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Differential equations and conformal structures. (English) Zbl 1082.53024
This highly interesting work explores the deep geometrical relations between differential equations and conformal geometry. The first example of this fact was obtained by a student of Engel, K. Wünschmann, who defined conformal Lorentzian metrics on the solution space of a certain class of third order differential equations. This classical result was later reinterpreted in terms of Lie algebra valued connections.
This article presents various interesting examples pointing out the common aspects of differential equations and conformal structures. First, an alternative derivation of Wünschmann’s result is given, analyzing the properties for a third order equation to satisfy the so called Wünschmann condition. The conformal invariants of the corresponding Lorentzian metrics are computed. Further, new examples motivated by Cartan’s approach to the problem are given. More specifically, the author recovers Cartan’s observation that an additional condition on the Wünschmann class defines a 3-dimensional Lorentzian Weyl geometry satisfying the Einstein-Weyl equations. Two interesting examples on this fact are exhibited.
The most relevant part of the paper is devoted to the study of geometries related to underdetermined ordinary differential equations of Monge type and conformal geometries of signature \((+++--)\). Using the \(NG_{2}\)-valued Cartan connection, where \(NG_{2}\) is the noncompact real form of the exceptional rank two Lie algebra \(G_{2}\), the Cartan normal conformal connection for the geometry appearing in this type of equations is reduced. Another quite interesting point is the fact that the square of the Weyl tensor for the type of metrics analyzed can be interpreted by a classical invariant, which presents some affinities with polynomials used in the NP-formalism.

MSC:
53B50 Applications of local differential geometry to the sciences
34A26 Geometric methods in ordinary differential equations
53B15 Other connections
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
58A15 Exterior differential systems (Cartan theory)
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