geometry of rank 2 distributions and differential equations i:...

Classical heritageGeometry of Monge equationsZoo of underdetermined ODEs

Geometry of rank 2 distributions

and differential equations I:

ODEs

Boris Kruglikov

University of Tromsø, Norway

(based on collaboration with Ian Anderson)

Kyoto-Hiroshima conference F Jan-Feb 2011 Rank 2 distributions � Monge Tanaka nowadays


Darboux integrabilityHilbert-Cartan equationGoursat parabolic equation

Closed form solutions

Solutions to a differential equation E can be parametrized bythe space S of ∴ many functions in ∵ number of arguments.Closed form solutions refer to parametrization of the genericstratum by a differential operator S : S

∼→ Sol(E).

Ex.1: Eikonal equation in 2 + 1-dimension:

x′(t)2 + y′(t)2 = 1 ⇔ dt2 − dx2 − dy2 = 0.

Obvious solution involves one arbitrary function and thequadrature: x =

∫cosφ(t)dt, y =

∫sinφ(t)dt.

But it can also be integrated in the closed form:

t = σ′′(τ)− σ(τ), x = σ′′(τ) cos τ + σ′(τ) sin τ,

y = −σ′′(τ) sin τ + σ′(τ) cos τ.

Eikonal equation in 3 + 1 dim cannot be solved in closed form.




ODE examples

Ex.2: Consider the Monge equation w′(t) = (z′(t))2. The generalsolution depends on 1 function of 1 variable and the form viaquadrature is obvious, but here is the closed form solution:

t = σ′′(τ), w = τ2σ′′(τ)− 2τσ′(τ) + 2σ(τ), z = τσ′′(τ)− σ′(τ)

However the next candidate w′(t) = (z′′(t))2 is no longer integrablein closed form.

The more general problem when a PDE is integrable in closed formvia solutions of a simpler equation (usually ODE) is known as themethod of Darboux. For instance, Liouville equation

uxy = eu =⇒ u = log2f ′(x)g′(y)

(f(x) + g(y))2

is Darboux integrable, while sin-Gordon uxy = sinu is not.




PDE examples

Ex.3: The following overdetermined scalar PDE on R2 appears in

[Cartan, 1910]:

uxx = 13λ

3, uxy = 12λ

2, uyy = λ. (†)

It is a compatible involutive system (1 common characteristic). Thegeneral solution is parametrized by 1 function of 1 argument

x = x, y = z′′(t) + xt,

u = xz(t) + z′(t)z′′(t)− 12w(t)− 1

2t z′′(t)2 − 1

2 t2x z′′(t)− 1

6t3x2,

where w′(t) = (z′′(t))2.




100 years ago...

In 1912 D.Hilbert demonstrated that the underdetermined equation(called Hilbert-Cartan equation)

w′ = (z′′)2

cannot be solved in closed form (without quadratures).

In 1914 E.Cartan gave a criterion for resolution of underdeterminedODEs in closed form. Below we associate to this equation a flag ofdistributions and the claim is that its dimensions must grow by 1.

Cartan arguably knew the HC equation in 1910 (even in 1893!), butconcentrated on overdetermined involutive 2nd order systems ofnon-linear PDEs and proved (†) is the most symmetric with thecontact symmetry group G2 of dim= 14.




More history

This was the first (geometric) representation of the exceptional Liegroup G2 from the Killing-Cartan abstract classification.To be more precise the above is the non-compact real G2. Thecompact version was realized in 1914 in another paper by E.Cartan.Namely we have learnt that

G2 = Aut(O) = Aut(R7,×) = Sym(S6, J).

In addition to the beautiful realization of G2 as the group ofsymmetries (which in retrospective turns to be the group of internalsymmetries of the Hilbert-Cartan equation) the paper by 1910contains the astonishing implementation of the equivalence methoddeveloped in 1908, which even today is considered as tour de forceby the practitioners of Cartan’s EDS method.




Goursat representation of involutive systems

General form of overdetermined compatible system of 2nd orderPDEs with a common characteristic is

r + 2λs+ λ2t = 2ψ, s+ λt = ψλ, t = ψλλ,

where we use the classical notations r = uxx, s = uxy, t = uyy andsuppose ψλλλ 6= 0 (nonlinearity).Removing the last equation we obtain a determined parabolic PDEof the 2nd order. It has the largest contact symmetry group forψ = λ3/3!, in which case excluding λ we obtain the equation

4(2s − t2)3 + (3r − 6st+ 2t3)2 = 0. (‡)

This is the Goursat equation with the symmetry group again G2.There is a geometric reduction of the solutions of this equation tothe solutions of the HC equation.




nowadays

Yes another representation of the Goursat equation is possible. Letus consider the 2D tangent cone ρ(λ) + µρ′(λ), where ρ(λ) is thetwisted cubic, and compose the corresponding differential equation

uxx = 13λ

3 + λ2µ, uxy = 12λ

2 + λµ, uyy = λ+ µ.

This is a single 2nd order PDE with double characteristic Dx−λDy.Excluding λ and µ we arrive to equation (‡) with G2 symmetry.

There was no much progress on the higher-dimensional analogs ofthe above results until recent.

In 2006 B.Doubrov and I.Zelenko solved the equivalenceproblem for rank 2 distributions of generic (maximal) class.

In 2009 I.Anderson and B.K. generalized the other – grouptheoretic aspect – of Cartan’s 1910 paper and constructed allmost symmetric models.



SymmetriesTanaka theory of rank 2 distributionsSketch of the proof

Symmetric models

Monge equations are underdetermined ODEs on 2 functionsy(x), z(x)

y(m) = F(x, y, y′, . . . , y(m−1), z, z′, . . . , z(n)

).

We fix 0 < m ≤ n and look for internal symmetries of such E .

Theorem (I. Anderson, BK; 2009)

Among equations E with Fz(n)z(n) 6= 0 the most symmetric areinternally equivalent to the equation

y(m) =(z(n)

)2.

Moreover the symmetry algebra Sym(E) for this equation is

sl2 nsolvalbe of dim = 2n + 5, for m = 1, n > 2;

solvable and dim = 2n+ 4 for m > 1.




External symmetries ...

What is a symmetry? Consider the space of mixed jets

Jm,n(R,R2) ' Rm+n+3(x, y, y1, . . . , ym, z, z1, . . . , zn).

The Cartan distribution is the following canonical rank 3sub-distribution of TJm,n

C = Ann{dyi − yi+1 dx, dzj − zj+1 dx | 1 ≤ i < m, 1 ≤ j < n}.

The Lie transformations are symmetries of the pair (Jm,n, C).

External symmetries are the Lie transformationsΦ : (Jm,n, Cm,n)←↩ preserving the equation E considered as asubmanifold in jets:

E ⊂ Jm,n(R,R2).

Infinitesimally these are Lie vector fields on the ambient mixed jetspace Jm,n(R,R2) tangent to E .




... versus internal symmetries

From internal viewpoint E ' Rm+n+2(x, y, . . . , ym−1, z, . . . , zn)

is a hypersurface ym = F (x, y, . . . , zn) equipped with the inducedrank 2 distribution

∆ = C|E = 〈Dx = ∂x+y1∂y+· · ·+F∂ym−1+z1∂z+· · ·+zn∂zn−1 , ∂zn〉.

Internal symmetries are sym(E) = sym(∆).

A fundamental problem in geometry is the comparison of internaland external geometry (think of Riemannian, symplectic andCauchy-Riemann geometries). For the geometry of differentialequations it corresponds to Lie-Bäcklund type theorems.

The classical Lie-Bäcklund theorem states that a Lie transformationof the jet space Jkπ is the lift of a contact transformation of J1πfor rankπ = 1, or a point transformation of J0π for rankπ > 1.




A Lie-Bäcklund type theorem

Various generalizations of the Lie-Bäcklund theorem to symmetriesof differential equations E ⊂ Jkπ are known (Krasilshchik-Lychagin-Vinogradov, Andersen-Kamran-Olver, Gardner-Kamran etc).

For the mixed jet space Jm,n(R,R2) a Lie transformation is theprolongation of a parametrized contact transformation of the spaceJ0,k(R,R2) ' R× Jk(R,R), k = n−m

(y, jkx(z)

)7→

(Y (y, jkx(z)), f (k−1)(jkx(z))

).


For Monge equations E with Fz(n)z(n) 6= 0, 0 < m ≤ n and (m,n)being different from the exceptional cases (1, 1) and (1, 2), thealgebras of internal and external symmetries coincide.




GNLA of a distribution

There are several ingredients in our approach but the most essentialis the Tanaka theory applied to rank 2 distributions.

A distribution is a subbundle ∆ ⊂ TM (also a Pfaffian systemAnn ∆ ⊂ T ∗M). The (weak) derived flag {∆i} is given via themodule of its sections by Γ(∆i+1) = [Γ(∆),Γ(∆i)] with ∆1 = ∆.

The quotient sheaf m = ⊕i<0 gi, gi = ∆−i/∆−i−1

has a natural structure of a graded nilpotent Lie algebra (GNLA) atany point x ∈M . The bracket on m is induced by the commutatorof vector fields on M . The growth vector of ∆ = g−1 is thesequence (dim g−1,dim g−2,dim g−3 . . . ).




Tanaka algebra

The Tanaka prolongation g = m is the graded Lie algebra

g = g−κ ⊕ · · · ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ . . .

with the negative graded part m and

gi = H1i (m,m⊕ g0 ⊕ . . . gi−1) for i ≥ 0.

If g = gx is independent of the point on M , then sym(∆) ismajorized by g [Tanaka, 1970]. We generalize this to

Theorem (BK, 2010)

dim sym(∆) ≤ supx∈M dim gx and the equality is attained only inthe case of flat structure, i.e. when ∆ is the left-invariantdistribution on the Lie group exp(m).




Stages of the proof

1. Consider the following basis of the vector fields:

e1 = −Dx, e′1 = ∂zn , e′i = Fznzn∂ym−i+2 (2 < i ≤ m+ 2),

ej = ∂zn−j+1 + Fzn∂ym−j+1 + cj∂ym−j+2 (2 ≤ j ≤ m+ 2),

ek = ∂zn−k+1(m+ 2 < k ≤ n+ 1)

The commutator table for m is (we show the case m < n− 1):

e1 e′1 e2 e3 e′3 e4 e′4 . . . em+2 e′m+2 em+3 em+4 ...

e1 e2 e3 e4 e′4 e5 e′5 . . . em+3 em+4 em+5 ...e′1 e′3 e′4 e′5 . . .e2...

The GNLA structure is: deg ei = −i = deg e′i.




2. Then we calculate the Tanaka prolongation.For n > 2, m = 1 we get the following string of dimensions:

(1(−n−1), . . . , 1(−4), 2(−3), 1(−2), 2(−1), 3(0), 2(1), 1(2), . . . , 1(n−2)).

For n > m > 1 we get:

(1(−n−1) . . . 1(−m−3), 2(−m−2) . . . 2(−3), 1(−2), 2(−1), 3(0), 1(1) . . . 1(n−m−1)).

3. Finally we calculate the actual symmetries of the models:

S0 = ∂x, Yi = xi

i! ∂y (0 ≤ i < m), Zj = xj

j! ∂z (0 ≤ j < n)

S1 = x ∂x + (m− 1) y ∂y + (n− 12 ) z ∂z, R = y ∂y + 1

2z ∂z.

Zn+k = 2k∑

p=0

(−1)p(m+p−1

p

)xk−p

(k−p)! zn−m−p ∂y + xn+k

(n+k)! ∂z

S2 = x2∂x + (nzn−1)2∂y + (2n − 1)xz ∂z [for m = 1],

and use the (generalized) Tanaka theorem.Kyoto-Hiroshima conference F Jan-Feb 2011 Rank 2 distributions � Monge Tanaka nowadays



Remark on prolongations

Let us understand what happens to theTanaka algebra upon prolongation ofequation/distribution (sym(E) not changed)on the example of the HC equation.

Its Tanaka algebra g is

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

The first prolongation gives the following graded Lie algebra

(1(−5), 1(−4), 1(−3), 1(−2), 2(−1), 2(0), 2(1), 1(2), 1(3), 1(4), 1(5)).

The next prolongation yields the infinite contact Lie algebra

(1(−6), 1(−5), 1(−4), 1(−3), 1(−2), 2(−1), 3(0), 3(1), 3(2), 3(3), 4(4), . . . ).



Monge equations with Tanaka flat modelsNon-degeneracy, realization and linearityOpen problems

Central extensions

The standard model of Tanaka provides a way to construct flatrank 2 distributions on manifolds. We realize them asunderdetermined ODE systems by relating the algebraic theory ofcentral extensions to the geometric theory of integrable extensions.

A d-dimensional central extension m of a Lie algebra m via ad-dimensional Abelian subalgebra a ⊂ Z(m) is defined by the exactsequence

0→ a ↪→ m→ m→ 0.

Let m be a fundamental GNLA with the group of gradingpreserving automorphisms H. Then d-dimensional nontrivial centralextensions of grading −k are bijective with GrdH

2k(m)/H.




Integrable extensions

An integrable extension (covering) E of an equation E is acompatible collection of additional differential relations such thatSol(E) are obtained from Sol(E) by ODEs.


There is a bijective correspondence between nontrivial centrald-dimensional extension of a fundamental GNLA m

of grading < −2 and non-trivial flat integrable extensionsπ : (M , ∆)→ (M,∆) with d-dimensional fibers V (as EDS).We can represent the latter as an underdetermined ODE system:F [x, y, y′, . . . , z, z′, . . . ] = 0, v′ = f(x, y, y′, . . . , z, z′, . . . , v).

In particular, any flat rank 2 distribution with growth vector(2, 1, 2, . . . ) can be represented as a result of successive integrableextensions of the Hilbert-Cartan equation y′ = (z′′)2.




Examples

There are 3 types of fundamental GNLAs with growth (2, 1, 2, 1).The corresponding Tanaka flat underdetermined ODEs are:

E1,3 : y′ = (v′′′)2 (parabolic case)

E2,2 : v′′ = (z′′)2 (hyperbolic case)

y′ = (z′′)2, v′ = z − (z′′)3 (elliptic case)

Eliminating z we can realize the latter as a degenerate Mongeequation

v(3) = ±(y′)1/2 − 34(y′)−1/2(y′′)2 − 3

2(y′)1/2y(3).




Examples

In 7D case there are 8 different fundamental GNLA withdim g−1 = 2. 4 of them are extensions of the most symmetricparabolic 6D. Here are the corresponding Tanaka flatunderdetermined models:

y′ = (ziv)2.

y′ = (z′′′)2, v′ = (z′′)2.

y′′ = (z′′′)2.

y′ = (z′′′)2, v′ = (z′′′)3.




Non-degeneracy condition

A flat rank 2 distributions is not always realizable as anon-degenerate (single) Monge equation. Example: flat elliptic 6D.More generally, no elliptic (2..6) distribution is realizable as aMonge ODE. On the other hand, a parabolic (2..6) distribution isalways realizable as a Monge ODE y′ = F (x, y, z, z′, z′′, z′′′).For hyperbolic ODEs the only target Monge ODEs can bey′′ = F (x, y, y′, z, z′, z′′). There are 2 differential invariants of thedistribution ∆ that control this realization.

Remark though that if we omit non-degeneracy conditionFznzn 6= 0, then realization as a Monge ODE is always possible, butfor the prolonged distribution which changes the growth from(2, 1, 2, . . . ) to (2, 1, 1, . . . ). This latter is equivalent toquasi-linearity of the equation, and more generally furtherprolongations boil down to linearity in several highest derivatives.




Linearization of the ODE E is equivalent to ∆ being the Goursatdistribution, i.e. the growth for both weak and strong flags is(2, 1, 1, . . . , 1). Only in this case the solutions can be expressed inclosed form, i.e. parametrized by a differential operatorP : C∞(π)→ Sol(E).

Consider an underdetermined operator ∇ : C∞(π)→ C∞(ν). If itis linear and generic, then it is epimorphic and we have an exactsequence for E = Ker(∇).

0→ Sol(E)→ C∞(π)∇→ C∞(ν)→ 0.

By Gromov theorem there exists a right inverse linear differentialoperator �, i.e. ∇ ·� = 1ν . Then P = 1π −� · ∇.

If ∇ is not linear, then existence of right inverse (no longer adifferential operator) follows from the Nash-Moser implicit functiontheorem. No closed form solutions exist. We can either prove theexistence results as in Nash-Green-Gromov theory, or constructalgorithmic solutions as in Laplace-Darboux method.




1. Most symmetric model. Investigation of lower complexitycases and some infinite chains lead to

Conjecture

Equation y′ = (z(n−3))2 yields the most symmetric among rank 2distributions ∆ on an n-dim manifold with dim Sym(∆) <∞.

2. Sub-symmetric cases. Even in low complexity cases not muchis known – Cartan’s 1910 complete treatment of 5 variables has nohigher analogs so far. In 6D we know the maximal model with 11Dsymmetry group, but do not know all sub-symmetric cases (9D?).They are not graded any more as in the Tanaka theory.

3. Differential invariants. The fundamental differential invariantswere found by Doubrov and Zelenko. They are the relativeinvariants for the equivalence problem. Calculation of absolutedifferential invariants, invariant derivatives and differential syzygieswould lead to resolution of Problem 2.


geometry of rank 2 distributions and differential equations i:...

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