Partially hyperbolic geodesic flows
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 5, pp. 985-1014.

We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such an example as the product metric and locally symmetric spaces of nonpositive curvature with rank bigger than one are not partially hyperbolic. We prove that if a metric of nonpositive curvature has a partially hyperbolic geodesic flow, then its rank is one. Other obstructions to partial hyperbolicity of a geodesic flow are also analyzed.

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Carneiro, Fernando; Pujals, Enrique. Partially hyperbolic geodesic flows. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 5, pp. 985-1014. doi : 10.1016/j.anihpc.2013.07.009. http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.009/

[1] D.V. Anosov, Geodesic Flows on Closed Riemann Manifolds with Negative Curvature, Proc. Steklov Inst. Math. vol. 90 , American Mathematical Society, Providence, RI (1969) | MR | Zbl

[2] Werner Ballmann, Lectures on Spaces of Nonpositive Curvature, DMV Seminar vol. 25 , Birkhäuser, Boston (1995) | MR | Zbl

[3] Werner Ballmann, Nonpositively curved manifolds of higher rank, Ann. Math. 122 (1985), 597 -609 | MR | Zbl

[4] Arthur L. Besse, Einstein Manifolds, Classics Math. , Springer-Verlag (1987) | MR | Zbl

[5] C. Bonatti, L.J. Diaz, Persistence of transitive diffeomorphisms, Ann. Math. 143 (1995), 367 -396

[6] A. Borel, Compact Clifford–Klein forms of symmetric spaces, Topology 2 (1963), 111 -122 | MR | Zbl

[7] M.I. Brin, Ja.B. Pesin, Flows of frames on manifolds of negative curvature, Usp. Mat. Nauk 28 no. 4 (172) (1973), 209 -210 | MR

[8] Keith Burns, Ralf Spatzier, Manifolds of nonpositive curvature and their buildings, Publ. Math. IHES 65 (1987), 35 -59 | EuDML | Numdam | MR | Zbl

[9] C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Isr. J. Math. 115 (2000), 157 -193 | MR | Zbl

[10] Manfredo Do Carmo, Geometria Riemanniana, Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro (1988) | MR | Zbl

[11] H.M.A. Castro, M.H. Kobayashi, W.M. Oliva, Partially hyperbolic Σ-geodesic flows, Special Issue in Celebration of Jack K. Hale's 70th Birthday, Part 3 Atlanta, GA/Lisbon, 1998 J. Differ. Equ. 169 no. 1 (2001), 142 -168 | Zbl

[12] Gonzalo Contreras, Partially hyperbolic geodesic flows are Anosov, C. R. Math. Acad. Sci. Paris Ser. I 334 (2002), 585 -590 | MR | Zbl

[13] Patrick Eberlein, When is a geodesic flow of Anosov type? I, J. Differ. Geom. 8 (1973), 437 -463 | MR | Zbl

[14] Patrick Eberlein, Structure of Manifolds of Nonpositive Curvature, Lect. Notes Math. vol. 1156 , Springer-Verlag (1985), 86 -153 | MR | Zbl

[15] Patrick Eberlein, Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics (1996) | MR | Zbl

[16] Patrick Eberlein, Geodesic flows in certain manifolds without conjugate points, Trans. Am. Math. Soc. 167 (May 1972) | MR

[17] William M. Goldman, Complex Hyperbolic Geometry, Clarendon Press, Oxford (1999) | MR | Zbl

[18] Alfred Gray, A note on manifolds whose holonomy group is a subgroup of Sp(n).Sp(1), Mich. Math. J. 16 no. 2 (1969), 125 -128 | MR | Zbl

[19] Ernst Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 no. 1 (1974), 23 -34 | EuDML | MR | Zbl

[20] Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math. , Academic Press, New York, London (1978) | MR | Zbl

[21] Boris Hasselblatt, Yakov Pesin, Partially hyperbolic dynamical systems, Handbook of Dynamical Systems, vol. 1B, Elsevier, North-Holland (2006) | MR | Zbl

[22] Jurgen Jost, Riemannian Geometry and Geometric Analysis, Universitext , Springer-Verlag (2002) | MR | Zbl

[23] W. Klingenberg, Lectures on Closed Geodesics, Springer-Verlag (1978) | MR | Zbl

[24] Ricardo Mañé, Contributions to the stability conjecture, Topology 17 no. 4 (1978), 383 -396 | MR | Zbl

[25] Ricardo Mañé, Oseledec's theorem from the generic viewpoint, Proceedings of the International Congress of Mathematicians, vol. 1, 2, Warsaw (1983), 1269 -1276 | MR | Zbl

[26] Ricardo Mañé, On a theorem of Klingenberg, Dynamical Systems and Bifurcation Theory, Proc. Meet., Rio de Janeiro/Braz. 1985, Pitman Res. Notes Math. Ser. vol. 160 (1987), 319 -345 | MR

[27] Sheldon E. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math. 99 no. 5 (1977), 1061 -1087 | MR | Zbl

[28] Gabriel Paternain, Geodesic Flows, Prog. Math. , Birkhäuser, Boston (1999) | MR | Zbl

[29] E.R. Pujals, M. Sambarino, Topics on homoclinic bifurcation, dominated splitting, robust transitivity and related results, Handbook of Dynamical Systems, vol. 1B, Elsevier (2005), 327 -378 | MR | Zbl

[30] Rafael Ruggiero, Persistently expansive geodesic flows, Commun. Math. Phys. 140 no. 1 (1991), 203 -215 | MR | Zbl

[31] Rafael Ruggiero, On the creation of conjugate points, Math. Z. 208 (1991), 41 -55 | EuDML | MR | Zbl

[32] M. Shub, Topologically transitive diffeomorphism of 𝕋 4 , Symposium on Differential Equations and Dynamical Systems, University of Warwick, 1968/1969, Lect. Notes Math. vol. 206 , Springer-Verlag (1971), 39 -40

[33] Steven Smale, Differentiable dynamical systems, Bull. Am. Math. Soc. 73 (1967), 747 -817 | MR | Zbl

[34] Norman Steenrod, Topology of fiber bundles, Princeton Landmarks Mathematics, Princeton University Press (1999) | Zbl

[35] Maciej Wojtjowski, Magnetic flows and Gaussian thermostats on manifolds of negative curvature, Fundam. Math. 163 no. 2 (2000), 177 -191 | EuDML | MR | Zbl

[36] Joseph A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. Mech. 14 no. 6 (1965), 1033 -1047 | MR | Zbl

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