Corps diff\'erentiels et flots g\'eod\'esiques I: Orthogonalit\'e aux constantes pour les \'equations diff\'erentielles autonomes

Abstract

We study the properties of orthogonality to the constants and disintegration for autonomous algebraic differential equations. We present a criterion of orthogonality to the constants for absolutely irreducible real $D$-varieties relying on the topological dynamic of the associated real analytic flow. More precisely, we prove that if there exists Zariski-dense invariant compact region of the smooth locus of real points of $X$ where the dynamic of the real analytic flow is topologically weakly mixing, then the generic type of $(X,v)$ is orthogonal to the constants. This criterion will be applied in a second part of this article to establish some transcendance properties for the geodesics of a compact algebraically presented compact Riemannian manifold with negative curvature.