Summary
The topological aspects of the Bel-Petrov classification of the curvature tensor are examined for compact orientable space-times in which the Einstein equations for the exterior case are satisfied. It is shown that for such space-times of Bel Case III the metric tensor is singularity-free and that the Pontrjagin number identically vanishes. Bel Cases I and II are examined and conditions are given for which the metric is singularity-free and the Pontrjagin number vanishes. Applications to gravitional radiation in general relativity are discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Bibliography
P. Alexandroff-H. Hopf,Topologie, Springer-Verlag (1935), Chelsea reprint (1966).
A. Avez,Formule de Gauss-Bonnet-Chern en métrique de signature quelconque, C. R. Acad. Sci.255 (1962), pp. 2049–2051.
—— ——,Index des variétés de dimensíon 4, C. R. Acad. Sci.259 (1964), pp. 1934–1937.
L. Bel,Étude algebrique d’un certain type de tenseurs de courbure, Le cas 3 de Petrov, C. R. Acad. Sci.247 (1958), pp. 2096–2099.
—— ——,Quelques remarques sur la classification de Petrov, Étude du cas 2, C. R. Acad. Sci.248 (1959), pp 2561–2563.
-- --,La radiation gravitationelle, Thèse (1960) Paris.
H. Bondi-F. A. E. Pirani-I. Robinson,Gravitational waves in general relativity III, Exact plane waves, Proc. Roy. Soc. (London) A251 (1959), pp. 519–533.
S. S. Chern,Pseudo-Riemannian geometry and the Gauss-Bonnet formula, An. Acad. Bras. de Ciências 35 (1963), pp. 17–26.
A. Einstein,The meaning of relativity, 5th edition, Princeton Univ Press (1955).
J. Géhéniau-R. Debever,Les invariants de l’espace de Riemann à quatre dimensions, Bull. Acad. Roy. de Belg. Cl. Sc.42 (1956), pp. 114–123.
F. Hirzebruch,Topological methods in algebraic geometry, Springer-Verlag (1966).
W. V. D. Hodge,The theory and applications of harmonic integrals, 2nd edition, Cambridge Univ. Press (1959).
R. P. Kerr-J. N. Goldberg,Einstein spaces with four-parameter holonomy groups, Jour. Math. Phys2 (1961) pp. 332–335.
C. Lanczos,A remarkable property of the Riemann tensor in four dimension, Ann. Math.39 (1938), pp. 842–850.
A. Lichnerowicz,Théorie globale des connexions et des groupes d’holonomie, Cremonese (1962).
—— ——,Ondes et radiations électromagnétiques et gravitationelles en relativité générale, Ann. di Mat. Pura e. App.50 (1960), pp. 1–95.
L. Markus,Line element fields and Lorentz structures on differentiable manifolds, Ann. Math.62 (1955), pp. 411–417.
J. W. Milnor,On simply connected 4-manifolds, Symposium International de Topologia Algebraica, La Universidad Nacional Autónoma de Mexico and UNESCO, (1958) pp. 122–128.
G. de Rham,Variétés differentiables, formes courants, formes harmoniques, ASI 1222, Hermann (1960).
I. Robinson-A. Trautman,Some spherical gravitational waves in general relativity, Proc. Roy. Soc (London) A265 (1961) pp. 463–473.
H. S. Ruse,On the line geometry of the Riemann tensor, Proc. Roy. Soc. (Edinburgh)62 (1944), pp. 64–73.
R. K. Sachs,Gravitational waves in general relativity VI, the outgoing radiation condition, Proc. Roy. Soc. (London) A264 (1961), pp. 309–338.
N. E. Steenrod,The topology of fibre bundles, Princeton Univ. Press. (1951).
J. D. Zund,Pontrjagin numbers and pseudo-Riemannian geometry, Ann. di Mat. Pura e App.72 (1966), pp. 319–324.
Author information
Authors and Affiliations
Additional information
Research supported in part by National Science Foundation grant GP-7401 to North Carolina State University.
Rights and permissions
About this article
Cite this article
Zund, J.D. Topological aspects of the bel-petrov classification. Annali di Matematica 78, 365–375 (1968). https://doi.org/10.1007/BF02415122
Issue Date:
DOI: https://doi.org/10.1007/BF02415122