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

@article{AIHPC_2014__31_5_985_0,
     author = {Carneiro, Fernando and Pujals, Enrique},
     title = {Partially hyperbolic geodesic flows},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {31},
     number = {5},
     year = {2014},
     pages = {985-1014},
     doi = {10.1016/j.anihpc.2013.07.009},
     zbl = {1298.53089},
     mrnumber = {3258363},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2014__31_5_985_0}
}
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/item/AIHPC_2014__31_5_985_0/

[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 242194 | Zbl 0176.19101

[2] Werner Ballmann, Lectures on Spaces of Nonpositive Curvature, DMV Seminar vol. 25 , Birkhäuser, Boston (1995) | MR 1377265 | Zbl 0834.53003

[3] Werner Ballmann, Nonpositively curved manifolds of higher rank, Ann. Math. 122 (1985), 597 -609 | MR 819559 | Zbl 0585.53031

[4] Arthur L. Besse, Einstein Manifolds, Classics Math. , Springer-Verlag (1987) | MR 867684 | Zbl 0613.53001

[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 146301 | Zbl 0116.38603

[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 418159

[8] Keith Burns, Ralf Spatzier, Manifolds of nonpositive curvature and their buildings, Publ. Math. IHES 65 (1987), 35 -59 | Numdam | MR 908215 | Zbl 0643.53037

[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 1749677 | Zbl 0996.37033

[10] Manfredo Do Carmo, Geometria Riemanniana, Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro (1988) | MR 651516 | Zbl 0505.53001

[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 0978.37020

[12] Gonzalo Contreras, Partially hyperbolic geodesic flows are Anosov, C. R. Math. Acad. Sci. Paris Ser. I 334 (2002), 585 -590 | MR 1903768 | Zbl 1154.37334

[13] Patrick Eberlein, When is a geodesic flow of Anosov type? I, J. Differ. Geom. 8 (1973), 437 -463 | MR 380891 | Zbl 0285.58008

[14] Patrick Eberlein, Structure of Manifolds of Nonpositive Curvature, Lect. Notes Math. vol. 1156 , Springer-Verlag (1985), 86 -153 | MR 824064 | Zbl 0569.53020

[15] Patrick Eberlein, Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics (1996) | MR 1441541 | Zbl 0883.53003

[16] Patrick Eberlein, Geodesic flows in certain manifolds without conjugate points, Trans. Am. Math. Soc. 167 (May 1972) | MR 295387

[17] William M. Goldman, Complex Hyperbolic Geometry, Clarendon Press, Oxford (1999) | MR 1695450 | Zbl 0939.32024

[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 244913 | Zbl 0177.50001

[19] Ernst Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 no. 1 (1974), 23 -34 | MR 353210 | Zbl 0273.53042

[20] Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math. , Academic Press, New York, London (1978) | MR 514561 | Zbl 0451.53038

[21] Boris Hasselblatt, Yakov Pesin, Partially hyperbolic dynamical systems, Handbook of Dynamical Systems, vol. 1B, Elsevier, North-Holland (2006) | MR 2186241 | Zbl 1130.37355

[22] Jurgen Jost, Riemannian Geometry and Geometric Analysis, Universitext , Springer-Verlag (2002) | MR 1871261 | Zbl 1034.53001

[23] W. Klingenberg, Lectures on Closed Geodesics, Springer-Verlag (1978) | MR 478069 | Zbl 0397.58018

[24] Ricardo Mañé, Contributions to the stability conjecture, Topology 17 no. 4 (1978), 383 -396 | MR 516217 | Zbl 0405.58035

[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 804776 | Zbl 0584.58007

[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 907897

[27] Sheldon E. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math. 99 no. 5 (1977), 1061 -1087 | MR 455049 | Zbl 0379.58011

[28] Gabriel Paternain, Geodesic Flows, Prog. Math. , Birkhäuser, Boston (1999) | MR 1712465 | Zbl 0930.53001

[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 2186244 | Zbl 1130.37354

[30] Rafael Ruggiero, Persistently expansive geodesic flows, Commun. Math. Phys. 140 no. 1 (1991), 203 -215 | MR 1124267 | Zbl 0746.58066

[31] Rafael Ruggiero, On the creation of conjugate points, Math. Z. 208 (1991), 41 -55 | MR 1125731 | Zbl 0749.58042

[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 228014 | Zbl 0202.55202

[34] Norman Steenrod, Topology of fiber bundles, Princeton Landmarks Mathematics, Princeton University Press (1999) | Zbl 0942.55002

[35] Maciej Wojtjowski, Magnetic flows and Gaussian thermostats on manifolds of negative curvature, Fundam. Math. 163 no. 2 (2000), 177 -191 | MR 1752103 | Zbl 0997.37011

[36] Joseph A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. Mech. 14 no. 6 (1965), 1033 -1047 | MR 185554 | Zbl 0141.38202