Advertisement

Geodesic rigidity of Levi-Civita connections admitting essential projective vector fields

  • Tianyu MaEmail author
Original Paper
  • 3 Downloads

Abstract

In this paper, it is proved that a connected 3-dimensional Riemannian manifold or a closed connected semi-Riemannian manifold \(M^n\) (\(n>1\)) admitting a projective vector field with a non-linearizable singularity is projectively flat.

Keywords

Geodesic rigidity Essential projective vector field Metrizable projective structure Local dynamics 

Mathematics Subject Classification (2000)

53A20 

Notes

References

  1. 1.
    Kobayashi, S., Nagano, T.: On projective connections. J. Math. Mech. 13, 215–236 (1964)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Matveev, V.: Proof of the projective Lichnerowicz–Obata conjecture. J. Differ. Geom. 75(3), 459–502 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bolsinov, A., Matveev, V.: Local normal forms for geodesically equivalent pseudo-Riemannian metrics. Trans. Am. Math. Soc. 367, 6719–6749 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bolsinov, A., Matveev, V.: Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics. Trans. Am. Math. Soc. 363, 4081–4107 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Matveev, V., Rosemann, S., Bolsinov, A.: 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. arXiv:1510.00275
  6. 6.
    Kiosak, V., Matveev, V.: Proof of the projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two. Commun. Mat. Phys. 297, 401–426 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Nagano, T., Ochiai, T.: On compact Riemannian manifolds admitting essential projective transformations. J. Fac. Sci. Univ. Tokyo. Sect. 1 A Math. 33(2), 233–246 (1986)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Frances, C.: Local dynamics of conformal vector fields. C. Geom. Dedicata 158, 35–59 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Melnick, K., Frances, C.: Formes normales pour les champs conformes pseudoriemanniens [Normal forms for pseudo-Riemannian conformal vector fields]. Bull. Soc. Math. Fr. 141(3), 377–421 (2013)CrossRefzbMATHGoogle Scholar
  10. 10.
    Čap, A., Melnick, K.: Essential Killing fields of parabolic geometries: projective and conformal structures. Cent. Eur. J. Math. 11(12), 2053–2061 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Zeghib, A.: On discrete projective transformation groups of Riemannian manifolds. Adv. Math. 297, 26–53 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sharpe, R.: Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program. Springer, New York (1997)zbMATHGoogle Scholar
  13. 13.
    Matveev, V., Eastwood, M.: Metric Connections in Projective Differential Geometry, Symmetries and Overdetermined Systems of Partial Differential Equations, Volume 144 of the Series the IMA Volumes in Mathematics and Its Applications, pp. 339–350 (2008)Google Scholar
  14. 14.
    Mike, J.: Geodesic mappings of affine-connected and Riemannian spaces. J. Math. Sci. 78(3), 311–333 (1996)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Matveev, V., Mounoud, P.: Gallot–Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications. Ann. Glob. Anal. Geom. 38(3), 259–271 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsFriedrich Schiller University JenaJenaGermany

Personalised recommendations