 ppp

− 4 ∆FpppF 2

ppFpppp + 9 ∆FppFppFpppFpppp − ∆FppF 2 ppFppppp

+ 5 Fp F 3

ppFppppz + 6 F 4 ppFpppz − 20 FpF 2 ppFpppFpppz − 12 F 3 ppFpppFppz

+ 36 FpFppF 2

pppFppz − 12 FpF 2 ppFppppFppz + 8 F 2 ppF 2 pppFpz − 24 FpF 3 pppFpz

− 3 F 3

ppFppppFpz + 18 FpFppFpppFppppFpz − 2 FpF 2 ppFpppppFpz.

Two equations zy = F(x, y, z, zx) and zy = F

• x, y, z, zx
• satisfying

DF = 0 = Fzxzx and DF ≡ 0 = F zxzx are locally point equivalent if and

• nly if there exists a bundle isomorphism Φ: P7

∼

− → P 7 between the corresponding principal bundles H3 − → P7 − → J4 and H3 − → P 7 − → J4 satisfying: Φ∗ θµ = θµ and Φ∗ Ωi = Ωi (µ = 1, 2, 3, 4; i = 1, 2, 3).

68

This theorem shows that the geometry of 3rd order ODEs y′′′ = H(x, y, y′, y′′) considered modulo point transformations of variables is the same as the geometry of PDEs zy = F(x, y, z, zx) with DF ≡ 0 = Fzxzx, also considered modulo point transformations. Thus provided that ▼(F) ≡ 0, there should exist a conformal Lorentzian metric on the leaf space of the integrable distribution in P7 annihilated by

• θ1, θ3, θ4

, and when moreover ❑(F) ≡ 0, all this should produce (new) Einstein-Weyl geometries. Actually, we gain the following

• Theorem. A PDE zy = F(x, y, z, zx) defines a bilinear form

g of signature (+, +, −, 0, 0, 0, 0) on the bundle P7 ∋ (x, y, z, p, u3, u5, u8):

• g = θ3 ⊗ θ3 + θ1 ⊗ θ4 + θ4 ⊗ θ1

= u2

5

9 F 2

pp

• 3 Fpp
• dx + Fp dy
• − Fppp
• dz − p dx − F dy

2 + +

• dz − p dx − F dy

18 F 3

pp dy

+

• 4 F 2

ppp − 3 Fpp Fpppp

dz − p dx − F dy

• ,

degenerate along the rank 4 integrable distribution D4 which is the an- nihilator of θ1, θ3, θ4. The PDE zy = F(x, y, z, zx) also defines the 1-form: Ω3 := rx dx + ry dy + rz dz + 1

3 d

• log
• u3

5 Fpp

• ,

where:

rx := 1 3 F 4

pp

• ∆FpppF 2

pp − ∆FppFppFppp + 3 FpF 2 ppFppz − F 3 ppFpz

− 2 FpFppFpppFpz − ∆FppppF 2

ppp + 3 ∆FpppFppFpppp

− 3 ∆FppF 2

pppp + ∆FppFppFppppp − 4 FpF 2 ppFpppzp

− 2 F 3

ppFppzp + 9 FpFppFpppFppzp + F 2 ppFpppFpzp

− 6 FpF 2

pppFpzp + 2 FpFppFppppFpzp

• ry =

1 3 F 4

pp

• − ∆FppppFF 2

pp + ∆FpppFpF 2 pp − ∆FppF 3 pp + 3 ∆FpppFFppFppp

− ∆FppFpFppFppp − 3 ∆FppFF 2

ppp + ∆FppFFppFpppp − 4 FFpF 2 ppFpppz

+ 3 F 2

p F 2 ppFppz − 2 FF 3 ppFppz + 9 FFpFppFpppFppz − 3 FpF 3 ppFpz

− 2 F 2

p FppFpppFpz + FF 2 ppFpppFpz − 6 FFpF 2 pppFpz + 2 FFpFppFppppFpz

+ 3 F 4

ppFz

• ,

70

rz = 1 3 F 4

pp

• ∆FppppF 2

pp − 3 ∆FpppFppFppp + 3 ∆FppF 2 ppp − ∆FppFppFpppp

+ 4 FpF 2

ppFpppz + 2 F 3 ppFppz − 9 FpFppFpppFppz − F 2 ppFpppFpz

+ 6 FpF 2

pppFpz − 2 FpFppFppppFpz

• .

The degenerate bilinear form g descends to a Lorentzian conformal class [g] on the leaf space M3 of the distribution D4, if and only if the Monge invariant ▼(F) ≡ 0 vanishes identically. When ▼(F) ≡ 0, the local coordinates on M3 are (x, y, z) with the projection: P7 − → M3

• x, y, z, p, u3, u5, u8
• −

→ (x, y, z), and the conformal class [g] has a representative which is explicitly ex- pressed in terms of dx, dy, dz, with coefficients depending only on (x, y, z). Next, Ω3 descends to a 1-form denoted A given up to the differential

• f a function on M3 ∋ (x, y, z), if and only if ❑(F) ≡ 0.

Moreover, the pair

• g, Ω3
• descends to a representative of a

Einstein-Weyl structure

• (g, A)
• n M3, if and only if both ▼(F) ≡ 0

and ❑(F) ≡ 0. Finally, this Weyl structure is actually Einstein-Weyl, namely it sat- isfies the Einstein-Weyl equations, and all Einstein-Weyl structures in 3-dimensions emerge from this construction.

72

Transformation of the Wünschmann Invariant Into the Monge Invariant In particular, PDEs with ❆1 ≡ 0 ≡ ❈1 always define an Einstein-Weyl geometry on the leaf space M3 of the integrable distribution in P7 annihilated by

• θ1, θ3, θ4

. The advantage of looking at this Weyl geometry from the PDE point of view zy = F(x, y, z, zx) rather than from the ODE side y′′′ = H(x, y, y′, y′′), is that now the Wünschmann invariant of the ODE becomes the much simpler and classical Monge invariant: ❆1(H) ∼ ▼(F) = 9 F 2

pp Fppppp − 45 Fpp Fpppp Fppp + 40 F 3 ppp.

Serendipitously, the identical vanishing ▼(F) ≡ 0 is well known to be equivalent to the condition that the graph of p − → F(p) is contained in a conic

• f the (p, F)-plane, with parameters (x, y, z). More precisely:

0 ≡ ▼(F) ⇐ ⇒

A F 2 + 2 B F p + C p2 + 2 K F + 2 L p + M ≡ 0,

(0.8) for some functions A, B, C, K, L, M depending only on (x, y, z).

Thus, passing from the formulation of Einstein-Weyl’s equations in terms

• f a 3rd order ODE y′′′ = H(x, y, y′, y′′) to the — equivalent! — formulation

in terms of a PDE zy = F(x, y, z, zx), we are able to find the general solution to the equation: ❲(H) ≡ 0 ! By replacing ❲(H) ▼(F), the general solution is just conical!

74

Differential Invariants of Parabolic Surfaces

[Joint with Zhangchi Chen]

We consider the special affine group: SA3(R) = SL3(R) ⋉ R3, which consists of invertible linear transformations (x, y, u) − → (s, t, v) cou- pled with translations: s = a x + b y + c u + d, t = k x + l y + m u + n, v = p x + q y + r u + s, 1 =

• a b c

k l m p q r

• ,

preserving volume and orientation. We have: dim SA3(R) = 3 · 3 − 1 + 3 = 11. We will always consider special affine transformations not far from the iden- tity, hence we may view SA3(R) as a local Lie group. The full affine group will be denoted A3(R) = GL3(R) ⋉ R3. In the source space (x, y, u), we consider surfaces S2 ⊂ R2

x,y×R1 u graphed

as

• u = F(x, y)
• with convergent power series F ∈ R{x, y}, and similarly, in

the target space (s, t, v), we consider graphed analytic surfaces

• v = G(s, t)
• :

u =

∞

• j=0

∞

• k=0

Fj,k xj

j! yk k!,

v =

∞

• l=0

∞

• m=0

Gl,m sl

l! tm m!.

• Problem. Determine when two given surfaces
• u = F(x, y)
• and
• v =

G(s, t)

• are SA3-equivalent.

When this holds, by a special affine transformation, every point

• x, y, F(x, y)
• is mapped to a point
• s, t, G(s, t)
• , and a fundamental equation

holds in R{x, y}: p x+q y+r F(x, y)+s ≡ G

• a x+b y+c F(x, y)+d,

k x+l y+m F(x, y)+n

• .

We denote a general element of the special affine group by g ∈ SA3(R), and the general transformation as: s = s

• g, x, y, u
• ,

t = t

• g, x, y, u
• ,

v = v

• g, x, y, u
• .

76

y(g, x)

• y(g, x), v
• g, y(g, x)
• x

Jn

x,u

g· g·

• x, u(x)
• Rp

y

Rp

x

• x, uα(x), uβ

xJ (x)

• y(g, x), vγ

g, y(g, x)

• , vδ

yK

• g, y(g, x)
• Jn

y,v

• Definition. A differential invariant of order n is a function of the horizontal

coordinates and the partial derivatives of the graphing function up to order n: ■

• s, t,
• Gsltm(s, t)
• 0l+mn
• ≡ ■
• x, y,
• Fxjyk(x, y)
• 0j+kn
• ,

which is unchanged after replacement of (s, t, v) in terms of

• g, x, y, u
• , for

every g ∈ SA3(R).

RN−s Rs T n z(n) straightening z(n)

78

z(n)

ρ(z(n)) ∈ G

T n

z(n)

ρ(z(n)) · z(n)

w(n1)

ν1

z(n) T n w(n) g, z(n) w(n)

ν

T n

target

w(nr)

νr

c1 cr

• Problem. Describe the structure of the algebra of differential invariants of

surfaces under the action of SA3(R). To a graphed surface

• u = F(x, y)
• is associated its Hessian matrix:

HessianF = Fxx Fxy Fyx Fyy

• .

80

• Definition. A pseudo-invariant is a function satisfying:

P

• s, t,
• Gsltm(s, t)
• 0l+mn
• ≡ nonzero·P
• x, y,
• Fxjyk(x, y)
• 0j+kn
• ,

with a nowhere vanishing factor, at least when g ∈ SA3(R) is not far from the identity. A starting observation is that the Hessian determinant is a pseudo-invariant: Gss Gtt − G2

st = nonzero ·

• Fxx Fyy − F 2

xy

• ,

even under general affine transformations. Moreover, the rank of the Hessian matrix remains unchanged through any (special) affine transformation. For the general theory of surfaces, this implies an elementary initial branch- ing: HessianF ≡

• {u = 0}
• u = F(x, y)
• rank HessianF ≡ 1
• Root Hypothesis in this Talk

Fxx = 0 ≡ FxxFyy − F 2

xy

rank HessianF ≡ 2

Not treated here.

Geometrically, it is clear that the case where the Hessian matrix is identi- cally zero: 0 ≡ Fxx ≡ Fxy ≡ Fyy, is flat in the proper sense, hence there exists a special affine transformation which maps any such

• u = F(x, y)
• to a reference plane {v = 0}. This

branch is hence trivial. The rank 2 case is a wide story in itself, it conducts to the so-called Pick invariant, of order 3, and to further order 4 differential invariants. In this talk, we will study the middle branch only. After a rotation in the (x, y) space, we can assume that Fxx(x, y) = 0 is nowhere vanishing (our reasonings are local). Then our main root hypothesis will constantly be: Fxx = 0 ≡ Fxx Fyy − F 2

xy.

82

(1,0)(2,0) (0,0) (0,1)(1,1) (n−1,1) (n,0)

j+k=n

k j

Solving: Fyy ≡ F 2

xy

Fxx , we may differentiate once: Fxyy = 2 Fxy Fxxy Fxx − F 2

xy Fxxx

F 2

xx

, Fyyy = 3 F 2

xy Fxxy

F 2

xx

− 2 F 3

xy Fxxx

F 3

xx

, and so on.

(0,1)(1,1)(2,1)(3,1) (1,0)(2,0) (0,0) (3,0)(4,0) k+l=4

k l

(1,1) (1,0)(2,0) (0,0) (3,0)(4,0) (0,1)

k l

(1,0)(2,0) (0,0) (3,0)(4,0) (1,1) (0,1) (2,1)

k l

84

It is easy to convince oneself that every partial derivative Fxjyk with k 2 expresses in terms of the partial derivatives:

• Fxj′
• j′j+k,
• Fxj′′y
• j′′j+k−1.

This conducts us to introduce the parabolic jet spaces of any order n 2: PJn

2,1 ∋

• x, y,

uy, ... ... uxn−1y, u, ux, ..., uxn−1, uxn

• ∈ R3+2n.

In effective differential invariant theory, for instance in the case of (not necessarily parabolic) surfaces, under any action of a (local) Lie group G, certain pseudo-invariants are encountered, call them: P = P

• x, y, u,
• uxjyk
• 1j+kn
• ,

◗, ❘, . . . According to the Definition, their zero-sets

• P = 0
• ,
• ◗ = 0
• , . . . , are invari-

ant under G. They are responsible for the creation of branches and of further

subbranches: ◗ ≡ 0, P ≡ 0

• ◗ = 0,

(∗)

• P = 0
• ❘ ≡ 0,

❘ = 0. We adopt Lie’s principle of thought, which admits that either a (pseudo- )differential invariant is identically zero, or it is assumed to be nowhere zero, after restriction to an appropriate open subset. Mixed cases where some (pseudo-)invariant is nonzero on some nonempty open subset and vanishes

• n a nonempty closed subset are excluded from exploration.

Importantly, as soon as some (pseudo-)invariant vanishes identically, like

• ur Hessian determinant:

❍F := Fxx Fyy − F 2

xy,

86

• ne must express all differential consequences of this assumption in order to

explore properly the concerned branch. When, on some (sub)branch, there

• ccurs a simultaneous vanishing of two or more (pseudo)invariants, one must

at first express the differential consequences under a closed workable form, like setting up a meaningful Gröbner basis for the differential ideal generated. We can now start to present our results. At first, if we abbreviate: root := 0 = Fxx 0 ≡ Fxx Fyy − F 2

xy

= Fxx = 0 ≡ ❍F the branching diagram which summarizes everything is:

P ≡ 0 ❈ ≡ 0 ❙ ≡ 0

• P = 0
• ❈ = 0

root

• ❙ = 0
• ❲ ≡ 0
• ❳ ≡ 0

❳ = 0

• ❨ ≡ 0

❲ = 0

• ▼ ≡ 0

❨ = 0 ▼ = 0

This tree decomposes in 3 main branches, extracted in three diagrams below, just before the statements of 3 associated theorems.

88

In the first, top branch, ❙ and P are pseudo-invariants: ❙ := Fxx Fxxy − Fxy Fxxx F 2

xx

, P := 1 3 − 5 F 3

xxx + 3 Fxx Fxxxx

F 2

xx

, while ❈ is a differential invariant: ❈ := 1 √ 3 9 F 2

xx Fxxxxx − 45 Fxx Fxxx Fxxxx + 40 F 3 xxx

• ± 3 Fxx Fxxxx ∓ 5 F 3

xxx

3/2 . In the second, middle branch, ❲ is a differential invariant, but it is assumed to vanish identically, hence it is trivial, and further, ❳ and ❨ are differential invariants: ❳ := 1 9

• uxx uxxy − uxy uxxx

9 u2

xx uxxxxx − 45 uxx uxxx uxxxxx + 40 u3 xxx

• u6

xx

,

❨ := 1 18 (FxxyFxx − FxxxFxy)5/3 Fxx

10

9 FxxxxxFxx

2 − 45 FxxxFxxxxFxx + 40 Fxxx 3

• 11200 Fxxx

8 − 12600 Fxxx 3FxxxxxFxx 3Fxxxx + 13230 FxxxFxxxxxFxx 4Fxxxx 2 + 1134 FxxxFxxxxxFxx 5

− 3150 Fxxx

2FxxxxFxx 4Fxxxxxx − 810 FxxxxxxxFxx 5FxxxFxxxx − 33600 Fxxx 6FxxxxFxx − 7875 Fxxx 2

− 756 Fxxx

2Fxxxxx 2Fxx 4 + 6720 Fxxx 5FxxxxxFxx 2 + 31500 Fxxx 4Fxxxx 2Fxx 2 − 4725 Fxxxx 4Fxx 4

− 189 Fxxxxxx

2Fxx 6 + 1890 Fxxxx 2Fxx 5Fxxxxxx − 2835 FxxxxFxx 5Fxxxxx 2 + 162 FxxxxxxxFxx 6Fxxxxx

+ 720 FxxxxxxxFxx

4Fxxx 3

• .

In the third, last, bottom branch, ❲ is a nontrivial differential invariant: ❲ := F 2

xx Fxxxy − Fxx Fxy Fxxxx + 2 Fxy F 2 xxx − 2 Fxx Fxxx Fxxy

(Fxx)2 Fxx Fxxy − Fxy Fxxx 3/2 ,

90

and ▼ is also a differential invariant:

▼ := 1 36 1 Fxx

6 (− FxxyFxx + FxxxFxy)

• FxxFxxxxFxy − Fxx

2Fxxxy + 2 FxxxFxxyFxx − 2 Fxxx 2Fxy

• − 1280 Fxxx

7Fxy 3 + 270 Fxx 6FxxxxyFxxy 2Fxxxx − 72 FxxxFxxxxxFxx 5Fxxy 3 + 820 FxxxFxx 3Fxxxx 3Fxy 3

− 2195 Fxxx

3Fxx 2Fxxxx 2Fxy 3 + 2560 Fxxx 5FxxFxxxxFxy 3 + 2000 Fxxx 2Fxx 5Fxxxy 2Fxxy − 2000 Fxxx 3Fxx 4Fxxxy 2Fxy

− 3040 Fxxx

3Fxx 4FxxxyFxxy 2 − 3040 Fxxx 5Fxx 2FxxxyFxy 2 − 3840 Fxxx 5Fxxy 2Fxx 2Fxy + 3840 Fxxx 6FxxyFxxFxy 2

− 420 Fxxxx

3Fxx 4FxxyFxy 2 + 480 FxxxxFxx 4Fxxy 3Fxxx 2 − 420 FxxyFxx 6FxxxxFxxxy 2 + 192 Fxxx 4Fxx 2FxxxxxFxy 3

− 120 Fxxx

2Fxx 5FxxxxyFxxy 2 − 120 Fxxx 4Fxx 3FxxxxyFxy 2 + 36 FxxxxxFxx 6Fxxy 2Fxxxy + 45 Fxx 6Fxxxxy 2FxxxFxy

− 45 Fxx

5Fxxxxx 2Fxy 2Fxxy + 45 Fxx 4Fxxxxx 2Fxy 3Fxxx − 120 Fxx 4FxxxxxFxy 3Fxxxx 2 − 120 Fxx 6FxxxxxFxyFxxxy 2

+ 120 Fxx

5FxxxxyFxxxx 2Fxy 2 − 45 Fxx 7Fxxxxy 2Fxxy − 405 Fxxy 3Fxx 5Fxxxx 2 + 1280 Fxxx 4Fxxy 3Fxx 3

+ 120 Fxx

7FxxxxyFxxxy 2 − 400 FxxxFxx 6Fxxxy 3 + 432 FxxxFxx 4FxxxxFxy 2FxxxxxFxxy

− 360 FxxxFxx

5FxxxxFxyFxxxxyFxxy + 108 FxxxFxx 5FxxxyFxxxxxFxyFxxy − 2040 FxxxFxx 4Fxxxx 2Fxy 2Fxxxy

+ 1985 Fxxx

2Fxx 3Fxxxx 2Fxy 2Fxxy + 1620 FxxxFxx 5FxxxxFxyFxxxy 2 + 4600 Fxxx 3Fxx 3FxxxxFxy 2Fxxxy

+ 1600 Fxxx

3Fxx 3FxxxxFxyFxxy 2 − 4640 Fxxx 4Fxx 2FxxxxFxy 2Fxxy + 6080 Fxxx 4Fxx 3FxxxyFxxyFxy

+ 840 Fxxxx

2Fxx 5FxxyFxyFxxxy + 615 Fxxxx 2Fxx 4Fxxy 2FxyFxxx + 600 FxxxxFxx 5Fxxy 2FxxxyFxxx

+ 336 Fxxx

2Fxx 4FxxxxxFxyFxxy 2 − 456 Fxxx 3Fxx 3FxxxxxFxy 2Fxxy − 126 Fxxx 2Fxx 3FxxxxFxy 3Fxxxxx

+ 90 Fxxx

2Fxx 4FxxxxFxy 2Fxxxxy − 144 Fxxx 2Fxx 4FxxxyFxxxxxFxy 2 − 306 Fxx 5FxxxxxFxyFxxxxFxxy 2

+ 240 Fxxx

3Fxx 4FxxxxyFxxyFxy − 180 FxxxFxx 6FxxxyFxxxxyFxxy + 180 Fxxx 2Fxx 5FxxxyFxxxxyFxy

+ 90 Fxx

6FxxxxxFxyFxxxxyFxxy − 90 Fxx 5FxxxxxFxy 2FxxxxyFxxx + 240 Fxx 5FxxxxxFxy 2FxxxxFxxxy

− 240 Fxx

6FxxxxyFxxxxFxyFxxxy − 5200 Fxxx 2Fxx 4FxxxxFxyFxxxyFxxy

• .

It is important to show these invariants, because they not only cause the branchings, but also, they will constitute generating collections for the full algebras of differential invariants. We may now state our results for the three kinds of branches. We always start from our root assumption. P ≡ 0 ❈ ≡ 0 ❙ ≡ 0

• P = 0
• ❈ = 0

Fxx = 0 ≡ ❍F

• The full affine group in two dimensions is A2(R) = GL2(R) ⋉ R2.
• Theorem. Within the branch ❙ ≡ 0:

(1) Every surface S2 ⊂ R3 is special affinely equivalent to the product

• f a curve in R1+1

x,u times R1 y, and SA3(R)-equivalences amount to A2(R)-

equivalences of such curves; (2) There is a pseudo-invariant P of order 4;

92

(3) When P ≡ 0, the surface is SA3-equivalent to

• u = x2

, the product

• f a parabola times R1

y, and conversely;

(4) When P = 0, the surface is, in a unique way, SA3-equivalent to: u = x2

2! ± x4 4! + F5,0 x5 5! +

• j6

Fj,0 xj

j! ,

and the collection of coefficients F5,0,

• Fj,0
• j6 is in one-to-one corre-

spondence with equivalent classes. Here: F5,0 = Fxxxxx(0) = value of ❈ at the origin. Infinitely many differential invariants correspond to these coefficients Fj,0, as we will soon explain.

• Question. How to compute explicitly differential invariants?

It is clear that SA3(R) contains all translations: s = x + d, t = y + n, v = u + s.

This implies — exercise — that every differential invariant: ■

• x, y, u

absent

,

• uxjyk
• 1j+kn
• ,

must depend only on jet derivatives of order 1. To compute these invariants ■, we start from a power series at the origin: u =

• j+k1

Fj,k xj j! yk k!, and we progressively perform (several) ‘simple’, ‘natural’, special affine trans- formations in order to annihilate

• normalize as much as possible Taylor coef-

ficients Fj,k. One main feature of the process is its progressivity. At the end, we reach a certain ‘normal form’: v =

• l+m1

Gl,m sl l! tm m!, in which several coefficients Gl,m are ‘simplified’, for instance as above: G1,0 = G0,1 = 0, G2,0 = 1, G3,0 = 0, etc.

94

Certainly, the full composition of all the progressively normalizing maps be- longs to SA3(R), hence is of the form above for some specific constants a, . . . ,

• s. These constants are complicated at the end, but step-by-step they are simple,
• nly the full composition of normalizing maps creates complexity.

After the process is pushed at its farthest point, the identity map of SA3(R) is the only transformation which leaves untouched the ‘normal form’ of the power series v = G(s, t). While normalizing low order Taylor coefficients, we also keep track (on a computer) of the way how the other (higher order) Taylor coefficients are

• modified. At the end, we receive formulas:

Gl,m = Πl,m

• Fj,k
• 1j+kl+m
• (l + m 1).

Then granted that: Fj,k = uxjyk(0, 0), all the desired genuine differential invariants are obtained simply by replacing in these formulas Taylor coefficients by jet coordinates: ■l,m := Πl,m

• uxjyk(x, y)
• 1j+kl+m
• (l + m 1).

Importantly, the hypothesis that the group contains all translations guarantees that we obtain the expressions of all differential invariants at every point (x, y) near the origin. This process could be explained abstractly in any dimension (forthcoming). During normalizations, pseudo-invariants play a crucial role.

• Observation. Any (pseudo)invariant P:
• Either creates a new branch P ≡ 0 to be explored farther;
• or is absorbed, when P = 0, into some normalization.

This is, for instance, true of ❙, P, ❲, ❳: when they are nonzero, they will be used to normalize some Taylor coefficients.

• Theorem. With the assumption ❙ = 0, there is exactly one differential

invariant of fourth order, ❲.

96

We can now state our second result for the second, middle branch. (The third, bottom branch will also assume ❙ = 0.) Fxx = 0 ≡ ❍F

❙ = 0 ❲ ≡ 0

• ❳ ≡ 0

❳ = 0

• ❨ ≡ 0

❨ = 0

• Theorem. Within the branch ❙ = 0, ❲ ≡ 0:

(1) There is a single pseudo-invariant, ❳, of order 5;

x y y

(2) When ❳ ≡ 0, every surface S2 ⊂ R3 is SA3-equivalent to the model: u = 1 2 x2 1 − y = x2

2 + x2 y 2 + x2 y2 2

+ x2 y3

2

+ · · · + x2 yk

2

+ · · · ; (3) When ❳ = 0, every surface is SA3-equivalent to: = x2

2 + x2 y 2 + x2 y2 2

+ F5,0 x5 120 + x2 y3

2

+ 4 F5,0 x5y 120 + x2 y4

2

+ F7,0 x7 5 040 + x2 y5

2

+ 20 F5,0 +

• j+k8

with: F5,0 = value of ❳ at the origin, F7,0 = value of ❨ at the origin; (4) The collection of coefficients F5,0, F7,0,

• Fj,0
• j8 is in one-to-one

correspondence with equivalent classes.

98

Lastly, we treat the main (thickest) branch: Fxx = 0 ≡ ❍F

❙ = 0

• ❲ = 0
• ▼ ≡ 0

▼ = 0.

• Theorem. Within the main branch S = 0, ❲ = 0:

(1) There is a single differential invariant ▼, of order 5, differentiably independent of ❲; (2) Every surface S2 ⊂ R3 is SA3-equivalent to: u = x2

2 + x2 y 2 + F3,1

x3 y 6 + x2 y2

2

+ F5,0 x5 120 + 6 F3,1 x3 y2 12 + x2 y3

2

+ +

• j+k6

Fj,k xjyk, with: F3,1 = value of ❲ at the origin, F5,0 = value of ▼ at the origin;

(3) Any other surface

• v = G(s, t)
• within the same branch similarly put

into the form: v = s2

2 + s2 t 2 + G3,1

s3 t 6 + s2 t2

2

+ G5,0 s5 120 + 6 G3,1 s3 t2 12 + s2 t3

2 +

+

• l+m6

Gl,m sltm, is SA3-equivalent to

• u = F(x, y)
• above if and only if all (independent)

Taylor coefficients in the parabolic jet space match: G3,1 = F3,1, G5,0 = F5,0, Gl,0 = Fl,0

(l 6),

Gl,1 = Fl,1

(l 5).

In these three Theorems, there always exist two invariant differential oper- ators D1 and D2 satisfying: Di

• differential invariant
• = differential invariant

(i = 1, 2),

and they are non-commuting, in general. Invariantly derivating an invariant means applying D1 and D2 several times, in any order.

• Theorem. The full algebra of differential invariants under the action of

SA3(R) is minimally generated by:

100

➀ In the branch ❙ ≡ 0, P = 0: ❈ and its invariant derivatives; ➁ In the branch ❙ = 0, ❲ ≡ 0: ❳, ❨ and their invariant derivatives; ➂ In the branch ❙ = 0, ❲ = 0: ❲, ▼ and their invariant derivatives.