Projective relatedness and conformal flatness

• Autorzy: Graham Hall
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Projective Structure; Conformal flatness; Holonomy

Kategorie tematyczne

• 53A30: Conformal differential geometry
• 53A20: Projective differential geometry

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 5
• Strony: 1763-1770

Daty

wydano

2012-10-01

online

2012-07-24

Twórcy

autor

Graham Hall

• University of Aberdeen

Bibliografia

• [1] Eisenhart L.P., Riemannian Geometry, 2nd ed., Princeton University Press, Princeton, 1949
• [2] de Felice F., Clarke C.J.S., Relativity on Curved Manifolds, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 1990
• [3] Hall G.S., Symmetries and Curvature Structure in General Relativity, World Sci. Lecture Notes Phys., 46, World Scientific, River Edge, 2004
• [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
• [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
• [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] Hall G.S., Lonie D.P., Projective equivalence of Einstein spaces in general relativity, Classical Quantum Gravity, 2009, 26(12), #125009
• [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
• [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
• [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
• [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
• [12] Kobayashi S., Nomizu K., Foundations of Differential Geometry, I, Interscience, New York-London, 1963
• [13] Mikeš J., Kiosak V., Vanžurová A., Geodesic Mappings of Manifolds with Affine Connection, Palacký University Olomouc, Olomouc, 2008
• [14] Mikeš J., Vanžurová A., Hinterleitner I., Geodesic Mappings and Some Generalizations, Palacký University Olomouc, Olomouc, 2009
• [15] Petrov A.Z., Einstein Spaces, Pergamon, Oxford-Edinburgh-New York, 1969
• [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
• [17] Sinyukov N.S., Geodesic Mappings of Riemannian Spaces, Nauka, Moscow, 1979 (in Russian)

Identyfikatory

DOI

10.2478/s11533-012-0087-6