Three-dimensional metrics as deformations of a constant curvature metric

General Relativity and Gravitation

Volumn 34, Issue 2, 2002, Pages 269-282

  Coll, B ^a   Llosa, J ^b   Soler, D ^c  

^a OBSERVATOIRE DE PARIS   (France)

^b UNIVERSITY OF BARCELONA   (Spain)

^c NONE

Author keywords

Cauchy problem; Flat deformation; Riemannian geometry

Indexed keywords

EID: 0036251258     PISSN: 00017701     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1015391411214     Document Type: Article

References (22)

1. Riemann, G. F. B., (1953). Über die Hypothesen, welche der Geometrie zu Grunde liegen, Abhand. K. Ges. Wiss. Göttingen, 13, 133, 1868; English translation by Clifford, W. K. Nature 8, 14, 1873; reprinted and edited by Weyl, H., Springer, Berlin, 1920. Included in its Gesammelte Mathematische Werke, wissenschaftlicher Nachlaß und Nachträge, eds. Weber, H., Dedekind, R., Teubner, B. G., Leipzig, 1892; 2d ed. Dover Publ., New York.
2. Riemann, G. F. B., (1953). Über die Hypothesen, welche der Geometrie zu Grunde liegen, Abhand. K. Ges. Wiss. Göttingen, 13, 133, 1868; English translation by Clifford, W. K. Nature 8, 14, 1873; reprinted and edited by Weyl, H., Springer, Berlin, 1920. Included in its Gesammelte Mathematische Werke, wissenschaftlicher Nachlaß und Nachträge, eds. Weber, H., Dedekind, R., Teubner, B. G., Leipzig, 1892; 2d ed. Dover Publ., New York.
3. Riemann, G. F. B., (1953). Über die Hypothesen, welche der Geometrie zu Grunde liegen, Abhand. K. Ges. Wiss. Göttingen, 13, 133, 1868; English translation by Clifford, W. K. Nature 8, 14, 1873; reprinted and edited by Weyl, H., Springer, Berlin, 1920. Included in its Gesammelte Mathematische Werke, wissenschaftlicher Nachlaß und Nachträge, eds. Weber, H., Dedekind, R., Teubner, B. G., Leipzig, 1892; 2d ed. Dover Publ., New York.
4. Roughly speaking, a geometric object is a mathematical object which remains unchanged under changes of local charts.
5. Eisenhart, L. P., (1960). A Treatise on the Differential Geometry of Curves and Surfaces, Dover Publications, Inc. New York, p 93.
6. ∞ proof; Walberer, P., Abhandl. Math. Sem. Univ. Hamburg, 10, p 169-79, 1934 for the decomposition in a given orthogonal frame; Bel, L., Gen. Rel. Grav., 28, p 1139-50, (1996) for decompositions in principal frames; Tanno, S., J. Differential Geometry, 11, p 467-74, (1976) for other particular decompositions.
7. ∞ proof; Walberer, P., Abhandl. Math. Sem. Univ. Hamburg, 10, p 169-79, 1934 for the decomposition in a given orthogonal frame; Bel, L., Gen. Rel. Grav., 28, p 1139-50, (1996) for decompositions in principal frames; Tanno, S., J. Differential Geometry, 11, p 467-74, (1976) for other particular decompositions.
8. ∞ proof; Walberer, P., Abhandl. Math. Sem. Univ. Hamburg, 10, p 169-79, 1934 for the decomposition in a given orthogonal frame; Bel, L., Gen. Rel. Grav., 28, p 1139-50, (1996) for decompositions in principal frames; Tanno, S., J. Differential Geometry, 11, p 467-74, (1976) for other particular decompositions.
9. ∞ proof; Walberer, P., Abhandl. Math. Sem. Univ. Hamburg, 10, p 169-79, 1934 for the decomposition in a given orthogonal frame; Bel, L., Gen. Rel. Grav., 28, p 1139-50, (1996) for decompositions in principal frames; Tanno, S., J. Differential Geometry, 11, p 467-74, (1976) for other particular decompositions.
10. ∞ proof; Walberer, P., Abhandl. Math. Sem. Univ. Hamburg, 10, p 169-79, 1934 for the decomposition in a given orthogonal frame; Bel, L., Gen. Rel. Grav., 28, p 1139-50, (1996) for decompositions in principal frames; Tanno, S., J. Differential Geometry, 11, p 467-74, (1976) for other particular decompositions.
11. Coll, B., (1999). A Universal Law of Gravitational Deformation for General Relativity, in Proceedings of the Spanish Relativity Meeting in honour of the 65th Birthday of Lluis Bel "Gravitation and Relativity in General" ed. by Martin, J., Ruiz, E., Atrio, F., Molina, A., World Scientific.
12. See for example O'Neill, B., Elementary Differential Geometry, Academic Press, 1966.
13. In this case, the deformation of material coordinates does not change the flat character of the metric, only its tensor expression (Saint Venant equations).
14. Lichnerowicz, A., (1944). J. Math. Pures Appl. 23, p 37-63.
15. i as in Eisenhart, L. P., Riemannian Geometry, eq. (8.3), Princeton University Press (Princeton, N.J. 1997).
16. i as in Eisenhart, L. P., Riemannian Geometry, eq. (8.3), Princeton University Press (Princeton, N.J. 1997).
17. Eisenhart, L. P., Op. cit., eq. (28.12).
18. Eisenhart, L. P., Op. cit., p. 83.
19. Eisenhart, L. P., Op. cit., eq. (26.2) Princeton University Press (Princeton, N.J. 1997).
20. John, F., (1971). Partial Differential Equations, Chap. 3, Springer (New York).
21. Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973). Gravitation, p. 877, Freeman, San Francisco.
22. Kramer, D., Stephani, H., Herlt, E., MacCallum, M., (1979). Exact solutions of Einstein's field equations, p. 299, Cambridge University Press, Cambridge.