Differential equations and conformal structures

Abstract

We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal Lorentzian geometries. The second example shows that every point equivalent class of 3rd order ODEs satisfying the Wuenschmann and the Cartan conditions define a 3-dimensional Lorentzian Einstein-Weyl geometry. The third example associates to each point equivalence class of 3rd order ODEs a 6-dimensional conformal geometry of neutral signature. The fourth example exhibits the one-to-one correspondence between point equivalent classes of 2nd order ODEs and 4-dimensional conformal Fefferman-like metrics of neutral signature. The fifth example shows the correspondence between undetermined ODEs of the Monge type and conformal geometries of signature $(3,2)$. The Cartan normal conformal connection for these geometries is reducible to the Cartan connection with values in the Lie algebra of the noncompact form of the exceptional group $G_2$. All the examples are deeply rooted in Elie Cartan's works on exterior differential systems.

Some typos in formulae concerning (3,2)-signature conformal metrics of Section 5.3 were corrected

References in corpus (2)

• 3-dimensional Cauchy-Riemann structures and 2nd order ordinary differential equations
• Differential Geometry from Differential Equations

Cited by in corpus (59)

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• Geometric realizations, curvature decompositions, and Weyl manifolds
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• Conformal geometry and 3-plane fields on 6-manifolds
• On Nurowski's conformal structure associated to a generic rank two distribution in dimension five
• Geometry of third-order ODEs
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• Nearly Kähler Geometry and $(2,3,5)$-Distributions via Projective Holonomy
• Geometry of Third-Order Ordinary Differential Equations and Its Applications in General Relativity
• Third order ODEs and four-dimensional split signature Einstein metrics
• The Geometry of Almost Einstein (2,3,5) Distributions
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• Conformal structures with $G_{2(2)}$-ambient metrics
• Flat $(2,3,5)$-Distributions and Chazy's Equations
• On deformations of the dispersionless Hirota equation
• New relations between $G_2$-geometries in dimensions 5 and 7
• Analytic torsion of generic rank two distributions in dimension five
• Nurowski's conformal structures for (2,5)-distributions via dynamics of abnormal extremals
• A new example of a generic 2-distribution on a 5-manifold with large symmetry algebra
• On 5-manifolds admitting rank two distributions of Cartan type
• Pure spinors, intrinsic torsion and curvature in odd dimensions
• A non-normal Fefferman-type construction of split-signature conformal structures admitting twistor spinors
• Dispersionless integrable hierarchies and GL(2,R) geometry
• Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions
• A Projective-to-Conformal Fefferman-Type Construction
• GL(2)-geometry and complex structures
• Parallel tractor extension and ambient metrics of holonomy split G_2
• New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions
• The Cayley cubic and differential equations
• Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional Heisenberg group holonomy
• Towards a Classification of pseudo-Riemannian Geometries Admitting Twistor Spinors
• Causal geometries and third-order ordinary differential equations
• Geometry of rank 2 distributions with nonzero Wilczynski invariants and affine control systems with one input
• Variations on the Tait-Kneser theorem
• Subtleties Concerning Conformal Tractor Bundles
• Homogeneous Real (2,3,5) Distributions with Isotropy
• Causal geometries, null geodesics, and gravity
• Inclusions between parabolic geometries
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• The ambient obstruction tensor and conformal holonomy
• Duality of (2,3,5)-distributions and Lagrangian cone structures
• Ants and bracket generating distributions in dimension 5 and 6
• Revisiting Kepler: new symmetries of an old problem
• Regularized determinants of the Rumin complex in irreducible unitary representations of the (2,3,5) nilpotent Lie group
• Analytic torsion of nilmanifolds with (2,3,5) distributions
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• Equivalence of variational problems of higher order
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• Duality on geodesics of Cartan distributions and sub-Riemannian pseudo-product structures
• Transitive conformal holonomy groups
• Connected subgroups of SO(2,n) acting irreducibly on ${\mathbb{R}}^{2,n}$
• Rolling balls and Octonions
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• Parabolic quasi-contact cone structures with an infinitesimal symmetry