Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 066, 23 pages      arXiv:0906.5227
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds

Graham S. Hall a and David P. Lonie b
a) Department of Mathematical Sciences, University of Aberdeen, Meston Building, Aberdeen, AB24 3UE, Scotland, UK
b) 108e Anderson Drive, Aberdeen, AB15 6BW, Scotland, UK

Received March 18, 2009, in final form June 11, 2009; Published online June 29, 2009

A study is made of 4-dimensional Lorentz manifolds which are projectively related, that is, whose Levi-Civita connections give rise to the same (unparameterised) geodesics. A brief review of some relevant recent work is provided and a list of new results connecting projective relatedness and the holonomy type of the Lorentz manifold in question is given. This necessitates a review of the possible holonomy groups for such manifolds which, in turn, requires a certain convenient classification of the associated curvature tensors. These reviews are provided.

Key words: projective structure; holonomy; Lorentz manifolds; geodesic equivalence.

pdf (364 kb)   ps (213 kb)   tex (33 kb)


  1. Geroch R.P., Spinor structure of space-times in general relativity, J. Math. Phys. 9 (1968), 1739-1744.
  2. Hall G.S., Symmetries and curvature structure in general relativity, World Scientific Lecture Notes in Physics, Vol. 46, World Scientific Publishing Co., Inc., River Edge, NJ, 2004.
  3. Hall G.S., McIntosh C.G.B., Algebraic determination of the metric from the curvature in general relativity, Internat. J. Theoret. Phys. 22 (1983), 469-476.
  4. Hall G.S., Curvature collineations and the determination of the metric from the curvature in general relativity, Gen. Relativity Gravitation 15 (1983), 581-589.
  5. McIntosh C.G.B., Halford W.D., Determination of the metric tensor from components of the Riemann tensor, J. Phys. A: Math. Gen. 14 (1981), 2331-2338.
  6. Hall G.S., Lonie D.P., On the compatibility of Lorentz metrics with linear connections on four-dimensional manifolds, J. Phys. A: Math. Gen. 39 (2006), 2995-3010, gr-qc/0509067.
  7. Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. 1, Interscience Publishers, New York, 1963.
  8. Schell J.F., Classification of four-dimensional Riemannian spaces, J. Math. Phys. 2 (1961), 202-206.
  9. Hall G.S., Lonie D.P., Holonomy groups and spacetimes, Classical Quantum Gravity 17 (2000), 1369-1382, gr-qc/0310076.
  10. Hall G.S., Covariantly constant tensors and holonomy structure in general relativity, J. Math. Phys. 32 (1991), 181-187.
  11. Besse A., Einstein manifolds, Springer-Verlag, Berlin, 1987.
  12. Wu H., On the de Rham decomposition theorem, Illinois J. Math. 8 (1964), 291-311.
  13. Hall G.S., Kay W., Holonomy groups in general relativity, J. Math. Phys. 29 (1988), 428-432.
  14. Ambrose W., Singer I.M., A theorem on holonomy, Trans. Amer. Math. Soc. 75 (1953), 428-443.
  15. Hall G.S., Connections and symmetries in space-times, Gen. Relativity Gravitation 20 (1988), 399-406.
  16. Eisenhart L.P., Riemannian geometry, Princeton University Press, Princeton, 1966.
  17. Thomas T.Y., Differential invariants of generalised spaces, Cambridge, 1934.
  18. Weyl H., Zur Infinitesimalgeometrie: Einordnung der projectiven und der konformen Auffassung, Gött. Nachr. (1921), 99-112.
  19. Petrov A.Z., Einstein spaces, Pergamon Press, Oxford - Edinburgh -New York, 1969.
  20. Sinyukov N.S., Geodesic mappings of Riemannian spaces, Nauka, Moscow, 1979 (in Russian).
  21. Mikes J., Hinterleitner I., Kiosak V.A., On the theory of geodesic mappings of Einstein spaces and their generalizations, in The Albert Einstein Centenary International Conference, Editors J.-M. Alini and A. Fuzfa, AIP Conf. Proc., Vol. 861, American Institute of Physics, 2006, 428-435.
  22. Hall G.S., Lonie D.P., The principle of equivalence and projective structure in spacetimes, Classical Quantum Gravity 24 (2007), 3617-3636, gr-qc/0703104.
  23. Kiosak V., Matveev V.A., Complete Einstein metrics are geodesically rigid, Comm. Math. Phys. 289 (2009), 383-400, arXiv:0806.3169.
  24. Hall G.S., Lonie D.P., Projective equivalence of Einstein spaces in general relativity, Classical Quantum Gravity 26 (2009), 125009, 10 pages.
  25. Hall G.S., Lonie D.P., Holonomy and projective structure in space-times, Preprint, University of Aberdeen, 2009.
  26. Hall G.S., Lonie D.P., The principle of equivalence and cosmological metrics, J. Math. Phys. 49 (2008), 022502, 13 pages.

Previous article   Next article   Contents of Volume 5 (2009)