Projective relatedness and conformal flatness

Graham Hall

Open Mathematics (2012)

  • Volume: 10, Issue: 5, page 1763-1770
  • ISSN: 2391-5455

Abstract

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This paper discusses the connection between projective relatedness and conformal flatness for 4-dimensional manifolds admitting a metric of signature (+,+,+,+) or (+,+,+,−). It is shown that if one of the manifolds is conformally flat and not of the most general holonomy type for that signature then, in general, the connections of the manifolds involved are the same and the second manifold is also conformally flat. Counterexamples are provided which place limitations on the potential strengthening of the results.

How to cite

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Graham Hall. "Projective relatedness and conformal flatness." Open Mathematics 10.5 (2012): 1763-1770. <http://eudml.org/doc/269654>.

@article{GrahamHall2012,
abstract = {This paper discusses the connection between projective relatedness and conformal flatness for 4-dimensional manifolds admitting a metric of signature (+,+,+,+) or (+,+,+,−). It is shown that if one of the manifolds is conformally flat and not of the most general holonomy type for that signature then, in general, the connections of the manifolds involved are the same and the second manifold is also conformally flat. Counterexamples are provided which place limitations on the potential strengthening of the results.},
author = {Graham Hall},
journal = {Open Mathematics},
keywords = {Projective Structure; Conformal flatness; Holonomy; projective structure; conformal flatness; holonomy},
language = {eng},
number = {5},
pages = {1763-1770},
title = {Projective relatedness and conformal flatness},
url = {http://eudml.org/doc/269654},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Graham Hall
TI - Projective relatedness and conformal flatness
JO - Open Mathematics
PY - 2012
VL - 10
IS - 5
SP - 1763
EP - 1770
AB - This paper discusses the connection between projective relatedness and conformal flatness for 4-dimensional manifolds admitting a metric of signature (+,+,+,+) or (+,+,+,−). It is shown that if one of the manifolds is conformally flat and not of the most general holonomy type for that signature then, in general, the connections of the manifolds involved are the same and the second manifold is also conformally flat. Counterexamples are provided which place limitations on the potential strengthening of the results.
LA - eng
KW - Projective Structure; Conformal flatness; Holonomy; projective structure; conformal flatness; holonomy
UR - http://eudml.org/doc/269654
ER -

References

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  1. [1] Eisenhart L.P., Riemannian Geometry, 2nd ed., Princeton University Press, Princeton, 1949 
  2. [2] de Felice F., Clarke C.J.S., Relativity on Curved Manifolds, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 1990 
  3. [3] Hall G.S., Symmetries and Curvature Structure in General Relativity, World Sci. Lecture Notes Phys., 46, World Scientific, River Edge, 2004 
  4. [4] Hall G.S., Lonie D.P., Holonomy groups and spacetimes, Classical Quantum Gravity, 2000, 17(6), 1369–1382 http://dx.doi.org/10.1088/0264-9381/17/6/304 Zbl0957.83014
  5. [5] Hall G.S., Lonie D.P., The principle of equivalence and cosmological metrics, J. Math. Phys., 2008, 49(2), #022502 http://dx.doi.org/10.1063/1.2837431 Zbl1153.81370
  6. [6] Hall G.S., Lonie D.P., Holonomy and projective equivalence in 4-dimensional Lorentz manifolds, SIGMA Symmetry Integrability Geom. Methods Appl., 2009, 5, #066 
  7. [7] Hall G.S., Lonie D.P., Projective equivalence of Einstein spaces in general relativity, Classical Quantum Gravity, 2009, 26(12), #125009 Zbl1170.83443
  8. [8] Hall G.S., Lonie D.P., Projective structure and holonomy in four-dimensional Lorentz manifolds, J. Geom. Phys., 2011, 61(2), 381–399 http://dx.doi.org/10.1016/j.geomphys.2010.10.007 Zbl1208.83035
  9. [9] Hall G.S., Lonie D.P., Projective structure and holonomy in general relativity, Classical Quantum Gravity, 2011, 28(8), #083101 http://dx.doi.org/10.1088/0264-9381/28/8/083101 Zbl1213.83008
  10. [10] Hall G., Wang Z., Projective structure in 4-dimensional manifolds with positive definite metrics, J. Geom. Phys., 2012, 62(2), 449–463 http://dx.doi.org/10.1016/j.geomphys.2011.10.007 Zbl1237.53012
  11. [11] Kiosak V., Matveev V.S., Complete Einstein metrics are geodesically rigid, Comm. Math. Phys., 2009, 289(1), 383–400 http://dx.doi.org/10.1007/s00220-008-0719-7 Zbl1170.53025
  12. [12] Kobayashi S., Nomizu K., Foundations of Differential Geometry, I, Interscience, New York-London, 1963 Zbl0119.37502
  13. [13] Mikeš J., Kiosak V., Vanžurová A., Geodesic Mappings of Manifolds with Affine Connection, Palacký University Olomouc, Olomouc, 2008 Zbl1176.53004
  14. [14] Mikeš J., Vanžurová A., Hinterleitner I., Geodesic Mappings and Some Generalizations, Palacký University Olomouc, Olomouc, 2009 Zbl1222.53002
  15. [15] Petrov A.Z., Einstein Spaces, Pergamon, Oxford-Edinburgh-New York, 1969 Zbl0174.28305
  16. [16] Schell J.F., Classification of four-dimensional Riemannian spaces, J. Math. Phys., 1961, 2, 202–206 http://dx.doi.org/10.1063/1.1703700 Zbl0096.21903
  17. [17] Sinyukov N.S., Geodesic Mappings of Riemannian Spaces, Nauka, Moscow, 1979 (in Russian) 

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