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.
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Partially supported by ValCoMo (ANR-13-BS01-0006).
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Jaoui, R. Differential fields and geodesic flows II: Geodesic flows of pseudo-Riemannian algebraic varieties. Isr. J. Math. 230, 527–561 (2019). https://doi.org/10.1007/s11856-018-1820-z
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DOI: https://doi.org/10.1007/s11856-018-1820-z