# Corps différentiels et flots géodésiques I: Orthogonalité aux constantes pour les équations différentielles autonomes

Abstract: Orthogonality to the constants is property of an algebraic differential equation that originated from the model-theoretic study of differential fields and that expresses remarkable independence properties for its solutions

In this article, we study the property of orthogonality to the constants in a differential algebraic language for autonomous differential equations and describe some effective methods to establish this property. The main result is a criterion for orthogonality to the constants (and its version for families) for real absolutely irreducible D-varieties $$(X,v)$$ based on the dynamical properties of the associated real analytic flow $$(M,\phi)$$. More precisely, we show that if there exists a compact region $$K$$ of $$M$$, Zariski-dense in $$X$$ and such that the restriction of the flow $$\phi$$ to $$K$$ 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 to study from this model-theoretic point of view the geodesic flow of compact Riemannian varieties (presented algebraically) with negative curvature.

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