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Dierential equations and conformal

structures

Paweª NurowskiInstytut Fizyki TeoretycznejUniwersytet Warszawski

Trieste, 15 December 2006

History

History

• Wünschmann K, (1905) Über Beruhrungsbedingungen beiDierentialgleichungen, Dissertation, Greifswald:

History


? Considered a general 3rd order ODE

y′′′ = F (x, y, y′, y′′), (∗)

asking what one has to assume about F = F (x, y, y′, y′′) to be able todene a null distance between the solutions.

History


? Considered a general 3rd order ODE

y′′′ = F (x, y, y′, y′′), (∗)

asking what one has to assume about F = F (x, y, y′, y′′) to be able todene a null distance between the solutions.

? Denoting by D the total dierential, D = ∂x + p∂y + q∂p + F∂q, wherep = y′, q = y′′, he found that the solution space of (∗) is naturallyequipped with a conformal Lorentzian metric i

Fy + (D − 23Fq) (1

6DFq − 19F

2q − 1

2Fp)︸︷︷︸K

≡ 0. (W )

? The metric reads:

g = [dy−pdx][dq− 13Fqdp+Kdy+(1

3qFq−F−pK)dx]− [dp−qdx]2.

? The metric reads:



? Condition (W ) is invariant with respect to contact transformations ofvariables and contact transformations of the variables result in a conformalchange of the metric.

? The metric reads:




? Wünschman: There is a one-to-one correspondence between equivalenceclasses of 3rd order ODEs satisfying (W ) considered modulo contacttransformations of variables and 3-dimensional Lorentzian conformalgeometries.

? The metric reads:




? Wünschman: There is a one-to-one correspondence between equivalenceclasses of 3rd order ODEs satisfying (W ) considered modulo contacttransformations of variables and 3-dimensional Lorentzian conformalgeometries.

? In particular: all contact invariants of such classes of equations areexpressible in terms of the conformal invariants of the associated conformalLorentzian metrics.

• Chern S S (1940) The geometry of the dierential equationsy′′′ = F (x, y, y′, y′′) Sci. Rep. Nat. Tsing Hua Univ. 4 97-111:


? Description of the invariants in terms of so(2, 3)-valued Cartan connection.


? Description of the invariants in terms of so(2, 3)-valued Cartan connection.? This may be identied with the Cartan normal conformal connection

associated with the conformal class [g].

• Cartan E (1941) La geometria de las ecuaciones diferenciales de tercerorden Rev. Mat. Hispano-Amer. 4 1-31:


? Considered equations (∗) modulo point transformations of the variables.


? Considered equations (∗) modulo point transformations of the variables.? If in addition to the Wünschmann condition (W ) equation (∗) satises

another point invariant condition

D2Fqq −DFqp + Fqy ≡ 0, (C)

then its solutions space is equipped with a 3-dimensional LorentzainEinstein-Weyl geometry.


? Considered equations (∗) modulo point transformations of the variables.? If in addition to the Wünschmann condition (W ) equation (∗) satises

another point invariant condition

D2Fqq −DFqp + Fqy ≡ 0, (C)

then its solutions space is equipped with a 3-dimensional LorentzainEinstein-Weyl geometry.

? There is a one-to-one correspondence between 3-dimensional LorentzianEinstein-Weyl geometries and 3rd order ODEs considered modulo pointtransformations and satisfying conditions (W ) and (C).

• Cartan E (1932) Sur la geometrie pseudo-conforme des hypersurfaces del'espace de deux variables complexes Part I Ann. Math. Pura Appl. 1117-90; Part II Ann. Sc. Norm. Pisa 1 333-54:


? Solved the local equivalence problem for 3-dimensional real hypersurfacesembedded in C2 and considered modulo biholomorphic transformations ofvariables



? If the hypersurface is not locally biholomorphically equivalent to C× R hefound all the invariants in terms of an su(2, 1)-valued Cartan connectionon an 8-dimensional ber bundle dened over the hypersurface.




• Feerman C L (1976) Monge-Ampere equations, the Bergman kernel, andgeometry of pseudoconvex domains Ann. of Math. 103, 395-416:




• Feerman C L (1976) Monge-Ampere equations, the Bergman kernel, andgeometry of pseudoconvex domains Ann. of Math. 103, 395-416:

? Dened a 4-dimensional Lorentzian class of metrics on an S1-bundle overthe hypersurface that transforms conformally when the hypersurfaceudergoes a biholomorphic transformation.

• Burns D Jr, Diederich K, Schnider S (1977) Distinguished curves inpseudoconvex boundariesDuke. Math. J. 44 407-31:


? understood that Cartan's su(2, 1)-valued connection associated with abiholomorphic class of strictly pseudoconvex hypersurfaces is a reduction ofthe Cartan normal conformal connection of the associated conformal classof Feermann metrics.



• Segre B (1931) Intorno al problema di Poincare dela representazionepseudo-conforme Rend. Acc. Lincei 131 676-83:




? biholomorphically equivalent pseudoconvex hypersurfaces in C2 are analogsof 2-nd order ODEs considered modulo point transformations





• Nurowski P, Sparling GAJ (2003) 3-dimensional Cauchy-Riemann structuresand 2nd order ODEs Class. Q. Grav. 20 4995-5016:





• Nurowski P, Sparling GAJ (2003) 3-dimensional Cauchy-Riemann structuresand 2nd order ODEs Class. Q. Grav. 20 4995-5016:

? What are the analogs of the Feerman metrics for 2nd order ODEs modulopoint transformations?

Conformal geometry of y′′ = Q(x, y, y′)


• Given 2nd order ODE: y′′ = Q(x, y, y′) consider a parametrization of therst jet space J1 by (x, y, p = y′).


• Given 2nd order ODE: y′′ = Q(x, y, y′) consider a parametrization of therst jet space J1 by (x, y, p = y′).

• on J1 × R consider a metric

g = 2[(dp−Qdx)dx−(dy−pdx)(dr+23Qpdx+1

6Qpp(dy−pdx))], (F )

where r is a coordinate along R in J1 × R.

Theorem:• If ODE undergoes a point transformation then the metric (F ) transforms

conformally.


conformally.

• The point invariants of a point equivalence class of ODEs y′′ = Q(x, y, y′)are expressible in terms of the conformal invariants of the associatedconformal class of metrics (F ).


conformally.

• The point invariants of a point equivalence class of ODEs y′′ = Q(x, y, y′)are expressible in terms of the conformal invariants of the associatedconformal class of metrics (F ).

• The metrics (F ) are very special among all the split signature metrics on4-manifolds. Their Weyl tensor C has algebraic type (N,N) in theCartan-Petrov-Penrose classication. Both, the selfdual C+ and theantiselfdual C−, parts of C are expressible in terms of only one component.

• C+ is proportional to

w1 = D2Qpp − 4DQpy −DQppQp + 4QpQpy − 3QppQy + 6Qyy

and C− is proportional tow2 = Qpppp,

whereD = ∂x + p∂y + Q∂p.

Each of the conditions w1 = 0 and w2 = 0 is invariant under pointtransformations.

• C+ is proportional to

w1 = D2Qpp − 4DQpy −DQppQp + 4QpQpy − 3QppQy + 6Qyy

and C− is proportional tow2 = Qpppp,

whereD = ∂x + p∂y + Q∂p.

Each of the conditions w1 = 0 and w2 = 0 is invariant under pointtransformations.

• Cartan normal conformal connection associated with any conformal class [g]of metrics (F ) is reducible to a certain SL(2 + 1,R) connection naturallydened on an 8-dimensional bundle over J1. This is uniquely associated withthe point equivalence class of corresponding ODEs via Cartan's equivalencemethod.

• The curvature of this connection has a very simple form0 w2 ∗

0 0 w1

0 0 0

.

• The curvature of this connection has a very simple form0 w2 ∗

0 0 w1

0 0 0

.

• If w1 = 0 or w2 = 0 this connection can be further understood as a Cartannormal projective connection over a certain two dimensional space Sequipped with a projective structure. S can be identied either with thesolution space of the ODE in the w1 = 0 case, or with the solution space ofits dual in the w2 = 0 case.

Conformal geometry of z′ = F (x, y, y′, y′′, z)


• Hilbert D (1912) Über den Begri der Klasse von DierentialgleichungenMathem. Annalen Bd. 73, 95-108:



? considered equations of the form z′ = F (x, y, y′, y′′, z) for two realfunctions y = y(x) and z = z(x).



? considered equations of the form z′ = F (x, y, y′, y′′, z) for two realfunctions y = y(x) and z = z(x).

? He observed that, contrary to the equationz′ = y′′F (x, y, y′, z) + G(x, y, y′, z), the general solution to theequation z′ = y′′2 can not be written in integral-free form:

x = x(t, w(t), w′(t), ....w(k)(t)),

y = y(t, w(t), w′(t), ....w(k)(t)),

z = z(t, w(t), w′(t), ....w(k)(t)).

• Cartan E (1910) Les systemes de Pfa a cinq variables et les equations auxderivees partielles du second ordre Ann. Sc. Norm. Sup. 27 109-192:


? solved an equivalence problem for equations

z′ = F (x, y, y′, y′′, z) with Fy′′y′′ 6= 0, (H)

by constructing a 14-dimensional Cartan bundle P → J over the5-dimensional space J parametrized by (x, y, y′, y′′, z).




by constructing a 14-dimensional Cartan bundle P → J over the5-dimensional space J parametrized by (x, y, y′, y′′, z).This bundle isequipped with a Cartan connection whose curvature gives all the localinvariants of the equation.




by constructing a 14-dimensional Cartan bundle P → J over the5-dimensional space J parametrized by (x, y, y′, y′′, z).This bundle isequipped with a Cartan connection whose curvature gives all the localinvariants of the equation.The connection has values in the Lie algebra ofthe nocompact form of the exceptional group G2 and is at i theequation is equivalent to the Hilbert's equation z′ = y′′2;




by constructing a 14-dimensional Cartan bundle P → J over the5-dimensional space J parametrized by (x, y, y′, y′′, z).This bundle isequipped with a Cartan connection whose curvature gives all the localinvariants of the equation.The connection has values in the Lie algebra ofthe nocompact form of the exceptional group G2 and is at i theequation is equivalent to the Hilbert's equation z′ = y′′2;in such case theequation has a symmetry group G2.

• PN (2003) Dierentail equations and conformal structures J. Geom. Phys55 19-49:

• PN (2003) Dierentail equations and conformal structures J. Geom. Phys55 19-49:

? Since G2 naturally seats in SO(3, 4), that is in a conformal group for(3, 2)-signature conformal metrics, is it possible to understand Cartan'sinvariants in terms of inavraints of some conformal structure in 5dimensions?

Cartan's construction


• Each equation (H) may be represented by formsω1 = dz − F (x, y, p, q, z)dx

ω2 = dy − pdx

ω3 = dp− qdx

on a 5-dimensional manifold J parametrized by (x, y, p = y′, q = y′′, z).



ω2 = dy − pdx

ω3 = dp− qdx


• every solution to the equation is a curve γ(t) = (x(t), y(t), p(t), q(t), z(t))in J on which the forms (ω1, ω2, ω3) simultaneously vanish.



ω2 = dy − pdx

ω3 = dp− qdx


• every solution to the equation is a curve γ(t) = (x(t), y(t), p(t), q(t), z(t))in J on which the forms (ω1, ω2, ω3) simultaneously vanish.

• Transformation that transforms solutions to solution may mix the forms(ω1, ω2, ω3) among themselves, thus:

DenitionTwo equations z′ = F (x, y, y′, y′′, z) and z′ = F (x, y, y′, y′′, z) represented bythe respective forms

ω1 = dz − F (x, y, p, q, z)dx, ω2 = dy − pdx, ω3 = dp− qdx;


ω1 = dz − F (x, y, p, q, z)dx, ω2 = dy − pdx, ω3 = dp− qdx;ω1 = dz − F (x, y, p, q, z)dx, ω2 = dy − pdx, ω3 = dp− qdx,



are (locally) equivalent i there exists a (local) dieomorphismφ : (x, y, p, q, z) → (x, y, p, q, z) such that



are (locally) equivalent i there exists a (local) dieomorphismφ : (x, y, p, q, z) → (x, y, p, q, z) such that

φ∗

ω1

ω2

ω2

=

α β γδ ε λκ µ ν

ω1

ω2

ω3

Solution for equivalence problem for eqs.

z′ = F (x, y, y′, y′′, z)

Theorem


z′ = F (x, y, y′, y′′, z)

Theorem

• There are two main branches of nonequivalent equationsz′ = F (x, y, y′, y′′, z). They are distinguished by vanishing or not of therelative invariant Fqq, q = y′′.


z′ = F (x, y, y′, y′′, z)

Theorem


• If Fqq ≡ 0 then such equations have integral-free solutions.


z′ = F (x, y, y′, y′′, z)

Theorem


• If Fqq ≡ 0 then such equations have integral-free solutions.

• There are nonequivalent equations among the equations having Fqq 6= 0. Allthese equations are beyond the class of equations with integral-free solutions.

Equations z′ = F (x, y, y′, y′′, z) with Fy′′y′′ 6= 0

Theorem


Theorem

An equivalence class of equations z′ = F (x, y, y′, y′′, z) with Fy′′y′′ 6= 0uniquely denes a 14-dimensional manifold P → J and


Theorem

An equivalence class of equations z′ = F (x, y, y′, y′′, z) with Fy′′y′′ 6= 0uniquely denes a 14-dimensional manifold P → J and a preferred coframe(θ1, θ2, θ3, θ4, θ5,Ω1,Ω2,Ω3,Ω4,Ω5,Ω6,Ω7,Ω8,Ω9) on it such that


Theorem


dθ1 = θ1 ∧ (2Ω1 + Ω4) + θ2 ∧ Ω2 + θ3 ∧ θ4

dθ2 = θ1 ∧ Ω3 + θ2 ∧ (Ω1 + 2Ω4) + θ3 ∧ θ5

dθ3 = θ1 ∧ Ω5 + θ2 ∧ Ω6 + θ3 ∧ (Ω1 + Ω4) + θ4 ∧ θ5

dθ4 = θ1 ∧ Ω7 + 43θ

3 ∧ Ω6 + θ4 ∧ Ω1 + θ5 ∧ Ω2

dθ5 = θ2 ∧ Ω7 − 43θ

3 ∧ Ω5 + θ4 ∧ Ω3 + θ5 ∧ Ω4.


Theorem


dθ1 = θ1 ∧ (2Ω1 + Ω4) + θ2 ∧ Ω2 + θ3 ∧ θ4

dθ2 = θ1 ∧ Ω3 + θ2 ∧ (Ω1 + 2Ω4) + θ3 ∧ θ5

dθ3 = θ1 ∧ Ω5 + θ2 ∧ Ω6 + θ3 ∧ (Ω1 + Ω4) + θ4 ∧ θ5

dθ4 = θ1 ∧ Ω7 + 43θ

3 ∧ Ω6 + θ4 ∧ Ω1 + θ5 ∧ Ω2

dθ5 = θ2 ∧ Ω7 − 43θ

3 ∧ Ω5 + θ4 ∧ Ω3 + θ5 ∧ Ω4.

We also have formulae for the dierentials of the forms Ωµ, µ = 1, 2, ..., 9.

For example:dΩ1 = Ω3 ∧ Ω2 + 1

3θ3 ∧ Ω7 − 2

3θ4 ∧ Ω5+

13θ

5 ∧ Ω6 + θ1 ∧ Ω8 + 38c2θ

1 ∧ θ2+b2θ

1 ∧ θ3 + b3θ2 ∧ θ3+

a2θ1 ∧ θ4 + a3θ

1 ∧ θ5 + a3θ2 ∧ θ4 + a4θ

2 ∧ θ5.where a2, a3, a4, b2, b3, c2 are functions on P uniquely dened by theequivalence class of equations.


3θ3 ∧ Ω7 − 2

3θ4 ∧ Ω5+

13θ

5 ∧ Ω6 + θ1 ∧ Ω8 + 38c2θ

1 ∧ θ2+b2θ

1 ∧ θ3 + b3θ2 ∧ θ3+

a2θ1 ∧ θ4 + a3θ

1 ∧ θ5 + a3θ2 ∧ θ4 + a4θ


The other dierentials, when decomposed on the basis θi, Ωµ, dene morefunctions, which Cartan denoted by a1, a2, a3, a4, a5, b1, b2, b3, b4, c1, c2, c3,δ1, δ2, e, h1, h2, h3, h4, h5, h6, k1, k2, k3.


3θ3 ∧ Ω7 − 2

3θ4 ∧ Ω5+

13θ

5 ∧ Ω6 + θ1 ∧ Ω8 + 38c2θ

1 ∧ θ2+b2θ

1 ∧ θ3 + b3θ2 ∧ θ3+

a2θ1 ∧ θ4 + a3θ

1 ∧ θ5 + a3θ2 ∧ θ4 + a4θ



The system provides all the local invariants for the equivalence class of equationssatisfying Fqq 6= 0.


3θ3 ∧ Ω7 − 2

3θ4 ∧ Ω5+

13θ

5 ∧ Ω6 + θ1 ∧ Ω8 + 38c2θ

1 ∧ θ2+b2θ

1 ∧ θ3 + b3θ2 ∧ θ3+

a2θ1 ∧ θ4 + a3θ

1 ∧ θ5 + a3θ2 ∧ θ4 + a4θ



The system provides all the local invariants for the equivalence class of equationssatisfying Fqq 6= 0.

We pass to the interpretetion in terms of Cartan connection:

P is a principal bre bundle over J with the 9-dimensional parabolic subgroup Hof G2 as its structure group.

P is a principal bre bundle over J with the 9-dimensional parabolic subgroup Hof G2 as its structure group.

On this bre bundle the following matrix of 1-forms:

ω =

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

−Ω1 − Ω4 −Ω8 −Ω9 − 1√3Ω7

13Ω5

13Ω6 0

θ1 Ω1 Ω21√3θ4 −1

3θ3 0 1

3Ω6

θ2 Ω3 Ω41√3θ5 0 −1

3θ3 −1

3Ω5

2√3θ3 2√

3Ω5

2√3Ω6 0 1√

3θ5 − 1√

3θ4 − 1√

3Ω7

θ4 Ω7 0 2√3Ω6 −Ω4 Ω2 Ω9

θ5 0 Ω7 − 2√3Ω5 Ω3 −Ω1 −Ω8

0 θ5 −θ4 2√3θ3 −θ2 θ1 Ω1 + Ω4

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

,

is a Cartan connection with values in the Lie algebra of G2.

The curvature of this connection R = dω + ω ∧ ω `measures' how much a givenequivalence class of equations is `distorted' from the at Hilbert casecorresponding to F = q2.

(3, 2)-signature conformal metric

Given an equivalence class of equation z′ = F (x, y, y′, y′′, z) consider itscorresponding bundle P with the coframe(θ1, θ2, θ3, θ4, θ5,Ω1,Ω2,Ω3,Ω4,Ω5,Ω6,Ω7,Ω8,Ω9).



Dene a bilinear form

g = 2θ1θ5 − 2θ2θ4 +43θ3θ3




g = 2θ1θ5 − 2θ2θ4 +43θ3θ3

This form is degenerate on P and has signature (3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0).




g = 2θ1θ5 − 2θ2θ4 +43θ3θ3

This form is degenerate on P and has signature (3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0).

The 9 degenerate directions generate the vertical space of P .

Theorem

Theorem

• The bilinear forms g transforms conformally when Lie transported along anyof the vertical directions.

Theorem


• It descends to a well dened conformal (3, 2)-signature metric on the5-dimensional space J on which the equation z′ = F (x, y, y′, y′′, z) isdened.

Theorem



• The Cartan normal conformal connection associated with this conformalmetric yields all the invariant information about the equivalence class ofthe equation.

Theorem



• The Cartan normal conformal connection associated with this conformalmetric yields all the invariant information about the equivalence class ofthe equation.

• This so(4, 3)-valued connection is reducible and, after reduction, can beidentied with the g2 Cartan connection ω on P .

In the quoted paper I gave the explicit formula for the (3, 2)-siganture metric interms of the function F dening the equation.

In the quoted paper I gave the explicit formula for the (3, 2)-siganture metric interms of the function F dening the equation. Here I present a simple example.


The Hilbert equation z′ = y′′2 has maximal symmetry group: G2 of dimension14.



Cartan knew that z′ = F (x, y, y′, y′′, z) is either equivalent to the Hilbertequation or its group of transitive symmetries is at most 7-dimensional.




The equations with 7-dimensional group of transitive symmetries are amongthose equivalent to z′ = F (y′′) with Fy′′y′′ 6= 0.




The equations with 7-dimensional group of transitive symmetries are amongthose equivalent to z′ = F (y′′) with Fy′′y′′ 6= 0.

For such F 's the (3, 2)-signature conformal metric reads:

g = 30(F ′′)4 [ dqdy − pdqdx ] + [ 4F (3)2 − 3F ′′F (4) ] dz2+

2 [−5(F ′′)2F (3) − 4F ′F (3)2 + 3F ′F ′′F (4) ] dpdz+

2 [15(F ′′)3 + 5q(F ′′)2F (3) − 4FF (3)2 + 4qF ′F (3)2+

3FF ′′F (4) − 3qF ′F ′′F (4) ] dxdz+

[−20(F ′′)4 + 10F ′(F ′′)2F (3) + 4(F ′)2F (3)2 − 3(F ′)2F ′′F (4) ] dp2+

2 [−15F ′(F ′′)3 + 20q(F ′′)4 + 5F (F ′′)2F (3) − 10qF ′(F ′′)2F (3)+

4FF ′F (3)2 − 4q(F ′)2F (3)2 − 3FF ′F ′′F (4) + 3q(F ′)2F ′′F (4) ] dpdx+

[−30F (F ′′)3 + 30qF ′(F ′′)3 − 20q2(F ′′)4 − 10qF (F ′′)2F (3)+

10q2F ′(F ′′)2F (3) + 4F2F (3)2 − 8qFF ′F (3)2 + 4q2(F ′)2F (3)2−3F2F ′′F (4) + 6qFF ′F ′′F (4) − 3q2(F ′)2F ′′F (4) ] dx2.


2 [−5(F ′′)2F (3) − 4F ′F (3)2 + 3F ′F ′′F (4) ] dpdz+

2 [15(F ′′)3 + 5q(F ′′)2F (3) − 4FF (3)2 + 4qF ′F (3)2+

3FF ′′F (4) − 3qF ′F ′′F (4) ] dxdz+

[−20(F ′′)4 + 10F ′(F ′′)2F (3) + 4(F ′)2F (3)2 − 3(F ′)2F ′′F (4) ] dp2+

2 [−15F ′(F ′′)3 + 20q(F ′′)4 + 5F (F ′′)2F (3) − 10qF ′(F ′′)2F (3)+


[−30F (F ′′)3 + 30qF ′(F ′′)3 − 20q2(F ′′)4 − 10qF (F ′′)2F (3)+


It is always conformal to an Einstein metric g = e2Υg with the conformal factorΥ = Υ(q) satisfying

10(F ′′)2 [ Υ′′ − (Υ′)2 ]− 40F ′′F (3)Υ′ + 17F ′′F (4) − 56F (3)2 = 0.


2 [−5(F ′′)2F (3) − 4F ′F (3)2 + 3F ′F ′′F (4) ] dpdz+

2 [15(F ′′)3 + 5q(F ′′)2F (3) − 4FF (3)2 + 4qF ′F (3)2+

3FF ′′F (4) − 3qF ′F ′′F (4) ] dxdz+

[−20(F ′′)4 + 10F ′(F ′′)2F (3) + 4(F ′)2F (3)2 − 3(F ′)2F ′′F (4) ] dp2+

2 [−15F ′(F ′′)3 + 20q(F ′′)4 + 5F (F ′′)2F (3) − 10qF ′(F ′′)2F (3)+


[−30F (F ′′)3 + 30qF ′(F ′′)3 − 20q2(F ′′)4 − 10qF (F ′′)2F (3)+



10(F ′′)2 [ Υ′′ − (Υ′)2 ]− 40F ′′F (3)Υ′ + 17F ′′F (4) − 56F (3)2 = 0.

Any conformal metric originated from our construction, has special conformalholonomy HC ⊆ G2.


2 [−5(F ′′)2F (3) − 4F ′F (3)2 + 3F ′F ′′F (4) ] dpdz+

2 [15(F ′′)3 + 5q(F ′′)2F (3) − 4FF (3)2 + 4qF ′F (3)2+

3FF ′′F (4) − 3qF ′F ′′F (4) ] dxdz+

[−20(F ′′)4 + 10F ′(F ′′)2F (3) + 4(F ′)2F (3)2 − 3(F ′)2F ′′F (4) ] dp2+

2 [−15F ′(F ′′)3 + 20q(F ′′)4 + 5F (F ′′)2F (3) − 10qF ′(F ′′)2F (3)+


[−30F (F ′′)3 + 30qF ′(F ′′)3 − 20q2(F ′′)4 − 10qF (F ′′)2F (3)+



10(F ′′)2 [ Υ′′ − (Υ′)2 ]− 40F ′′F (3)Υ′ + 17F ′′F (4) − 56F (3)2 = 0.

Any conformal metric originated from our construction, has special conformalholonomy HC ⊆ G2.It is therefore interesting to look for the ambient metrics for them. These, inturn, may have special pseudo-riemanian holonomy HψR ⊆ G2.

In the present example it is particularly easy, since g is conformal to Einstein. Wend that the Feerman-Graham ambient metric for this example is:

g = t2g + 2drdt +2rt

10F ′′2(56F (3)3 − 17F ′′F (4))dq2.



10F ′′2(56F (3)3 − 17F ′′F (4))dq2.

By construction it is Ricci at and has pseudo-riemannian holonomyHψR ⊆ G2.



10F ′′2(56F (3)3 − 17F ′′F (4))dq2.

By construction it is Ricci at and has pseudo-riemannian holonomyHψR ⊆ G2. Since the metric g is conformal to Einstein, the equalityHψR = G2 is excluded. We therefore have HψR ( G2.



10F ′′2(56F (3)3 − 17F ′′F (4))dq2.


Conformal metrics from our construction are rarely conformal to Einstein.



10F ′′2(56F (3)3 − 17F ′′F (4))dq2.


Conformal metrics from our construction are rarely conformal to Einstein.

Thus, evaluation of the ambient metrics for them should lead to quite nontrivial(4, 3)-signature metrics which might have strict noncompact G2

pseudo-riemannian holonomy.

Cartan classied various types of nonequivalent equations z′ = F (x, y, y′, y′′, z)according to the roots of Ψ(z) = a1z

4 + 4a2z3 + 6a3z

2 + 4a4z + a5, where(a1, a2, a3, a4, a5) are the scalar invariants of the equation.


4 + 4a2z3 + 6a3z


This polynomial encodes partial information of the Weyl tensor of the associatedconformal (3, 2)-signature metric. In particular, the well known invariantIΨ = 6a2

3 − 8a2a4 + 2a1a5 of this polynomial is, modulo a numerical factor,proportional to the square of the Weyl tensor C2 = CµνρσCµνρσ of theconformal metric.


4 + 4a2z3 + 6a3z




Vanishing of IΨ means that Ψ = Ψ(z) has a root with multiplicity no smallerthan 3.


4 + 4a2z3 + 6a3z





Our example above corresponds to the situation when this multiplicity is equal to4.


4 + 4a2z3 + 6a3z





Our example above corresponds to the situation when this multiplicity is equal to4. According to Cartan, all nonequivalent equations for which Ψ has quartic rootare covered by this example.

In this example nonequivalent equations are distinguished by the onlynonvanishing scalar invariant a5 to which the Weyl tensor of the metric g isproportional.


If a5 =const the equation has a 7-dimensional group of symmetries.



Further relations:



Further relations:

Bryant R L (2005) Conformal geometry and 3-plane elds on 6 manifolds,DG/0511110:



Further relations:

Bryant R L (2005) Conformal geometry and 3-plane elds on 6 manifolds,DG/0511110:

Equations z′ = F (x, y, y′, y′′, z) are in relations with 2-plane elds onmanifolds of dimension 5. Bryant found description of certain 3-plane elds indimension 6 in terms of conformal (3, 3)-signature geometries.

• Bobienski M, Nurowski P (2006) Irreducible SO(3) geometries in dimensionve J. reine angew. Math. in print, math.DG/0507152:


• Motivated by type IIB string theory we considered Riemannian manifolds(M5, g) equipped with a tensor eld Υ s.t.

i) Υijk = Υ(ijk), (totally symmetric)ii) Υijj = 0, (trace-free)iii) ΥjkiΥlmi + ΥljiΥkmi + ΥkliΥjmi = gjkglm + gljgkm + gklgjm.




• Tensor Υ reduces the structure group of the frame bundle via O(5) to theirreducible (maximal) SO(3) ⊂ GL(5, R)




• Tensor Υ reduces the structure group of the frame bundle via O(5) to theirreducible (maximal) SO(3) ⊂ GL(5, R)

• A 5-dimensional Riemannian manifold (M5, g) equipped with a tensor eld Υsatisfying conditions i)-iii) and admitting a unique decompositionLC

Γ = Γ + 12T, with T ∈

∧3 R5 and Γ ∈ so(3)⊗ R5 is called nearlyintegrable irreducible SO(3) structure.

• Such structures are classied according to the decomposition of the totallyskew symmetric torsion T onto the irreducibles w.r.t. the SO(3) action:∧3 R5 = R3

⊕R7


⊕R7

• We have examples of such geometries. All our examples admit transitivesymmetry group (which may be of dimension 8, 6 and 5)


⊕R7


• In particular, T = 0 corresponds to the symmetric spaces SU(3)/SO(3),SL(3, R)/SO(3), or the at (SO(3)×ρ R5)/SO(3).


⊕R7



• We also have nontrivial examples with torsion T 6= 0 being one of the puretypes R3 or R7.


⊕R7



• We also have nontrivial examples with torsion T 6= 0 being one of the puretypes R3 or R7.

• Some of these examples satisfy Strominger equations of type IIB string theory.

M Godlinski+PN; idea from E Ferapontov


Consider a 5th order ODE y(5) = F (x, y, y′, y′′, y(3), y(4)) modulo contacttransformation of the variables.


Consider a 5th order ODE y(5) = F (x, y, y′, y′′, y(3), y(4)) modulo contacttransformation of the variables.Suppose that the equation satsies three, contactinvariant conditions:



50D2F4 − 75DF3 + 50F2 − 60DF4F4 + 30F3F4 + 8F 34 = 0, (∗)



50D2F4 − 75DF3 + 50F2 − 60DF4F4 + 30F3F4 + 8F 34 = 0, (∗)

375D2F3 − 1000DF2 + ... = 0, 1250D2F2 − 6250DF1 + ... = 0

where D = ∂x + y′∂y + y′′∂y′ + y(3)∂y′′ + y(4)∂y(3) + F∂y(4).



50D2F4 − 75DF3 + 50F2 − 60DF4F4 + 30F3F4 + 8F 34 = 0, (∗)

375D2F3 − 1000DF2 + ... = 0, 1250D2F2 − 6250DF1 + ... = 0


Then the solution space of the equation is naturally equipped with a class ofpairs [(g,Υ)] with representatives satisfying our conditions i)-iii).



50D2F4 − 75DF3 + 50F2 − 60DF4F4 + 30F3F4 + 8F 34 = 0, (∗)

375D2F3 − 1000DF2 + ... = 0, 1250D2F2 − 6250DF1 + ... = 0


Then the solution space of the equation is naturally equipped with a class ofpairs [(g,Υ)] with representatives satisfying our conditions i)-iii).The metric g ofsignature (+,+,−,−,−) and the tensor Υ are determined by the contactequivalence class of the ODE up to g → e2φg, Υ → e3φΥ.

Theorem

Theorem

There is a one to one correspondence between the conformal (3, 2)-signaturegeometries (M5, [(g,Υ)]) having totally skew symmetric torsion of the puretype R3

Theorem

There is a one to one correspondence between the conformal (3, 2)-signaturegeometries (M5, [(g,Υ)]) having totally skew symmetric torsion of the puretype R3 and contact equivalent classes of 5th order ODEs satisfyingconditions (∗).

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