  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.
