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1997 | 41 | 1 | 109-118
Tytuł artykułu

TT-tensors and conformally flat structures on 3-manifolds

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We study TT-tensors on conformally flat 3-manifolds (M,g). The Cotton-York tensor linearized at g maps every symmetric tracefree tensor into one which is TT. The question as to whether this is the general solution to the TT-condition is viewed as a cohomological problem within an elliptic complex first found by Gasqui and Goldschmidt and reviewed in the present paper. The question is answered affirmatively when M is simply connected and has vanishing 2nd de Rham cohomology.
Słowa kluczowe
Rocznik
Tom
41
Numer
1
Strony
109-118
Opis fizyczny
Daty
wydano
1997
Twórcy
autor
  • Institut für Theoretische Physik, Universität Wien, Austria, Boltzmanngasse 5, A-1090 Wien, Austria
Bibliografia
  • [1] L. Bérard-Bergery, J. P. Bourguignon and J. Lafontaine (1975), Déformations localement triviales des variétés riemanniennes, Differential Geometry, Proc. Sympos. Pure Math., vol. XXVII, Part 1, Amer. Math. Soc., Providence, R.I., 3-32.
  • [2] A. L. Besse (1987), Einstein Manifolds, Springer, Berlin.
  • [3] E. Calabi (1971), On compact Riemannian manifolds with constant curvature. I, Differential Geometry, Proc. Sympos. Pure Math., vol. III, Amer. Math. Soc., Providence, R.I., 155-180.
  • [4] S.-S. Chern, L. Simons (1974), Characteristic forms and geometric invariants, Ann. Math. 99, 48-69, and S.-S. Chern (1986), On a conformal invariant of three-dimensional manifolds, Aspects of Mathematics and its Applications, J. A. Barroso (Ed.), Elsevier Science Publishers B.V., 245-252.
  • [5] Y. Choquet-Bruhat, J. W. York Jr. (1980), The Cauchy Problem, General Relativity and Gravitation, Vol. 1, A. Held (Ed.), Plenum, N.Y., 99-172.
  • [6] S. Deser (1967), Covariant decomposition of symmetric tensors and the gravitational Cauchy problem, Ann. Inst. Henri Poincaré, VII, 149-188.
  • [7] S. Deser, R. Jackiw and S. Templeton (1982), Topologically Massive Gauge Theories, Ann. Phys. 140, 372-411.
  • [8] D. Ferus (1981), A remark on Codazzi tensors in constant curvature spaces, Global Differential Geometry and Global Analysis, D. Ferus et al. (Eds.) LNM 838, Springer, Berlin, 257.
  • [9] A. E. Fischer, J. E. Marsden (1977), The manifold of conformally equivalent metrics, Can. J. Math. XXIX, 193-209.
  • [10] J. Gasqui, H. Goldschmidt (1984), Déformations Infinitésimales des Structures Conformes Plates, Birkhäuser, Basel.
  • [11] G. Hall (1989), The global extension of local symmetries in general relativity, Class. Quant. Grav. 6, 157-161.
  • [12] S. Kobayashi, K. Nomizu (1963), Foundations of Differential Geometry Vol. 1, Interscience, Wiley, London.
  • [13] N. H. Kuiper (1949), On conformally-flat spaces in the large, Ann. Math. 50, 916-924, and N. H. Kuiper (1950), On compact conformally Euclidean spaces of dimension > 2, Ann. Math. 52, 478-490.
  • [14] J. P. Lafontaine (1983), Modules de structures conformes plates et cohomologie de groupes discrets, C.R. Acad. Sc. t. 297, Ser.. I, 655-658.
  • [15] G. D. Mostow (1973), Strong rigidity of locally symmetric spaces, Ann. of Math. Studies 78, Princeton.
  • [16] J. Schouten (1921), Über die konforme Abbildung n-dimensionaler Mannigfaltigkeiten mit quadratischer Maß bestimmung auf eine Mannigfaltigkeit mit euklidischer Maß bestimmung, Math. Z. 11, 58-88.
  • [17] P. Sommers (1978), The geometry of the gravitational field at spacelike infinity, J. Math. Phys. 19, 549-554.
  • [18] D. C. Spencer (1969), Overdetermined systems of linear partial differential equations, Bull. AMS 75, 179-239.
  • [19] F. Warner (1983), Foundations of Differentiable Manifolds and Lie Groups, Springer, Berlin.
  • [20] J. W. York Jr. (1973), Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial-value problem of general relativity, J. Math. Phys. 14, 456-464, and J. W. York Jr. (1974), Covariant decompositions of symmetric tensors in the theory of gravitation, Ann. Inst. Henri Poincaré 21, 319-332.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-bcpv41z1p109bwm
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