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Conformal diffeomorphisms preserving the Ricci tensor

Authors: W. Kühnel and H.-B. Rademacher
Journal: Proc. Amer. Math. Soc. 123 (1995), 2841-2848
MSC: Primary 53C20; Secondary 53A30, 53C50, 58G30
MathSciNet review: 1260173
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Abstract: We characterize semi-Riemannian manifolds admitting a global conformal transformation such that the difference of the two Ricci tensors is a constant multiple of the metric. Unless the conformal transformation is homothetic, the only possibilities are standard Riemannian spaces of constant sectional curvature and a particular warped product with a Ricci flat Riemannian manifold.

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  • [Be] A. L. Besse, Einstein manifolds, Ergebnisse Math. Grenzgeb., 3. Folge, Band 10, Springer, Berlin, Heidelberg, and New York, 1987. MR 867684 (88f:53087)
  • [Br] H. W. Brinkmann, Einstein spaces which are mapped conformally on each other, Math. Ann. 94 (1925), 119-145. MR 1512246
  • [Fe] J. Ferrand, Sur une classe de morphismes conformes, C. R. Acad. Sci. Paris 296 (1983), 681-684. MR 705690 (85g:53045)
  • [Fi] A. Failkow, Conformal geodesics, Trans. Amer. Math. Soc. 45 (1939), 443-473. MR 1501998
  • [H] J. Haantjes, Conformal representations of an n-dimensional euclidean space with a non-dimensional euclidean space with a non-definite fundamental form on itself, Proc. Kon. Nederl. Akad. Amsterdam 40 (1937), 700-705.
  • [Hf] W. D. Halford, Brinkmann's theorem in general relativity, Gen. Relativity Gravitation 14 (1982), 1193-1195. MR 683685 (83m:83022)
  • [Kan] M. Kanai, On a differential equation characterizing a Riemannian structure of a manifold, Tokyo J. Math. 6 (1983), 143-151. MR 707845 (85c:58103)
  • [Kb] Y. Kerbrat, Transformations conformes des variétés pseudo-Riemanniennes, J. Differential Geom. 11 (1976), 547-571. MR 0478083 (57:17576)
  • [Ke1] M. G. Kerckhove, Conformal transformations of pseudo-Riemannian Einstein manifolds, thesis, Brown Univ., 1988.
  • [Ke2] -, The structure of Einstein spaces admitting conformal motions, Classical Quantum Gravity 8 (1991), 819-825. MR 1104754 (92e:53064)
  • [Kü] W. Kühnel, Conformal transformations between Einstein spaces, Conformal Geometry (R. S. Kulkarni and U. Pinkall, eds.), Aspects of Math., vol. E12, Braunschweig, 105-146, Vieweg-Verlag, 1988, pp. 105-146. MR 979791 (90b:53055)
  • [Lie] S. Lie, Komplexe, insbesondere Linien und Kugelkomplexe mit Anwendung auf die Theorie partieller Differentialgleichungen, Math. Ann. 5 (1872), 145-246. MR 1509773
  • [Liv] J. Liouville, Extension au cas des trois dimensions de la question du tracé géographique, Note VI, Applications de l'Analyse à la Géométrie (G. Monge, ed.), Paris, 1850, pp. 609-617.
  • [O] B. O'Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983. MR 719023 (85f:53002)
  • [S] J. A. Schouten, Der Ricci-Kalkül, Springer-Verlag, Berlin, 1924. MR 516659 (80a:53001)
  • [Tb] S. Tachibana, On concircular geometry and Riemann spaces with constant scalar curvatures, Tôhoku Math. J. 3 (1951), 149-158. MR 0045425 (13:581b)
  • [T] Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251-275. MR 0174022 (30:4229)
  • [V] P. Venzi, Klassifikation der geodätischen Abbildungen mit $ \overline {{\text{Ric}}} - {\text{Ric}} = \Delta \cdot g$, Tensor (N.S.) 37 (1982), 137-147. MR 831155 (87c:53046)
  • [X] X. Xu, Prescribing a Ricci tensor in a conformal class of Riemannian metrics, Proc. Amer. Math. Soc. 115 (1992), 455-459; corrigenda, ibid. 118 (1993), 333. MR 1093607 (92i:53036)
  • [Y] K. Yano, Concircular geometry I-V, Proc. Imperial Acad. Japan 16 (1940), 195-200, 354-360, 442-448, 505-511; ibid. 18 (1942), 446-451.
  • [YN] K. Yano and T. Nagano, Einstein spaces admitting a one-parameter group of conformal transformations, Ann. of Math. (2) 69 (1959), 451-460. MR 0101535 (21:345)

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Keywords: Semi-Riemannian manifolds, Ricci tensor, conformal mapping, Hessian
Article copyright: © Copyright 1995 American Mathematical Society

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