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 G2. All the examples are deeply rooted in Elie Cartan's works on exterior differential systems.MSC 2000: 34A26, 53B15, 53B30, 53B50.

Cartan’s incomplete classification and an explicit ambient metric of holonomy $$\mathrm{G}_2^*$$

“…Nurowski showed that any real (2, 3, 5) distribution (M, D) determines a canonical conformal structure c D on the underlying manifold M [26,Section 5.3], and the argument in that reference shows that Nurowski's construction applies just as well to 3 Since G * 2 is semisimple and Q is parabolic, this geometry is an example of an important class of structures called parabolic geometries. For any such geometry one can encode any structure on a manifold M in a bundle E → M equipped with a canonically determined Cartan connection; see the standard reference [12] for a general treatment of parabolic geometries and Section 4.3.2 there for discussion of the geometry of (2, 3, 5) distributions in this setting.…”

Twistor Space for Rolling Bodies

“…[4,9,13], facts about rank two distributions in dimensions five, which will be needed in the next Section. This part of the paper is purely expository, and it is based on Ref.…”

“…This part of the paper is purely expository, and it is based on Ref. [13]. The reader is referred to this paper for details.…”

“…This gives us an important corollary of the identification theorem (2.6). In the procedure below, which is implicit in [13], and more explicit in [11], we summarize how to effectively calculate C(ζ) given a (2, 3, 5) distribution D on M 5 . In particular, we show how to calculate the functions A i .…”

Duality of Singular Paths for (2, 3, 5)-Distributions

“…For spaces with (2, 3, 5)-distributions, the construction of the space of singular paths and the existence of natural double fibration was mentioned for the first time in unpublished lecture notes by Bryant [5]. On such spaces, indefinite conformal structures of signature (3,2) were constructed by Nurowski [17]. Then, for the alternative constructions to Nurowski's, the exactly same cone structure as in this paper was explicitly used in [3].…”