Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics


Authors: Alexey V. Bolsinov and Vladimir S. Matveev
Journal: Trans. Amer. Math. Soc. 363 (2011), 4081-4107
MSC (2010): Primary 53A20, 53A35, 53A45, 53B20, 53B30, 53C12, 53C21, 53C22, 37J35
DOI: https://doi.org/10.1090/S0002-9947-2011-05187-1
Published electronically: March 21, 2011
MathSciNet review: 2792981
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Two metrics $ g $ and $ \bar g$ are geodesically equivalent if they share the same (unparameterized) geodesics. We introduce two constructions that allow one to reduce many natural problems related to geodesically equivalent metrics, such as the classification of local normal forms and the Lie problem (the description of projective vector fields), to the case when the $ (1,1)-$tensor $ G^i_j:= g^{ik}\bar g_{kj}$ has one real eigenvalue, or two complex conjugate eigenvalues, and give first applications. As a part of the proof of the main result, we generalise the Topalov-Sinjukov (hierarchy) Theorem for pseudo-Riemannian metrics.


References [Enhancements On Off] (What's this?)

  • [Am1] A. V. Aminova, Pseudo-Riemannian manifolds with general geodesics, Russian Math. Surveys 48 (1993), no. 2, 105-160. MR 1239862
  • [Am2] A. V. Aminova, Projective transformations of pseudo-Riemannian manifolds. Geometry, 9. J. Math. Sci. (N. Y.) 113(2003), no. 3, 367-470. MR 1965077 (2004a:53089)
  • [Be] E. Beltrami, Resoluzione del problema: riportari i punti di una superficie sopra un piano in modo che le linee geodetische vengano rappresentante da linee rette, Ann. Mat., 1 (1865), no. 7, 185-204.
  • [BM] A. V. Bolsinov, V. S. Matveev, Geometrical interpretation of Benenti's systems, J. of Geometry and Physics, 44(2003), 489-506, MR 1943174.
  • [BMM] R. L. Bryant, G. Manno, V. S. Matveev, A solution of a problem of Sophus Lie: Normal forms of $ 2$-dim metrics admitting two projective vector fields, Math. Ann. 340(2008), no. 2, 437-463. MR 2368987 (2008m:53037)
  • [Ca] L. Carleson, Mergelyan's theorem on uniform polynomial approximation, Math. Scand. 15(1964), 167-175. MR 0198209 (33:6368)
  • [DR] G. de Rham, Sur la reductibilité d'un espace de Riemann, Comment. Math. Helv. 26(1952), 328-344. MR 0052177 (14:584a)
  • [EM] M. Eastwood, V. S. Matveev: Metric connections in projective differential geometry, Symmetries and Overdetermined Systems of Partial Differential Equations (Minneapolis, MN, 2006), 339-351, IMA Vol. Math. Appl., 144(2007), Springer, New York. MR 2384718 (2009a:53018)
  • [Ei] L. P. Eisenhart, Non-Riemannian Geometry, American Mathematical Society Colloquium Publications VIII(1927). MR 1466961 (98j:53001)
  • [Fu] G. Fubini, Sui gruppi transformazioni geodetiche, Mem. Acc. Torino 53(1903), 261-313.
  • [GWW] G. W. Gibbons, C. M. Warnick, M. C. Werner, Light-bending in Schwarzschild-de-Sitter: projective geometry of the optical metric, Class. Quant. Grav. 25(2008), 245009 (8 pages). MR 2461162
  • [Haa] J. Haantjes, On $ X_m$-forming sets of eigenvectors, Nederl. Akad. Wetensch. Proc. Ser. A. 58(1955) = Indag. Math. 17(1955), 158-162. MR 0070232 (16:1151a)
  • [Hal1] G. S. Hall, Some remarks on symmetries and transformation groups in general relativity, Gen. Relativity Gravitation 30(1998), no. 7, 1099-1110. MR 1632505 (99e:83013)
  • [Hal2] G. S. Hall, Projective symmetry in FRW spacetimes, Classical Quantum Gravity 17(2000), no. 22, 4637-4644. MR 1797960 (2001i:83020)
  • [Hal3] G. S. Hall, Symmetries and curvature structure in general relativity, World Scientific Lecture Notes in Physics, 46. World Scientific Publishing Co., Inc., River Edge, NJ, 2004. MR 2109072 (2005j:83001)
  • [HL1] G. S. Hall, D. P. Lonie, Projective collineations in spacetimes, Classical Quantum Gravity 12(1995), no. 4, 1007-1020. MR 1330299 (96b:83017)
  • [HL2] G. S. Hall, D. P. Lonie, The principle of equivalence and projective structure in spacetimes, Classical Quantum Gravity 24(2007), no. 14, 3617-3636. MR 2339411 (2008h:53124)
  • [HL3] G. S. Hall, D. P. Lonie, The principle of equivalence and cosmological metrics, J. Math. Phys. 49(2008), no. 2., 022502 (13 pages). MR 2392851 (2009b:53136)
  • [Hi] N. J. Higham, Functions of matrices. Theory and computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. MR 2396439 (2009b:15001)
  • [KM1] V. Kiosak, V. S. Matveev, Complete Einstein metrics are geodesically rigid, Comm. Math. Phys. 289(1), 383-400, 2009. MR 2504854
  • [KM2] V. Kiosak, V. S. Matveev, Proof of the projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two, Comm. Math. Phys. 297 (2010), 401-426. MR 2651904
  • [La] J.-L. Lagrange, Sur la construction des cartes géographiques, Nouveaux Mémoires de l'Académie des Sciences et Bell-Lettres de Berlin, 1779.
  • [LC] T. Levi-Civita, Sulle transformazioni delle equazioni dinamiche, Ann. di Mat., serie $ 2^a$, 24(1896), 255-300.
  • [Lie] S. Lie, Untersuchungen über geodätische Kurven, Math. Ann. 20(1882); Sophus Lie Gesammelte Abhandlungen, Band 2, erster Teil, 267-374. Teubner, Leipzig, 1935.
  • [MM] G. Manno, V. S. Matveev, $ 2$-dim metrics admitting two projective vector fields near the points where the vector fields are linearly dependent, in preparation.
  • [MT1] V. S. Matveev, P. J. Topalov, Trajectory equivalence and corresponding integrals, Regular and Chaotic Dynamics, 3(1998), no. 2, 30-45. MR 1693470
  • [MT2] V. S. Matveev, P. J. Topalov, Quantum integrability for the Beltrami-Laplace operator as geodesic equivalence, Math. Z. 238(2001), 833-866. MR 1872577 (2002k:58068)
  • [Ma1] V. S. Matveev, Hyperbolic manifolds are geodesically rigid, Invent. Math. 151(2003), 579-609. MR 1961339 (2004f:53044)
  • [Ma2] V. S. Matveev, Three-dimensional manifolds having metrics with the same geodesics, Topology 42(2003) no. 6, 1371-1395. MR 1981360.
  • [Ma3] V. S. Matveev, Lichnerowicz-Obata conjecture in dimension two, Comm. Math. Helv. 81(2005) no. 3, 541-570. MR 2165202 (2006g:53134)
  • [Ma4] V. S. Matveev, Beltrami problem, Lichnerowicz-Obata conjecture and applications of integrable systems in differential geometry, Tr. Semin. Vektorn. Tenzorn. Anal. 26(2005), 214-238.
  • [Ma5] V. S. Matveev, Proof of projective Lichnerowicz-Obata conjecture, J. Diff. Geom. 75(2007), 459-502. MR 2301453 (2007m:53030)
  • [Ma6] V. S. Matveev, On projectively equivalent metrics near points of bifurcation, In ``Topological methods in the theory of integrable systems''(Eds.: Bolsinov A.V., Fomenko A.T., Oshemkov A.A.; Cambridge scientific publishers), pp. 213 - 240. MR 2454556
  • [Ma7] V. S. Matveev, Two-dimensional metrics admitting precisely one projective vector field, to appear in Math. Ann., arXiv:math/0802.2344.
  • [Ma8] V. S. Matveev, 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., to appear, arXiv:math/1002.3934.
  • [Mi1] J. Mikes, Global geodesic mappings and their generalizations for compact Riemannian spaces. Differential geometry and its applications. Proceedings of the 5th international conference, Opava, Czechoslovakia, August 24-28, 1992. Silesian Univ. Math. Publ. 1(1993), 143-149. http://www.emis.de/proceedings/5ICDGA/III/mike.ps . MR 1255535 (94m:53059)
  • [Mi2] J. Mikes, Geodesic mappings of affine-connected and Riemannian spaces. Geometry, 2, J. Math. Sci. 78(1996), no. 3, 311-333. MR 1384327 (97b:53043)
  • [Pa] P. Painlevé, Sur les intégrales quadratiques des équations de la Dynamique, Compt. Rend., 124(1897), 221-224.
  • [Pe1] A. Z. Petrov, Einstein spaces, Pergamon Press. XIII, 1969. MR 0244912 (39:6225)
  • [Pe2] A. Z. Petrov, New methods in the general theory of relativity. (in Russian) Izdat. ``Nauka'', Moscow, 1966. MR 0207365 (34:7181)
  • [Sc] J. A. Schouten, Erlanger Programm und Übertragungslehre. Neue Gesichtspunkte zur Grundlegung der Geometrie, Rendiconti Palermo 50(1926), 142-169.
  • [Sh] I. G. Shandra, On the geodesic mobility of Riemannian spaces, Math. Notes 68(2000), no. 3-4, 528-532. MR 1823149 (2002b:53058)
  • [Si1] N. S. Sinjukov, Geodesic mappings of Riemannian spaces, (in Russian) ``Nauka'', Moscow, 1979. MR 0552022
  • [Si2] N. S. Sinjukov, On the theory of a geodesic mapping of Riemannian spaces, Dokl. Akad. Nauk SSSR 169(1966), 770-772. MR 0202088 (34:1962)
  • [So] A. S. Solodovnikov, Projective transformations of Riemannian spaces, Uspehi Mat. Nauk (N.S.) 11(1956), no. 4(70), 45-116. MR 0084826
  • [To] P. Topalov, Families of metrics geodesically equivalent to the analogs of the Poisson sphere, J. Math. Phys. 41(2000), no. 11, 7510-7520. MR 1788587 (2002h:37114)
  • [We1] H. Weyl, Zur Infinitisimalgeometrie: Einordnung der projektiven und der konformen Auflösung, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1921; ``Selecta Hermann Weyl'', Birkhäuser Verlag, Basel und Stuttgart, 1956.
  • [We2] H. Weyl, Geometrie und Physik, Die Naturwissenschafter 19(1931), 49-58; ``Hermann Weyl Gesammelte Abhandlungen'', Band 3, Springer-Verlag, 1968.
  • [Wu] H. Wu, On the de Rham decomposition theorem, Illinois J. Math. 8(1964), 291-311. MR 0161280 (28:4488)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53A20, 53A35, 53A45, 53B20, 53B30, 53C12, 53C21, 53C22, 37J35

Retrieve articles in all journals with MSC (2010): 53A20, 53A35, 53A45, 53B20, 53B30, 53C12, 53C21, 53C22, 37J35


Additional Information

Alexey V. Bolsinov
Affiliation: School of Mathematics, Loughborough University, Loughborough, LE11 3TU, United Kingdom
Email: A.Bolsinov@lboro.ac.uk

Vladimir S. Matveev
Affiliation: Institute of Mathematics, Friedrich-Schiller University Jena, 07737 Jena, Germany
Email: vladimir.matveev@uni-jena.de

DOI: https://doi.org/10.1090/S0002-9947-2011-05187-1
Received by editor(s): April 16, 2009
Published electronically: March 21, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society