Submission date: 1 March 2017

Abstract:

We define the notion of a smooth pseudo-Riemannian algebraic variety (X,g) over a field k of characteristic 0, which is an algebraic analogue of the notion of Riemannian manifold and we study, from a model-theoretic perspective, the algebraic differential equation describing the geodesics on (X,g).
When k is the field of real numbers, we prove that if the real points of X are Zariski-dense in X and if the real analytification of (X,g) is a compact Riemannian manifold with negative curvature, then the algebraic differential equation describing the geodesics on (X,g) is absolutely irreducible and its generic type is orthogonal to the constants.

Mathematics Subject Classification: 03C98

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Full text arXiv 1703.02890: pdf, ps.