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Three-dimensional Cauchy-Riemann structures and second-order ordinary differential equations. (English) Zbl 1051.32019

The paper explains the analogy between three-dimensional Cauchy-Riemann structures and second-order ordinary differential equations defined modulo point transformations. The first part consists of a review of the concept of three-dimensional Cauchy-Riemann structure and the quotation of Cartan’s theorem solving the equivalence problem for three-dimensional Cauchy-Riemann structures. Furthermore these results are described in terms of \(\text{SU}(2,1)\) connections and associated to the corresponding class of Fefferman metrics. The following chapter describes the analogy between the three-dimensional Cauchy-Riemann structures and second-order ordinary differential equations defined modulo point transformations. Thus the equivalence problem for second-order ordinary differential equations is solved by using again Cartan’s theorem but now interpreted in the sense of \(\text{SL}(3,\mathbb{R})\) connection. Point invariants are given for a local representation. The analogy of ordinary second-order differential equation to the Fefferman metrics are discussed. The metrics conformal to the Einstein metrics are discussed, especially the Robinson manifolds.

MSC:

32V05 CR structures, CR operators, and generalizations
34C14 Symmetries, invariants of ordinary differential equations
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
53C80 Applications of global differential geometry to the sciences
58A15 Exterior differential systems (Cartan theory)