Geodesically equivalent metrics on homogenous spaces

  • Neda BokanEmail author
  • Tijana Šukilović
  • Srdjan Vukmirović


Two metrics on a manifold are geodesically equivalent if the sets of their un-parameterized geodesics coincide. We show that if two G-invariant metrics of arbitrary signature on homogenous space G/H are geodesically equivalent, they are affinely equiva-lent, i.e. they have the same Levi-Civita connection. We also prove that the existence of nonproportional, geodesically equivalent, G-invariant metrics on homogenous space G/H implies that their holonomy algebra cannot be full. We give an algorithm for finding all left invariant metrics geodesically equivalent to a given left invariant metric on a Lie group. Us-ing that algorithm we prove that no two left invariant metrics of any signature on sphere S3 are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, nonproportional, left-invariant metrics.


invariant metric geodesically equivalent metric affinely equivalent metric 


53C22 22E15 53C30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. Bokan, T. Sukilovic, S. Vukmirovic: Lorentz geometry of 4-dimensional nilpotent Lie groups. Geom. Dedicata 177 (2015), 83–102.MathSciNetCrossRefGoogle Scholar
  2. [2]
    A. V. Bolsinov, V. Kiosak, V. S. Matveev: A Fubini theorem for pseudo-Riemannian geodesically equivalent metrics. J. Lond. Math. Soc., II. Ser. 80 (2009), 341–356.MathSciNetCrossRefGoogle Scholar
  3. [3]
    L. P. Eisenhart: Symmetric tensors of the second order whose first covariant derivatives are zero. Trans. Amer. Math. Soc. 25 (1923), 297–306.MathSciNetCrossRefGoogle Scholar
  4. [4]
    G. S. Hall, D. P. Lonie: Holonomy groups and spacetimes. Classical Quantum Gravity 17 (2000), 1369–1382.MathSciNetCrossRefGoogle Scholar
  5. [5]
    G. S. Hall, D. P. Lonie: Projective structure and holonomy in four-dimensional Lorentz manifolds. J. Geom. Phys. 61 (2011), 381–399.MathSciNetCrossRefGoogle Scholar
  6. [6]
    V. Kiosak, V. S. Matveev: Complete Einstein metrics are geodesically rigid. Commun. Math. Phys. 289 (2009), 383–400.MathSciNetCrossRefGoogle Scholar
  7. [7]
    V. Kiosak, V. S. Matveev: Proof of the projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two. Commun. Math. Phys. 297 (2010), 401–426.MathSciNetCrossRefGoogle Scholar
  8. [8]
    T. Levi-Civita: Sulle trasformazioni dello equazioni dinamiche. Annali di Mat. 24 (1896), 255–300. (In Italian.)CrossRefGoogle Scholar
  9. [9]
    N. S. Sinyukov: On geodesic mappings of Riemannian spaces onto symmetric Rieman-nian spaces. Dokl. Akad. Nauk SSSR, n. Ser. 98 (1954), 21–23. (In Russian.)MathSciNetGoogle Scholar
  10. [10]
    P. Topalov: Integrability criterion of geodesical equivalence. Hierarchies. Acta Appl. Math. 59 (1999), 271–298.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Z. Wang, G. Hall: Projective structure in 4-dimensional manifolds with metric signature (+,+,-,-). J. Geom. Phys. 66 (2013), 37–49.MathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  • Neda Bokan
    • 1
    Email author
  • Tijana Šukilović
    • 1
  • Srdjan Vukmirović
    • 1
  1. 1.University of BelgradeFaculty of MathematicsBelgradeSerbia

Personalised recommendations