Papers in preparation

• Open problems and questions about geodesics, (with K. Burns)

  Submitted Papers

• The Binet-Legendre Metric in Finsler Geometry, (with M. Troyanov), submitted to Geom. and Topol.
• Proof of the Yano-Obata Conjecture for holomorph-projective transformations, (with S. Rosemann), submitted to J. Diff. Geom.
• Some Remarks on Nijenhuis Bracket, Formality, and Kähler Manifolds, (with P. de Bartolomeis) submitted to Advances in Geometry.

Published or accepted papers (updated in Feb. 2012)

• 72. On projective equivalence and pointwise projective relation of Randers metrics, accepted to Int. J. Math.
• 71. Can we make a Finsler metric complete by a trivial projective change? accepted to proceedings of the VI International Meeting on Lorentzian Geometry (Granada, September 6--9, 2011).
• 70. On the dimension of the group of projective transformations of closed Randers and Riemannian manifolds, SIGMA, 8(2012), 007, 4 pages.
• 69. On integrable natural hamiltonian systems on the suspention of toric automorphisms, accepted to Qual. Theor. Dyn. Syst., doi: 10.1007/s12346-012-0067-z
• 68. Two remarks on $PQ^\epsilon$-projectivity of Riemannian metrics, (with S. Rosemann), accepted to Glasgow Math. Journal.
• 67. Every closed Kähler manifold with degree of mobility >2 is (CP(n), g_{Fubini-Study}), (with A. Fedorova, V. Kiosak, S. Rosemann), Proc. Lond. Math. Soc. doi: 10.1112/plms/pdr053
• 66. Geodesically equivalent metrics in general relativity, J. Geom. Phys. Volume 62, Issue 3, March 2012, Pages 675–691.
• 65. Two-dimensional superintegrable metrics with one linear and one cubic integral, (with V. Shevchishin),Journal of Geometry and Physics 61(2011), no 8, pp. 1353-1377
• 64. 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. Jpn. 64(2012) no. 1, 107--152.
• 63. On the Degree of Geodesic Mobility for Riemannian Metrics, (with V. A. Kiosak, J. Mikes, and I. G. Shandra) Math Notes 87(2010), no. 4, 628–629. Russian original
• 62. Differential invariants for cubic integrals of geodesic flows on surfaces, (with Vsevolod V. Shevchishin), J. Geom. Phys. Volume 60(2010) no. 6-8, 833-856
• 61. Gallot-Tanno Theorem for closed incomplete pseudo-Riemannian manifolds and applications, (with Pierre Mounoud) Global. Anal. Geom. 38(2010), no. 3, 259-271.
• 60. Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two, (with Volodymyr Kiosak), Comm. Mat. Phys. 297(2010), 401-426.
• 59. Two-dimensional metrics admitting precisely one projective vector field, with the Appendix Dini theorem for pseudoriemannian metrics (with A. Bolsinov and G. Pucacco), accepted to Math. Ann.
• 58. Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics, (with A. Bolsinov), Transactions of the American Mathematical Society 363(8), 4081-4107
• 57. Compatibility of Gauss maps with metrics, (with J.-H. Eschenburg, B. S. Kruglikov, R. Tribuzy), J. Differential Geometry and Its Applications. 28(2010) no. 2, 228-235.
• 56. Gallot-Tanno theorem for pseudo-Riemannian metrics and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist, J. Differential Geometry and Its Applications 28(2010) no. 2, 236-240.
• 55. There are no conformal Einstein rescalings of complete pseudo-Riemannian Einstein metrics, (with Volodymyr Kiosak), C. R. Acad. Sci. Paris, Ser. I 347(2009) 1067–1069
• 54. Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta (with A. Bolsinov and G. Pucacco), J. Geom. Phys. 59(2009), no. 7, 1048–1062
• 53. Fubini Theorem for pseudo-Riemannian metrics, (with Alexey V. Bolsinov, and Volodymyr Kiosak), Journal of the London Mathematical Society 80(2009) no. 2, 341–356
• 52. Complete Einstein metrics are geodesically rigid, (with Volodymyr Kiosak), Comm. Math. Phys. 289(2009), no. 1, 383–400
• 51. Riemannian metrics having the same geodesics with Berwald metrics, Publ. Math. Debrecen 74(2009) no. 3-4, 405-416
• 50. Conformal Lichnerowicz-Obata conjecture, (with Hans-Bert Rademacher, Marc Troyanov, and Abdelghani Zeghib), Annales de l'institut Fourier, 59(2009), no. 3, 937-949
• 49. On ``All regular Landsberg metrics are always Berwald" by Z. I. Szabo, Balkan Journ. Geom. 14(2009), No. 2
• 48. A solution of S. Lie Problem: Normal forms of 2-dim metrics admitting two projective vector fields, (with R. Bryant und G. Manno), Math. Ann. 340(2008), no. 2, 437-463
• 47. Proof of Projective Lichnerowicz-Obata Conjecture, J. of Differential Geometry, 75(2007), 459-502.
• 46. Metric Connections in Projective Differential Geometry, (with Michael Eastwood), Symmetries and Overdetermined Systems of Partial Differential Equations (Minneapolis, MN, 2006), 339--351, IMA Vol. Math. Appl., 144(2007), Springer, New York.
• 45. Geometric explanation of the Beltrami Theorem, Int. J. Geom. Methods Mod. Phys. 3(2006), no. 3, 623--629.
• 44. On vanishing of topological entropy for certain integrable systems, (with B. Kruglikov), Electron. Res. Announc. Amer. Math. Soc. 12(2006), 19--28
• 43. Lichnerowicz-Obata conjecture in dimension two, Comm. Math. Helv. 80(2005) no. 3, 541-570.
• 42. Strictly non-proportional geodesically equivalent metrics have zero topological entropy, (with B. Kruglikov), Ergodic Theory and Dynamical Systems 26 (2006) no. 1, 247-266.
• 41. On the rigidity of magnetic systems with the same magnetic geodesics, (with K. Burns) Proc. Amer. Math. Soc. 134 (2006), 427-434.
• 40. On degree of mobility of complete metrics, Compt. Math., 43(2006), 221--250.
• 39. Beltrami problem, Lichnerowicz-Obata conjecture and applications of integrable systems in differential geometry, Tr. Semin. Vektorn. Tenzorn. Anal, 26(2005), 214--238.
• 38. On projectively equivalent metrics near points of bifurcation, In the book "Topologival methods in the theory of integrable systems"'(Eds.: Bolsinov A.V., Fomen ko A.T., Oshemkov A.A.), Cambridge scientific publishers, pp. 213 -- 240, arXiv:0809.3602.
• 37. New integrable system on the sphere, Math. Res. Lett., Vol. 11 (2004), 715--722.
• 36. Solodovnikov's theorem in dimension two, Dokl. Math. 69(2004), no. 3, 338--341.
• 35. Closed manifolds admitting metrics with the same geodesics, Proceedings of SPT2004 (Cala Gonone). World Scientific (2005), 198-209.
• 34. Projectively equivalent metrics on the torus, Differential Geom. Appl., 20(2004) 251-265.
• 33. The eigenvalues of the Sinjukov mapping are globally ordered, Math. Notes 77(2005) no. 3-4. 380-390.
• 32. Die Vermutung von Obata für Dimension 2, Arch. Math. 82(2004) 273--281.
• 31. Three-dimensional manifolds having metrics with the same geodesics, Topology 42(2003) no. 6, 1371-1395.
• 30. Hyperbolic manifolds are geodesically rigid, Invent. math. 151(2003), 579-609.
• 29. Three-manifolds admitting metrics with the same geodesics, Math. Res. Lett. 9(2002), no. 2-3, 267--276.
• 28. Geometrical interpretation of Benenti systems, (with A. Bolsinov) Journal of Geometry and Physics, 44(2003), 489-506
• 27. Geodesic equivalence via integrability, (with P. Topalov), Geometriae Dedicata 96(2003) 91--115.
• 26. Geschlossene hyperbolische 3-Mannigflatigkeiten sind geodätisch starr. [Three-dimensional closed hyperbolic manifolds are geodesically rigid] , Manuscripta Math. 105(2001), no. 3, 343--352.
• 25. Quantum integrability and complete separation of variables for projectively equivalent metrics on the torus, Geometry, integrability and quantization (Varna, 2000), 228--244, Coral Press Sci. Publ., Sofia, 2001.
• 24. Integrability in theory of geodesically equivalent metrics, J. Phys. A., 34(2001), 2415--2433.
• 23. Metric with ergodic geodesic flow is completely determined by unparameterized geodesics, (with P. Topalov), ERA-AMS, 6(2000).
• 22. Quantum integrability for the Beltrami-Laplace operator as geodesic equivalence, (with P. Topalov), Math. Z. 238(2001), no. 4, 833--866.
• 21. Commuting operators and separation of variables for Laplacians of projectively equivalent metrics, Let. Math. Phys., 54, 193-201, 2000.
• 20. Geodesic equivalence of metrics as a particular case of integrability of geodesic flows, (with P. Topalov), Theor. Math. Phys. 123(2000) no 2. 285--293.
• 19. Dynamical and topological methods in theory of geodesically equivalent metrics, (with P. Topalov), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 266 (2000), Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 5, 155--168, 338.
• 18. Quantum integrability of the Beltrami-Laplace operator for geodesically equivalent metrics, Russian Math. Doklady 61(2000), no 2, 216--219.
• 17. Riemannian metrics with integrable geodesic flows on surfaces: local and global geometry, (with A. V. Bolsinov and A. T. Fomenko) - Mat. Sb. 189(1998), no. 10, 5--32. ( (extended version)
• 16. Algorithmic classification of invariant neighborhoods of saddle-saddle points, (with A. Oshemkov) - Vestnik Moskov. Univ. Ser. I Mat. Mekh. (Moscow University Math. Bull), 1999, no. 2, 62 -- 65.
• 15. On Integrals of Third Degree in Momenta, (with H. Dullin and P. Topalov) - Regular and Chaotic Dynamics, 4(1999), no. 3, 35--44.
• 14. Geodesic equivalence of metrics on surfaces as integrability, (with P. Topalov) - Doklady of Russian Academy of science (Russian Math. Doklady 60, 112--114), 367(1999), no. 6. 736--738.
• 13. Trajectory equivalence and corresponding integrals, (with P. Topalov) - Regular and Chaotic Dynamics, 3(1998), 30--45.
• 12. Integrable Hamiltonian Systems with two degrees of freedom. Topological structure of saturated neighborhoods of non-degenerate singular points, (with A.V. Bolsinov), - In the book Tensor and Vector Analysis, Gordon & Breach 1998, 31--57.
• 11. Geodesic flows on the Klein bottle, integrable linear or quadratic in velocities, - In the book: Topological methods in Theory of Hamiltonian Systems, Factorial 1998, 213--223.
• 10. Asymptotic eigenvalues of the operator \nabla D(x,y) \nabla, corresponding to Liouville metrics, and wave on water, caught by bottom non-homogeneity, - Mat. Zam.(Math. Notes) 64(1998), no. 3-4, 357--363.
• 9. If a metric on the sphere is geodesically equivalent to a metric of constant curvature, then it is a metric of constant curvature, (with P. Topalov), Vestnik Moskov. Univ. Ser. I Mat. Mekh (Moscow University Math. Bull), 1998, no. 5.
• 8. Topological structure of integrable geodesic flow on the Klein bottle, Regular and Chaotic Dynamics, 3(1998)
• 7. Conjugate points of hyperbolic geodesics of quadratically integrable geodesics flows, ( with Peter Topalov) - Vestnik Moskov. Univ. Ser. I Mat. Mekh (Moscow University Math. Bull), 1998, no. 1, 60--62.
• 6. Jacobi vector fields for integrable geodesic flows, (with P. Topalov) - Regular and Chaotic Dynamics, 2 (1997), 103--116.
• 5. An example of geodesic flow on the Klein Bottle, integrable in polynomial in momenta of fourth degree, - Vestnik Moskov. Univ. Ser. I Mat. Mekh (Moscow University Math. Bull), 1997, no. 4, 47--48.
• 4. Quadratically integrable geodesic flows on the torus and the Klein bottle, - Regular and Chaotic Dynamics, 2(1997), 96--102.
• 3. Singularities of momentum maps of integrable Hamiltonian systems with two degrees of freedom, (with A. Bolsinov) J. Math. Sci., New York 94(1999), No.4, 1477-1500; translation from Zap. Nauchn. Semin. POMI 235(1996), 54-86.
• 2. Integrable Hamiltonian Systems with two degrees of freedom. Topological structure of saturated neighborhoods of saddle-saddle and focus points, - Mat. Sb. 187(1996), no. 4, 29-58.
• 1. Computation of values of the Fomenko invariant for a point of the type ``saddle-saddle''of an integrable Hamiltonian system, Tr. Semin. Vektorn. Tenzorn. Anal, 25(1993), pp. 75-105 (in Russian).

Preprints

• Projective Lichnerowicz-Obata Conjecture , Freiburg University, Nr. 20/2004 .
• Three-manifolds admitting metrics with the same geodesics, Freiburg University, Nr. 24/2002.
• Hyperbolic manifolds are geodesically rigid, Freiburg University, Nr. 06/2002.
• Benenti systems and projective equivalence, Freiburg University, Nr. 11/2001.
• Projectively equivalent metrics on the torus, Isaac Newton Institute for Mathematical Sciences Preprint Series NI01045-ITS (2001).
• Geodesic equivalence via integrability, Isaac Newton Institute for Mathematical Sciences Preprint Series NI00018-SGT (2000).
• Integrabilities in theory of geodesically equivalent metrics, (with P. Topalov), Warwick Preprint Series 17/2000 (2000).
• Quantum integrability for the Beltrami-Laplace operator for geodesically equivalent metrics. Integrability criterium for geodesic equivalence. Separation of variables, (with P. Topalov), IHES Preprint Series M/00/17 (2000).
• Geodesic equivalence of metrics as a particular case of integrability of geodesic flows, (with P. Topalov), MPIM Preprint Series, 1999(47).
• Quantum integrability of Beltrami-Laplace operator for geodesically equivalent metrics, (with P. Topalov), MPIM Preprint Series, 1999(21).
• Riemannian metrics with integrable geodesic flows on surfaces: local and global geometry, (with A. V. Bolsinov and A. T. Fomenko), MPIM Preprint Series 1998(120).
• Geodesic equivalence and integrability, (with P. Topalov), MPIM Preprint Series 1998(74).