Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T18:22:41.122Z Has data issue: false hasContentIssue false
  • English
  • Français

Open problems and questions about geodesics

Part of: Differential geometry Global analysis, analysis on manifolds Global differential geometry

Published online by Cambridge University Press:  06 December 2019

KEITH BURNS Open the ORCID record for KEITH BURNS [Opens in a new window]  and
VLADIMIR S. MATVEEV Open the ORCID record for VLADIMIR S. MATVEEV [Opens in a new window]
KEITH BURNS
Affiliation:
Department of Mathematics, Northwestern University Evanston, IL60208-2730, USA email [email protected]
VLADIMIR S. MATVEEV
Affiliation:
Institut für Mathematik, Friedrich-Schiller-Universität Jena, 07743Jena, Germany email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper surveys open problems and questions related to geodesics defined by Riemannian, Finsler, semi-Riemannian and magnetic structures on manifolds. It is an extended report on problem sessions held during the International Workshop on Geodesics in August 2010 at the Chern Institute of Mathematics in Tianjin.

Keywords

geodesicsclosed geodesicentropyintegrable geodesic flowsprojectively equivalent metrics

MSC classification

Primary: 53-02: Research exposition (monographs, survey articles) 58-02: Research exposition (monographs, survey articles) 53C22: Geodesics
Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 3 , March 2021 , pp. 641 - 684
Check for updates
Copyright
© Cambridge University Press, 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alonso, J. M. and Bridson, M. R.. Semihyperbolic groups. Proc. Lond. Math. Soc. (3) 70(1) (1995), 56–114.CrossRefGoogle Scholar
Álvarez Paiva, J. C. and Berck, G.. Finsler surfaces with prescribed geodesics. Preprint, 2010,arXiv:1002.0243v1.Google Scholar
Álvarez Paiva, J. C. and Balacheff, F.. Contact geometry and isosystolic inequalities. Geom. Funct. Anal. 24 (2014), 648–669.CrossRefGoogle Scholar
Asselle, L. and Benedetti, G.. The Lusternik–Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles. J. Topol. Anal. 8 (2016), 545–570.CrossRefGoogle Scholar
Abbondandolo, A., Bramham, B., Hryniewicz, U. L. and Salomão, P. A. S.. A systolic inequality for geodesic flows on the two-sphere. Math. Ann. 367(1–2) (2017), 701–753.CrossRefGoogle Scholar
Abbondandolo, A., Bramham, B., Hryniewicz, U. L. and Salomão, P. A. S.. Sharp systolic inequalities for Reeb flows on the three-sphere. Invent. Math. 211(2) (2018), 687–778.CrossRefGoogle Scholar
Álvarez Paiva, J. C.. Some Problems on Finsler Geometry (Handbook of Differential Geometry, II) . Elsevier/North-Holland, Amsterdam, 2006, pp. 1–33.Google Scholar
Abbondandolo, A., Macarini, L., Mazzucchelli, M. and Paternain, G. P.. Infinitely many periodic orbits of exact magnetic flows on surfaces for almost every subcritical energy level. J. Eur. Math. Soc. (JEMS) 19(2) (2017), 551–579.CrossRefGoogle Scholar
Abbondandolo, A., Macarini, L. and Paternain, G. P.. On the existence of three closed magnetic geodesics for subcritical energies. Comment. Math. Helv. 90 (2015), 155–193.CrossRefGoogle Scholar
Avila, A., De Simoi, J. and Kaloshin, V.. An integrable deformation of an ellipse of small eccentricity is an ellipse. Ann. of Math. (2) 184 (2016), 527–558.CrossRefGoogle Scholar
Akbar-Zadeh, H.. Sur les espaces de Finsler à courbures sectionnelles constantes. Bull. Acad. Roy. Bel. Cl. Sci. 74 (1988), 281–322.Google Scholar
Ballmann, W.. Axial isometries of manifolds of nonpositive curvature. Math. Ann. 259 (1982), 131–144.CrossRefGoogle Scholar
Balacheff, F.. Sur la systole de la sphère au voisinage de la métrique standard. Geom. Dedicata 121 (2006), 61–71 (in French).CrossRefGoogle Scholar
Bangert, V.. Closed geodesics on complete surfaces. Math. Ann. 251(1) (1980), 83–96.CrossRefGoogle Scholar
Bangert, V.. On the existence of closed geodesics on two-spheres. Internat. J. Math. 4(1) (1993), 1–10.CrossRefGoogle Scholar
Bao, D.. On two curvature-driven problems in Riemann–Finsler geometry. Finsler Geometry, Sapporo 2005 (Advanced Studies in Pure Mathematics, 48) . Mathematical Society of Japan, Tokyo, 2007.Google Scholar
Ballmann, W. and Brin, M.. On the ergodicity of geodesic flows. Ergod. Th. & Dynam. Sys. 2 (1982), 311–315.CrossRefGoogle Scholar
Ballmann, W., Brin, M. and Eberlein, P.. Structure of manifolds of nonpositive curvature. I. Ann. of Math. (2) 122(1) (1985), 171–203.CrossRefGoogle Scholar
Besson, G., Courtois, G. and Gallot, S.. Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5 (1995), 731–799.CrossRefGoogle Scholar
Bryant, R. L., Dunajski, M. and Eastwood, M.. Metrisability of two-dimensional projective structures. J. Differential Geom. 83(3) (2009), 465–500.CrossRefGoogle Scholar
Bangert, V. and Emmerich, P.. On the flatness of Riemannian cylinders without conjugate points. Comm. Anal. Geom. 19 (2011), 773–805.CrossRefGoogle Scholar
Bangert, V. and Emmerich, P.. Area growth and rigidity of surfaces without conjugate points. J. Differential Geom. 94 (2013), 367–385.CrossRefGoogle Scholar
Benedetti, G.. The contact property for symplectic magnetic fields on S 2 . Ergod. Th. & Dynam. Sys. 36 (2016), 682–713.CrossRefGoogle Scholar
Berger, M.. Riemannian Geometry During the Second Half of the Twentieth Century (American Mathematical Society University Lecture Series, 17) . American Mathematical Society, Providence, RI, 2000.Google Scholar
Berger, M.. A Panoramic View of Riemannian Geometry. Springer, Berlin, 2003.CrossRefGoogle Scholar
Bernard, P.. On the Conley decomposition of Mather sets. Rev. Mat. Iberoam. 26(1) (2010), 115–132.CrossRefGoogle Scholar
Besse, A. L.. Manifolds all of whose geodesics are closed. Ergebnisse der Mathematik und ihrer Grenzgebiete ([Results in Mathematics and Related Areas], 93) . Springer, New York, 1978.Google Scholar
Foulon, P., Ivanov, S., Matveev, V. S. and Ziller, W.. Geodesic behavior for Finsler metrics of constant positive flag curvature on $S^{2}$ . J. Differential Geom., to appear. Preprint, 2017,  arXiv:1710.03736.Google Scholar
Benoist, Y., Foulon, P. and Labourie, F.. Flots d’Anosov a distributions stable et instable différentiables. J. Amer. Math. Soc. 5 (1992), 33–74.Google Scholar
Burns, K. and Gedeon, T.. Metrics with ergodic geodesic flow on spheres, unpublished.Google Scholar
Burns, K. and Gerber, M.. Ergodic geodesic flows on product manifolds with low-dimensional factors. J. Reine Angew. Math. 450 (1994), 1–35.Google Scholar
Burago, D. and Ivanov, S.. Riemannian tori without conjugate points are flat. Geom. Funct. Anal. 4 (1994), 259–269.CrossRefGoogle Scholar
Burago, D. and Ivanov, S.. Boundary rigidity and filling volume minimality of metrics close to a flat one. Ann. of Math. (2) 171 (2010), 1183–1211.CrossRefGoogle Scholar
Burago, D. and Ivanov, S.. Area minimizers and boundary rigidity of almost hyperbolic metrics. Duke Math. J. 162 (2013), 1205–1248.CrossRefGoogle Scholar
Burago, D. and Ivanov, S.. Boundary distance, lens maps and entropy of geodesic flows of Finsler metrics. Geom. Topol. 20 (2016), 469–490.CrossRefGoogle Scholar
Bialy, M.. First integrals that are polynomial in the momenta for a mechanical system on the two-dimensional torus. Funktsional. Anal. i Prilozhen 21(4) (1987), 64–65.Google Scholar
Bialy, M.. Convex billiards and a theorem by E. Hopf. Math. Z. 214 (1993), 147–154.CrossRefGoogle Scholar
Bialy, M.. Rigidity for periodic magnetic fields. Ergod. Th. & Dynam. Sys. 20 (2000), 1619–1626.CrossRefGoogle Scholar
Bialy, M.. Integrable geodesic flows on surfaces. Geom. Funct. Anal. 20 (2010), 357–367.CrossRefGoogle Scholar
Birkhoff, G. D.. Dynamical systems. Amer. Math. Soc. Colloq. Publ. 9 (1927), xii+305 pp.Google Scholar
Burns, K. and Katok, A.. Manifolds with non-positive curvature. Ergod. Theory & Dynam. Sys. 5(2) (1985), 307–317.CrossRefGoogle Scholar
Burns, K. and Knieper, G.. Rigidity of surfaces with no conjugate points. J. Differential Geom. 34(3) (1991), 623–650.CrossRefGoogle Scholar
Bolsinov, A. V., Kozlov, V. V. and Fomenko, A. T.. The Maupertuis’ principle and geodesic flows on S 2 arising from integrable cases in the dynamics of rigid body motion. Russian Math. Surveys 50 (1995), 473–501.CrossRefGoogle Scholar
Bangert, V. and Long, Y.. The existence of two closed geodesics on every Finsler 2-sphere. Math. Ann. 346(2) (2010), 335–366.CrossRefGoogle Scholar
Bavard, Ch. and Mounoud, P.. Sur les surfaces lorentziennes compactes sans points conjugués [On compact Lorentzian surfaces without conjugate points]. Geom. Topol. 17(1) (2013), 469–492 (in French).CrossRefGoogle Scholar
Bolsinov, A. V. and Matveev, V. S.. Local normal forms for geodesically equivalent pseudo-Riemannian metrics. Trans. Amer. Math. Soc. 367 (2015), 6719–6749.CrossRefGoogle Scholar
Bialy, M. and Mironov, A. E.. Angular billiard and algebraic Birkhoff conjecture. Adv. Math. 313 (2017), 102–126.CrossRefGoogle Scholar
Bolsinov, A. V., Matveev, V. S. and Fomenko, A. T.. Two-dimensional Riemannian metrics with an integrable geodesic flow. Local and global geometries. Sb. Math. 189(9–10) (1998), 1441–1466.CrossRefGoogle Scholar
Bryant, R. L., Manno, G. and Matveev, V. S.. A solution of a problem of Sophus Lie: normal forms of 2-dim metrics admitting two projective vector fields. Math. Ann. 340 (2008), 437–463.CrossRefGoogle Scholar
Bolsinov, A., Miranda, E., Matveev, V. and Tabachnikov, S.. Open problems, questions, and challenges in finite-dimensional integrable systems. Philos. Trans. A 376 (2018), 20170430.Google ScholarPubMed
Bolsinov, A. V., Matveev, V. S. and Rosemann, S.. Local normal forms for c-projectively equivalent metrics and proof of the Yano–Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics. Preprint, 2015, arXiv:1510.00275.Google Scholar
Bolotin, S. V.. Integrable Birkhoff billiards. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (1990), 33–36, 105 (in Russian); Engl. trans. Mosc. Univ. Mech. Bull. 45 (1990), 10–13.Google Scholar
Boubel, Ch.. The algebra of the parallel endomorphisms of a germ of pseudo-Riemannian metric. J. Differential Geom. 99 (2015), 77–123.CrossRefGoogle Scholar
Boubel, Ch.. The algebra of the parallel endomorphisms of a pseudo-Riemannian metric: semi-simple part. Math. Proc. Cambridge Philos. Soc. 159(2) (2015), 219–237.CrossRefGoogle Scholar
Bialy, M. and Polterovich, L.. Geodesic flows on the two-dimensional torus and ‘commensurability-incommensurability’ phase transitions. Funct. Anal. Appl. 20(4) (1986), 9–16, 96.Google Scholar
Bryant, R. L.. Finsler structures on the 2-sphere satisfying K = 1. Contemp. Math. 196 (1996), 27–41.CrossRefGoogle Scholar
Bryant, R. L.. Projectively flat Finsler 2-spheres of constant curvature. Selecta Math. (N.S.) 3(2) (1997), 161–203.CrossRefGoogle Scholar
Bryant, R. L.. Some remarks on Finsler manifolds with constant flag curvature. Houston J. Math. 28 (2002), 221–262.Google Scholar
Bryant, R. L.. Geodesically reversible Finsler 2-spheres of constant curvature. Inspired by S. S. Chern—A Memorial Volume in Honor of a Great Mathematician (Nankai Tracts in Mathematics, 11) . Ed. Griffiths, P. A.. World Scientific, Hackensack, NJ, 2006.Google Scholar
Bangert, V. and Schroeder, V.. Existence of flat tori in analytic manifolds of nonpositive curvature. Ann. Sci. Éc. Norm. Supér. (4) 24 (1991), 605–634.CrossRefGoogle Scholar
Balacheff, F. and Sabourau, S.. Diastolic inequalities and isoperimetric inequalities on surfaces. Ann. Sci. Éc. Norm. Supér. (4) 43(4) (2010), 579–605.CrossRefGoogle Scholar
Bryant, R. L. and Thurston, B.. mathoverflow.net/questions/54434/.Google Scholar
Burns, K.. Hyperbolic behaviour of geodesic flows on manifolds with no focal points. Ergod. Th. & Dynam. Sys. 3 (1983), 1–12.CrossRefGoogle Scholar
Burns, K.. The flat strip theorem fails for surfaces with no conjugate points. Proc. Amer. Math. Soc. 115 (1992), 199–206.CrossRefGoogle Scholar
Busemann, H.. The Geometry of Geodesics. Academic Press, New York, 1955, p. 422.Google Scholar
Buser, P.. Geometry and Spectra of Compact Riemann Surfaces (Progress in Mathematics, 106) . Birkhäuser Boston, Boston, MA, 1992.Google Scholar
Carrière, Y.. Autour de la conjecture de L. Markus sur les variétés affines. Invent. Math. 95 (1989), 615–628.CrossRefGoogle Scholar
Cooper, D. and Delp, K.. The marked length spectrum of a projective manifold or orbifold. Proc. Amer. Math. Soc. 138 (2010), 3361–3376.CrossRefGoogle Scholar
Croke, C., Dairbekov, N. and Sharafutdinov, V.. Local boundary rigidity of a compact Riemannian manifold with curvature bounded above. Trans. Amer. Math. Soc. 352 (2000), 3937–3956.CrossRefGoogle Scholar
Connell, C. and Farb, B.. Minimal entropy rigidity for lattices in products of rank one symmetric spaces. Comm. Anal. Geom. 11(5) (2003), 1001–1026.CrossRefGoogle Scholar
Connell, C. and Farb, B.. The degree theorem in higher rank. J. Differential Geom. 65(1) (2003), 19–59.CrossRefGoogle Scholar
Connell, C. and Farb, B.. Some recent applications of the barycenter method in geometry. Topology and Geometry of Manifolds (Athens, GA, 2001) (Proceedings of Symposia in Pure Mathematics, 71) . American Mathematical Society, Providence, RI, 2003, pp. 19–50.CrossRefGoogle Scholar
Croke, C., Fathi, A. and Feldman, J.. The marked length-spectrum of a surface of nonpositive curvature. Topology 31(4) (1992), 847–855.CrossRefGoogle Scholar
Contreras, G. and Iturriaga, R.. Global minimizers of autonomous Lagrangians. 22o Colóquio Brasileiro de Matemática. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999.Google Scholar
Croke, C. and Kleiner, B.. Conjugacy and rigidity for manifolds with a parallel vector field. J. Differential Geom. 39 (1994), 659–680.CrossRefGoogle Scholar
Chambers, G. R. and Liokumovich, Ye.. Optimal sweepouts of a Riemannian 2-sphere. J. Eur. Math. Soc. (JEMS) 21(5) (2019), 1361–1377.CrossRefGoogle Scholar
Cheng, J.-H., Marugame, T., Matveev, V. and Montgomery, R.. Chains in CR geometry as geodesics of a Kropina metric. Adv. Math. 350 (2019), 973–999.CrossRefGoogle Scholar
Contreras, G., Macarini, L. and Paternain, G. P.. Periodic orbits for exact magnetic flows on surfaces. Int. Math. Res. Not. IMRN 2004(8) (2004), 361–387.CrossRefGoogle Scholar
Contreras, G.. The Palais–Smale condition on contact type energy levels for convex Lagrangian systems. Calc. Var. Partial Differential Equations 27 (2006), 321–395.CrossRefGoogle Scholar
Contreras, G.. Geodesic flows with positive topological entropy, twist maps and hyperbolicity. Ann. of Math. (2) 172 (2010), 761–808.CrossRefGoogle Scholar
Contreras, G. and Paternain, G. P.. Genericity of geodesic flows with positive topological entropy on S 2 . J. Differential Geom. 61(1) (2002), 1–49.CrossRefGoogle Scholar
Carneiro, F. and Pujals, E.. Partially hyperbolic geodesic flows. Ann. Inst. H. Poincaré (C) Non Linear Anal. 31 (2014), 985–1014.CrossRefGoogle Scholar
Carneiro, F. and Pujals, E.. Curvature and partial hyperbolicity. Preprint, 2013, arXiv:1305.0678.Google Scholar
Croke, C.. Area and the length of the shortest closed geodesic. J. Differential Geom. 27(1) (1988), 1–21.CrossRefGoogle Scholar
Croke, C.. Rigidity for surfaces of nonpositive curvature. Comment. Math. Helv. 65(1) (1990), 150–169.CrossRefGoogle Scholar
Croke, Ch. and Schroeder, V.. The fundamental group of compact manifolds without conjugate points. Comment. Math. Helv. 61(1) (1986), 161–175.CrossRefGoogle Scholar
Cao, J. and Xavier, F.. A closing lemma for flat strips in compact surfaces of non-positive curvature. Preprint.Google Scholar
Dunajski, M. and Eastwood, M.. Metrisability of three-dimensional path geometries. Eur. J. Math. 2 (2016), 809–834.CrossRefGoogle Scholar
De Rham, G.. Sur la reductibilité d’un espace de Riemann. Comment. Math. Helv. 26 (1952), 328–344 (French).CrossRefGoogle Scholar
Denisova, N. V., Kozlov, V. V. and Treshchev, D. V.. Remarks on polynomial integrals of higher degree for reversible systems with a toral configuration space. Izv. Ross. Akad. Nauk Ser. Mat. 76(5) (2012), 57–72 (in Russian); Engl. trans. Izv. Math. 76(5) (2012), 907–921.Google Scholar
Donnay, V.. Geodesic flow on the two-sphere, Part I: Positive measure entropy. Ergod. Th. & Dynam. Sys. 8 (1988), 531–553.CrossRefGoogle Scholar
Donnay, V.. Geodesic flow on the two-sphere part II: Ergodicity. Dynamical Systems (College Park, MD, 1986/87) (Lecture Notes in Mathematics, 1342) . Springer, Berlin, 1988, pp. 112–153.Google Scholar
Douglas, J.. Solution of the inverse problem of the calculus of variations. Trans. Amer. Math. Soc. 50 (1941), 71–128.CrossRefGoogle Scholar
Donnay, V. and Pugh, C.. Finite horizon Riemann structures and ergodicity. Ergod. Th. & Dynam. Sys. 24 (2004), 89–106.CrossRefGoogle Scholar
Eberlein, P.. Geodesic flows in manifolds of nonpositive curvature. Smooth Ergodic Theory and its Applications (Seattle, WA, 1999) (Proceedings of Symposia in Pure Mathematics, 69) . American Mathematical Society, Providence, RI, 2001, pp. 525–571.CrossRefGoogle Scholar
Eisenhart, L. P.. Symmetric tensors of the second order whose first covariant derivatives are zero. Trans. Amer. Math. Soc. 25(2) (1923), 297–306.CrossRefGoogle Scholar
Eastwood, M. and Matveev, V. S.. Metric connections in projective differential geometry. Symmetries and Overdetermined Systems of Partial Differential Equations (Minneapolis, MN, 2006) (IMA Volumes in Mathematics and its Applications, 144) . Springer, New York, 2007, pp. 339–351.Google Scholar
Eschenburg, J.-H.. Horospheres and the stable part of the geodesic flow. Math. Z. 153 (1977), 237–251.CrossRefGoogle Scholar
Fet, A. I.. Variational problems on closed manifolds. Math. Sb. 30 (1952), 271–316 (in Russian). Engl. trans. Amer. Math. Soc. Transl. No. 90 (1953), 61 pp.Google Scholar
Flores, J. L., Javaloyes, M. A. and Piccione, P.. Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field. Math. Z. 267 (2011), 221–233.CrossRefGoogle Scholar
Flaminio, L.. Local entropy rigidity for hyperbolic manifolds. Comm. Anal. Geom. 3(3–4) (1995), 555–596.CrossRefGoogle Scholar
Freire, A. and Mañé, R.. On the entropy of the geodesic flow in manifolds without conjugate points. Invent. Math. 69(3) (1982), 375–392.CrossRefGoogle Scholar
Foulon, P.. Zermelo deformation of Finsler metrics by Killing vector fields. Electron. Res. Announc. Math. Sci. 25 (2018), 1–7.Google Scholar
Foulon, P.. Locally symmetric Finsler spaces in negative curvature. C. R. Acad. Sci. 324(10) (1997), 1127–1132.CrossRefGoogle Scholar
Frankel, T.. Manifolds of positive curvature. Pacific J. Math. 11 (1961), 165–174.CrossRefGoogle Scholar
Franks, J.. Geodesics on S 2 and periodic points of annulus homeomorphisms. Invent. Math. 108(2) (1992), 403–418.CrossRefGoogle Scholar
Gaidukov, E. V.. Asymptotic geodesics on a Riemannian manifold non-homeomorphic to a sphere. Dokl. Akad. Nauk SSSR 169 (1966), 999–1001.Google Scholar
Gromoll, D. and Grove, K.. On metrics on S 2 all of whose geodesics are closed. Invent. Math. 65 (1981), 175–177.CrossRefGoogle Scholar
Ginzburg, V.. New generalizations of Poincaré’s geometric theorem. Funct. Anal. Appl. 21 (1989), 575–611.Google Scholar
Ginzburg, V.. On closed trajectories of a charge in a magnetic field. An application of symplectic geometry. Contact and Symplectic Geometry (Publications of the Newton Institute, 8) . Ed. Thomas, C. B.. Cambridge University Press, Cambridge, 1996, pp. 131–148.Google Scholar
Guillarmou, C. and Lefeuvre, T.. The marked length spectrum of Anosov manifolds. Preprint, 2018, arXiv:1806.04218.CrossRefGoogle Scholar
Gomes, J. B. and Ruggiero, R. O.. On Finsler surfaces without conjugate points. Ergod. Th. & Dynam. Sys. 33(2) (2013), 455–474.CrossRefGoogle Scholar
Green, L. W.. Surfaces without conjugate points. Trans. Amer. Math. Soc. 76 (1954), 529–546.CrossRefGoogle Scholar
Green, L. W.. Geodesic instability. Proc. Amer. Math. Soc. 7 (1956), 438–448.CrossRefGoogle Scholar
Grjuntal, A. I.. The existence of convex spherical metrics all of whose closed nonselfintersecting geodesics are hyperbolic. Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 3–18, 237.Google Scholar
Gromov, M.. Homotopical effects of dilatation. J. Differential Geom. 13 (1978), 303–310.CrossRefGoogle Scholar
Gromov, M.. Filling Riemannian manifolds. J. Differential Geom. 18 (1983), 1–147.CrossRefGoogle Scholar
Gromov, M.. Metric Structures for Riemannian and Non-Riemannian Spaces (Progress in Mathematics, 152) . Birkhäuser Boston, Boston, MA, 1999, with appendices by M. Katz, P. Pansu and S. Semmes, translated from the French by S. M. Bates.Google Scholar
Gromov, M. and Thurston, W.. Pinching constants for hyperbolic manifolds. Invent. Math. 89 (1987), 1–12.CrossRefGoogle Scholar
Gutkin, E.. Billiard dynamics: an updated survey with the emphasis on open problems. Chaos 22 (2012), 026116, 13 pages.CrossRefGoogle ScholarPubMed
Hamenstädt, U.. A geometric characterization of negatively curved locally symmetric spaces. J. Differential Geom. 34(1) (1991), 193–221.CrossRefGoogle Scholar
Hasselblatt, B.. Regularity of the Anosov splitting and of horospheric foliations. Ergod. Th. & Dynam. Sys. 14 (1994), 645–666.CrossRefGoogle Scholar
Hedlund, G.. Geodesics on a two-dimensional Riemmnnian manifold with periodic coefficients. Ann. of Math. (2) 33 (1932), 719–739.CrossRefGoogle Scholar
Hirsch, M. W.. Differential Topology (Graduate Texts in Mathematics, 33) . Springer, New York, 1976.CrossRefGoogle Scholar
Hurder, S. and Katok, A.. Differentiability, rigidity and Godbillon–Vey classes for Anosov flows. Publ. Math. Inst. Hautes Études Sci. 72 (1990), 5–61.CrossRefGoogle Scholar
Hass, J. and Morgan, F.. Geodesic nets on the 2-sphere. Proc. Amer. Math. Soc. 124 (1996), 3843–3850.CrossRefGoogle Scholar
Hopf, E.. Closed surfaces without conjugate points. Proc. Natl Acad. Sci. USA 34 (1948), 47–51.CrossRefGoogle ScholarPubMed
Hersonsky, S. and Paulin, F.. On the rigidity of discrete isometry groups of negatively curved spaces. Comment. Math. Helv. 72 (1997), 349–388.CrossRefGoogle Scholar
Harris, A. and Paternain, G.. Dynamically convex Finsler metrics and J-holomorphic embedding of asymptotic cylinders. Ann. Global Anal. Geom. 34(2) (2008), 115–134.CrossRefGoogle Scholar
Hingston, N. and Rademacher, H.-B.. Resonance for loop homology of spheres. J. Differential Geom. 93 (2013), 133–174.CrossRefGoogle Scholar
Hasselblatt, B. and Wilkinson, A.. Prevalence of non-Lipschitz Anosov foliations. Ergod. Th. & Dynam. Sys. 19(3) (1999), 643–656.CrossRefGoogle Scholar
Hofer, H., Wysocki, K. and Zehnder, E.. Finite energy foliations of tight three-spheres and Hamiltonian dynamics. Ann. of Math. (2) 157(1) (2003), 125–257.CrossRefGoogle Scholar
Ivanov, S. and Kapovich, V.. Manifolds without conjugate points and their fundamental groups. J. Differential Geom. 96 (2014), 223–240.CrossRefGoogle Scholar
Katok, A. B.. Ergodic perturbations of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk SSSR 37 (1973), 775–778 (in Russian); Engl. trans. Math. USSR-Isv. 7 (1973), 535–571.Google Scholar
Katok, A. B.. Entropy and closed geodesics. Ergod. Th. & Dynam. Sys. 2(3–4) (1982), 339–365.CrossRefGoogle Scholar
Katok, A. B.. Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dynamical systems. Ergod. Th. & Dynam. Sys. 14 (1994), 757–785.CrossRefGoogle Scholar
Katok, A., Knieper, G. and Weiss, H.. Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows. Comm. Math. Phys. 138(1) (1991), 19–31.CrossRefGoogle Scholar
Klingenberg, W.. Riemannian manifolds with geodesic flow of Anosov type. Ann. of Math. (2) 99 (1974), 1–13.CrossRefGoogle Scholar
Klingler, B.. Complétude des variétés lorentziennes à courbure constante. Math. Ann. 306 (1996), 353–370.CrossRefGoogle Scholar
Kruglikov, B. and Matveev, V. S.. The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta. Nonlinearity 29 (2016), 1755–1768.CrossRefGoogle Scholar
Knieper, G.. The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds. Ann. of Math. (2) 148 (1998), 291–314.CrossRefGoogle Scholar
Kolokoltsov, V. N.. Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial with respect to velocities. Math. USSR-Izv. 21(2) (1983), 291–306.CrossRefGoogle Scholar
Kozma, L. and Peter, I. R.. Intersection theorems for Finsler manifolds. Publ. Math. Debrecen 57 (2000), 193–201.Google Scholar
Knieper, G. and Weiss, H. N.. C ∞ -genericity of positive topological entropy for geodesic flows on S 2 . J. Differential Geom. 62(1) (2002), 127–141.CrossRefGoogle Scholar
Lang, J.. Projectively equivalent Finsler metrics on surfaces of negative Euler characteristic. Preprint, 2019, arXiv:1908.02701.CrossRefGoogle Scholar
Lebedeva, N.. On the fundamental group of a compact space without conjugate points. PDMI Preprint 5/2002, 2002, http://www.pdmi.ras.ru/preprint/2002/02-05.html.Google Scholar
Levi-Civita, T.. Sulle trasformazioni delle equazioni dinamiche. Ann. Mat. Pura Appl. (2) 24 (1896), 255–300.CrossRefGoogle Scholar
Lyusternik, L. A. and Fet, A. I.. Variational problems on closed manifolds. Dokl. Akad. Nauk. SSSR 81 (1951), 17–18.Google Scholar
Lange, C. and Mettler, T.. On a duality between certain Finsler 2-spheres and Weyl orbifolds. Preprint, 2018, arXiv:1812.00827.Google Scholar
Long, Y.. Multiplicity and stability of closed geodesics on Finsler 2-spheres. J. Eur. Math. Soc. (JEMS) 8(2) (2006), 341–353.CrossRefGoogle Scholar
Lusternik, L. and Schnirelmann, L.. Existence de trois géodésiques fermées sur toute surface de genre 0. C. R. Acad. Sci. Paris 188 (1929), 269–271.Google Scholar
Lusternik, L. and Schnirelmann, L.. Sur le problème de trois géodésiques fermées sur toute surface de genre 0. C. R. Acad. Sci. Paris 189 (1929), 534–536.Google Scholar
Long, Y. and Wang, W.. Stability of closed geodesics on Finsler 2-spheres. J. Funct. Anal. 255(3) (2008), 620–641.CrossRefGoogle Scholar
Manning, A.. Topological entropy for geodesic flows. Ann. of Math. (2) 110(3) (1979), 567–573.CrossRefGoogle Scholar
Mañé, R.. On a theorem of Klingenberg. Dynamical Systems and Bifurcation Theory (Rio de Janeiro, 1985) (Pitman Research Notes in Mathematics Series, 160) . Longman Science and Technology, Harlow, 1987, pp. 319–345.Google Scholar
Mañé, R.. The Lyapunov exponents of generic area preserving diffeomorphisms. International Conference on Dynamical Systems (Montevideo, 1995) (Pitman Research Notes in Mathematics Series, 362) . Longman, Harlow, 1996, pp. 110–119.Google Scholar
Marsden, J. E.. On completeness of homogeneous pseudo-Riemannian manifolds. Indiana Math. J. 22 (1973/74), 1065–1066.CrossRefGoogle Scholar
Margulis, G.. On Some Aspects of the Theory of Anosov Systems, With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Translated from the Russian by Valentina Vladimirovna Szulikowska (Springer Monographs in Mathematics) . Springer, Berlin, 2004.CrossRefGoogle Scholar
Masur, H.. On a class of geodesics in Teichmüller space. Ann. of Math. (2) 102 (1975), 205–221.CrossRefGoogle Scholar
Mather, J. N.. Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207(2) (1991), 169–207.CrossRefGoogle Scholar
Matveev, V. S.. Hyperbolic manifolds are geodesically rigid. Invent. Math. 151 (2003), 579–609.CrossRefGoogle Scholar
Matveev, V. S.. Three-dimensional manifolds having metrics with the same geodesics. Topology 42(6) (2003), 1371–1395.CrossRefGoogle Scholar
Mather, J. N.. Examples of Aubry sets. Ergod. Th. & Dynam. Sys. 24(5) (2004), 1667–1723.CrossRefGoogle Scholar
Matveev, V. S.. Lichnerowicz–Obata conjecture in dimension two. Comment. Math. Helv. 81(3) (2005), 541–570.CrossRefGoogle Scholar
Matveev, V. S.. Proof of projective Lichnerowicz–Obata conjecture. J. Differential Geom. 75 (2007), 459–502.CrossRefGoogle Scholar
Matveev, V. S.. Geodesically equivalent metrics in general relativity. J. Geom. Phys. 62 (2012), 675–691.CrossRefGoogle Scholar
Matveev, V. S.. Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics. J. Math. Soc. Japan 64 (2012), 107–152.CrossRefGoogle Scholar
Matveev, V. S.. Can we make a Finsler metric complete by a trivial projective change? Proceedings of the VI International Meeting on Lorentzian Geometry (Granada, September 6–9, 2011) (Springer Proceedings in Mathematics & Statistics, 26) . Springer, New York, 2013, pp. 231–243.Google Scholar
Merlin, L.. Minimal entropy for uniform lattices in PSL(ℝ) × PSL(ℝ). Comment. Math. Helv. 91(1) (2016), 107–129.CrossRefGoogle Scholar
Michel, R.. Sur la rigidité imposée par la longueur des géodésiques. Invent. Math. 65 (1981/82), 71–83.CrossRefGoogle Scholar
Milnor, J.. Morse Theory (Annals of Mathematics Studies, 51) . Princeton University Press, Princeton, NJ, 1963, based on Lecture Notes by M. Spivak and R. Wells.CrossRefGoogle Scholar
Miller, S.. Geodesic knots in the figure-eight knot complement. Experiment. Math. 10(3) (2001), 419–436.CrossRefGoogle Scholar
Mirzakhani, M.. Growth of the number of simple closed geodesics on hyperbolic surfaces. Ann. of Math. (2) 168 (2008), 97–125.CrossRefGoogle Scholar
Mironov, A. E.. Polynomial integrals of a mechanical system on a two-dimensional torus. Izv. Ross. Akad. Nauk Ser. Mat. 74(4) (2010), 145–156 (in Russian); Engl. trans. Izv. Math. 74(4) (2010), 805–817.Google Scholar
Morse, H. M.. A fundamental class of geodesics on any closed surface of genus greater than one. Trans. Amer. Math. Soc. 26 (1924), 25–60.CrossRefGoogle Scholar
Mostow, G. D. and Siu, Y. T.. A compact Kähler surface of negative curvature not covered by the ball. Ann. of Math. (2) 112 (1980), 321–360.CrossRefGoogle Scholar
Mounoud, P. and Suhr, S.. Pseudo-Riemannian geodesic foliations by circles. Math. Z. 274(1–2) (2013), 225–238.CrossRefGoogle Scholar
Matveev, V. S. and Topalov, P. J.. Trajectory equivalence and corresponding integrals. Regul. Chaotic Dyn. 3(2) (1998), 30–45.CrossRefGoogle Scholar
Matveev, V. S. and Topalov, P. J.. Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. ERA-AMS 6 (2000), 98–104.Google Scholar
Matveev, V. S. and Trautman, A.. A criterion for compatibility of conformal and projective structures. Comm. Math. Phys. 329 (2014), 821–825.CrossRefGoogle Scholar
Nabutovsky, A. and Rotman, R.. The length of the shortest closed geodesic on a 2-dimensional sphere. Int. Math. Res. Not. IMRN 23 (2002), 1211–1222.CrossRefGoogle Scholar
Ontaneda, P.. Pinched smooth hyperbolization. Preprint, 2011, arXiv:1110.6374.Google Scholar
Ontaneda, P.. Riemannian hyperbolization. Preprint, 2014, arXiv:1406.1730.Google Scholar
Otal, J.-P.. Le spectre marqué des longueurs des surfaces à courbure négative. Ann. of Math. (2) 131(1) (1990), 151–162.CrossRefGoogle Scholar
Paternain, G. P.. Finsler structures on surfaces with negative Euler characteristic. Houston J. Math. 23(3) (1997), 421–426.Google Scholar
Pesin, Ya.. Geodesic flows in closed Riemannian manifolds without focal points. Izv. Akad. Nauk SSSR Ser. Mat. 41(6) (1977), 1252–1288, 1447.Google Scholar
Preissman, A.. Quelques propriétés globales des espaces de Riemann. Comment. Math. Helv. 15 (1943), 175–216 (in French).CrossRefGoogle Scholar
Pestov, L. and Uhlmann, G.. Two dimensional compact simple Riemannian manifolds are boundary distance rigid. Ann. of Math. (2) 161 (2005), 1093–1110.CrossRefGoogle Scholar
Rademacher, H.-B.. On the average indices of closed geodesics. J. Differential Geom. 29(1) (1989), 65–83.CrossRefGoogle Scholar
Rademacher, H.-B.. A sphere theorem for non-reversible Finsler metrics. Math. Ann. 328 (2004), 373–387.CrossRefGoogle Scholar
Rademacher, H.-B.. Bumpy metrics on spheres and minimal index growth. Preprint, 2016,  arXiv:1608.01937.CrossRefGoogle Scholar
Reid, A.. Closed hyperbolic 3-manifolds whose closed geodesics all are simple. J. Differential Geom. 38(3) (1993), 545–558.Google Scholar
Rivin, I.. Simple curves on surfaces. Geom. Dedicata 87 (2001), 345–360.CrossRefGoogle Scholar
Rodriguez Hertz, F.. On the geodesic flow of surfaces of nonpositive curvature. Preprint, 2003, arXiv:0301010.Google Scholar
Rotman, R.. The length of a shortest closed geodesic and the area of a 2-dimensional sphere. Proc. Amer. Math. Soc. 134 (2006), 3041–3047.CrossRefGoogle Scholar
Ruggiero, R. O.. On non-hyperbolic quasi-convex spaces. Trans. Amer. Math. Soc. 350 (1998), 665–687.CrossRefGoogle Scholar
Radeschi, M. and Wilking, B.. On the Berger conjecture for manifolds all of whose geodesics are closed. Preprint, 2015, arXiv:1511.07852.Google Scholar
Sabourau, S.. Filling radius and short closed geodesic of the sphere. Bull. Soc. Math. France 132 (2004), 105–136.CrossRefGoogle Scholar
Sanchez, M.. On the completeness of trajectories for some mechanical systems. Geometry, Mechanics, and Dynamics (Fields Institute Communications, 73) . Springer, New York, 2015, pp. 343–372.CrossRefGoogle Scholar
Schmidt, B. G.. Conditions on a connection to be a metric connection. Comm. Math. Phys. 29 (1973), 55–59.CrossRefGoogle Scholar
Sharafutdinov, V. and Uhlmann, G.. On deformation boundary rigidity and spectral rigidity of Riemannian surfaces with no focal points. J. Differential Geom. 56 (2000), 93–110.CrossRefGoogle Scholar
Suhr, S.. Closed geodesics in Lorentzian surfaces. Trans. Amer. Math. Soc. 365 (2013), 1469–1486.CrossRefGoogle Scholar
Suhr, S.. A Counterexample to Guillemin’s Zollfrei conjecture. J. Topol. Anal. 5 (2013), 251–261.CrossRefGoogle Scholar
Tabachnikov, S.. Geometry and Billiards (Student Mathematical Library, 30) . American Mathematical Society, Providence, RI, 2005.CrossRefGoogle Scholar
Taimanov, I. A.. Closed extremals on two-dimensional manifolds. Russian Math. Surveys 47(2) (1992), 163–211.CrossRefGoogle Scholar
Tao, T.. http://mathoverflow.net/questions/37651.Google Scholar
Taubes, C. H.. The Seiberg–Witten equations and the Weinstein conjecture. Geom. Topol. 11 (2007), 2117–2202.CrossRefGoogle Scholar
Thomas, T. Y.. On the projective and equi-projective geometries of paths. Proc. Natl. Acad. Sci. USA 11 (1925), 199–203.CrossRefGoogle Scholar
Vignéras, M. F.. Variétés Riemanniennes isopectrales et non isométriques. Ann. Math. 112 (1980), 21–32.CrossRefGoogle Scholar
Weinstein, A.. Sur la non-densité des géodésiques fermées. C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A504.Google Scholar
Weyl, H.. Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffassung, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1921; ‘Selecta Hermann Weyl’. Birkhäuser, Basel, 1956.Google Scholar
Wu, H.. Holonomy groups of indefinite metrics. Pacific J. Math. 20 (1967), 351–392.CrossRefGoogle Scholar
Wu, W.. On the ergodicity of geodesic flows on surfaces of nonpositive curvature. Ann. Fac. Sci. Toulouse Math. (6) 24 (2015), 625–639.CrossRefGoogle Scholar
Ziller, W.. Geometry of the Katok examples. Ergod. Th. & Dynam. Sys. 3 (1983), 135–157.CrossRefGoogle Scholar
Zoll, O.. Ueber Flächen mit Scharen von geschlossenen geodätischen Linien. Math. Ann. 57 (1903), 108–133.CrossRefGoogle Scholar