Abstract
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.
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References
V. I. Arnol’d and A. B. Givental’, Symplectic geometry, in Dynamical Systems. IV, Encyclopaedia of Mathematical Sciences, Vol. 4, Springer, Berlin, 2001, pp. 1–138.
D. V. Anosov, Geodesic flows on closed riemannian manifolds of negative curvature, Trudy Matematicheskogo Instituta imeni V. A. Steklova 90 (1967).
V. I. Arnold, Ordinary Differential Equations, Universitext, Springer-Verlag, Berlin, 2006.
J. Bochnak, M. Coste and M.-F. Roy, Géométrie algébrique réelle, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 12, Springer-Verlag, Berlin, 1987.
Y. Coudene, Topological dynamics and local product structure, Journal of the London Mathematical Society 69 (2004), 441–456.
F. Dal’bo, Remarques sur le spectre des longueurs d’une surface et comptages, Boletim da Sociedade Brasileira de Matemática 30 (1999), 199–221.
P. Eberlein, When is a geodesic flow of Anosov type, II, Journal of Differential Geometry 8 (1973), 437–463, 565–577.
E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, Vol. 101, American Mathematical Society, Providence, RI, 2003.
M. Gromov, Partial Differential Relations, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 9, Springer-Verlag, Berlin, 1986.
J. Hadamard, Les surfaces à courbures opposés et leurs lignes géodésiques, Journal de Mathématiques Pures et Appliquées 4 (1898), 27–74.
B. Hasselblatt, Hyperbolic dynamical systems, in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 239–319.
G. A. Hedlund, The dynamics of geodesic flows, Bulletin of the American Mathematical Society 45 (1939), 241–260.
R. Jaoui, Corps différentiels et Flots géodésiques I: Orthogonalité aux constantes pour les équations différentielles autonomes, arXiv: 1612.06222v2.
R. Jaoui, Rational factors of D-varieties of dimension 3 with real anosov flow, arXiv:1803.08811.
J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Graduate Texts in Mathematics, Vol. 176, Springer-Verlag, New York, 1997.
J. Milnor, Morse Theory, Annals of Mathematics Studies, Vol. 51. Princeton University Press, Princeton, NJ, 1963.
J. Nash, C1 isometric imbeddings, Annals of Mathematics 60 (1954), 383–396.
J. Nash, The imbedding problem for Riemannian manifolds, Annals of Mathematics 63 (1956), 20–63.
P. Pansu, Le flot géodésique des variétés riemanniennes à courbure negative, Astérisque 201–203 (1991), 269–298.
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Partially supported by ValCoMo (ANR-13-BS01-0006).
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Jaoui, R. Differential fields and geodesic flows II: Geodesic flows of pseudo-Riemannian algebraic varieties. Isr. J. Math. 230, 527–561 (2019). https://doi.org/10.1007/s11856-018-1820-z
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DOI: https://doi.org/10.1007/s11856-018-1820-z