Third order ordinary differential equations and Legendre connections. (English) Zbl 0987.34029

This article revisits a paper by S.-S. Chern [Sci. Rep. Nat. Tsing Hua Univ. Kunming, Ser. A 4, 97-111 (1940; Zbl 0024.19801)]. The authors prove that for any given third-order ordinary differential equation \(y'''= F(x, y,y',y'')\) one can construct a normal \(\text{sp}(2,\mathbb{R})\) Cartan connection on \(J^2(\mathbb{R},\mathbb{R})\). The structure equations for the connection produce two (relative) differential invariants, which are shown to generate all other invariants, a result that is not clear from Chern’s original paper. One of the invariants is the primary invariant found by Chern. The calculations of the curvature are done explicitly in terms of a specified basis for a grading of \(\text{sp}(2,\mathbb{R})\) and, from this, the above result follows.


34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.


Zbl 0024.19801
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