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

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.