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.