# 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.