# Differential fields and Geodesic flows II : Geodesic flows of pseudo-Riemannian algebraic varieties

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 $$(X,g)$$ 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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