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}, pages = {985--1014}, publisher = {Elsevier}, volume = {31}, number = {5}, year = {2014}, doi = {10.1016/j.anihpc.2013.07.009}, mrnumber = {3258363}, zbl = {1298.53089}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.009/} }
TY - JOUR AU - Carneiro, Fernando AU - Pujals, Enrique TI - Partially hyperbolic geodesic flows JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 985 EP - 1014 VL - 31 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.009/ DO - 10.1016/j.anihpc.2013.07.009 LA - en ID - AIHPC_2014__31_5_985_0 ER -
%0 Journal Article %A Carneiro, Fernando %A Pujals, Enrique %T Partially hyperbolic geodesic flows %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 985-1014 %V 31 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.07.009/ %R 10.1016/j.anihpc.2013.07.009 %G en %F 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/articles/10.1016/j.anihpc.2013.07.009/
[1] Geodesic Flows on Closed Riemann Manifolds with Negative Curvature, Proc. Steklov Inst. Math. vol. 90 , American Mathematical Society, Providence, RI (1969) | MR | Zbl
,[2] Lectures on Spaces of Nonpositive Curvature, DMV Seminar vol. 25 , Birkhäuser, Boston (1995) | MR | Zbl
,[3] Nonpositively curved manifolds of higher rank, Ann. Math. 122 (1985), 597 -609 | MR | Zbl
,[4] Einstein Manifolds, Classics Math. , Springer-Verlag (1987) | MR | Zbl
,[5] Persistence of transitive diffeomorphisms, Ann. Math. 143 (1995), 367 -396
, ,[6] Compact Clifford–Klein forms of symmetric spaces, Topology 2 (1963), 111 -122 | MR | Zbl
,[7] Flows of frames on manifolds of negative curvature, Usp. Mat. Nauk 28 no. 4 (172) (1973), 209 -210 | MR
, ,[8] Manifolds of nonpositive curvature and their buildings, Publ. Math. IHES 65 (1987), 35 -59 | EuDML | Numdam | MR | Zbl
, ,[9] SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Isr. J. Math. 115 (2000), 157 -193 | MR | Zbl
, ,[10] Geometria Riemanniana, Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro (1988) | MR | Zbl
,[11] 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] Partially hyperbolic geodesic flows are Anosov, C. R. Math. Acad. Sci. Paris Ser. I 334 (2002), 585 -590 | MR | Zbl
,[13] When is a geodesic flow of Anosov type? I, J. Differ. Geom. 8 (1973), 437 -463 | MR | Zbl
,[14] Structure of Manifolds of Nonpositive Curvature, Lect. Notes Math. vol. 1156 , Springer-Verlag (1985), 86 -153 | MR | Zbl
,[15] Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics (1996) | MR | Zbl
,[16] Geodesic flows in certain manifolds without conjugate points, Trans. Am. Math. Soc. 167 (May 1972) | MR
,[17] Complex Hyperbolic Geometry, Clarendon Press, Oxford (1999) | MR | Zbl
,[18] 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] On homogeneous manifolds of negative curvature, Math. Ann. 211 no. 1 (1974), 23 -34 | EuDML | MR | Zbl
,[20] Differential Geometry, Lie Groups, and Symmetric Spaces, Pure Appl. Math. , Academic Press, New York, London (1978) | MR | Zbl
,[21] Partially hyperbolic dynamical systems, Handbook of Dynamical Systems, vol. 1B, Elsevier, North-Holland (2006) | MR | Zbl
, ,[22] Riemannian Geometry and Geometric Analysis, Universitext , Springer-Verlag (2002) | MR | Zbl
,[23] Lectures on Closed Geodesics, Springer-Verlag (1978) | MR | Zbl
,[24] Contributions to the stability conjecture, Topology 17 no. 4 (1978), 383 -396 | MR | Zbl
,[25] Oseledec's theorem from the generic viewpoint, Proceedings of the International Congress of Mathematicians, vol. 1, 2, Warsaw (1983), 1269 -1276 | MR | Zbl
,[26] 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] Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math. 99 no. 5 (1977), 1061 -1087 | MR | Zbl
,[28] Geodesic Flows, Prog. Math. , Birkhäuser, Boston (1999) | MR | Zbl
,[29] Topics on homoclinic bifurcation, dominated splitting, robust transitivity and related results, Handbook of Dynamical Systems, vol. 1B, Elsevier (2005), 327 -378 | MR | Zbl
, ,[30] Persistently expansive geodesic flows, Commun. Math. Phys. 140 no. 1 (1991), 203 -215 | MR | Zbl
,[31] On the creation of conjugate points, Math. Z. 208 (1991), 41 -55 | EuDML | MR | Zbl
,[32] Topologically transitive diffeomorphism of , Symposium on Differential Equations and Dynamical Systems, University of Warwick, 1968/1969, Lect. Notes Math. vol. 206 , Springer-Verlag (1971), 39 -40
,[33] Differentiable dynamical systems, Bull. Am. Math. Soc. 73 (1967), 747 -817 | MR | Zbl
,[34] Topology of fiber bundles, Princeton Landmarks Mathematics, Princeton University Press (1999) | Zbl
,[35] Magnetic flows and Gaussian thermostats on manifolds of negative curvature, Fundam. Math. 163 no. 2 (2000), 177 -191 | EuDML | MR | Zbl
,[36] Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. Mech. 14 no. 6 (1965), 1033 -1047 | MR | Zbl
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