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

TL;DR

This paper investigates orthogonality to constants in autonomous algebraic differential equations, providing a criterion based on topological dynamics, and applies it to study transcendence properties of geodesics on negatively curved Riemannian manifolds.

Contribution

It introduces a new criterion for orthogonality to constants using topological dynamics of real analytic flows in algebraic differential equations.

Findings

Criterion links topological weak mixing to orthogonality to constants.

Application to geodesics shows transcendence properties on negatively curved manifolds.

Establishes a connection between dynamics and differential algebraic properties.

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.