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Conformal metrics and Ricci tensors on the sphere

Authors: Romildo Pina and Keti Tenenblat
Journal: Proc. Amer. Math. Soc. 132 (2004), 3715-3724
MSC (2000): Primary 53C21, 53C50, 53C80
Published electronically: July 22, 2004
MathSciNet review: 2084096
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Abstract: We consider tensors $T=fg$ on the unit sphere $S^n$, where $n\geq 3$, $g$ is the standard metric and $f$ is a differentiable function on $S^n$. For such tensors, we consider the problems of existence of a Riemannian metric $\bar{g}$, conformal to $g$, such that $\mbox{Ric }\bar{g} = T$, and the existence of such a metric that satisfies $\mbox{Ric }\bar{g} - {\bar{K}}\bar{g}/2 = T$, where $\bar{K}$ is the scalar curvature of $\bar{g}$. We find the restrictions on the Ricci candidate for solvability, and we construct the solutions $\bar{g}$ when they exist. We show that these metrics are unique up to homothety, and we characterize those defined on the whole sphere. As a consequence of these results, we determine the tensors $T$ that are rotationally symmetric. Moreover, we obtain the well-known result that a tensor $T=\alpha g$, $\alpha>0 $, has no solution $\bar{g}$ on $S^n$if $\alpha\neq n-1$ and only metrics homothetic to $g$ admit $(n-1)g$ as a Ricci tensor. We also show that if $\alpha\neq -(n-1)(n-2)/2$, then equation $\mbox{Ric }\bar{g} - \displaystyle {\bar{K}}\bar{g}/2 = \alpha g $ has no solution $\bar{g}$, conformal to $g$ on $S^n$, and only metrics homothetic to $g$ are solutions to this equation when $\alpha= -(n-1)(n-2)/2$. Infinitely many $C^\infty$ solutions, globally defined on $S^n$, are obtained for the equation

\begin{displaymath}- \varphi \Delta_g \varphi +\displaystyle\frac{n}{2} \vert\na... ...displaystyle\frac{n}{2}\left( \lambda + \varphi^2 \right) = 0, \end{displaymath}

where $\lambda\in R$. The geometric interpretation of these solutions is given in terms of existence of complete metrics, globally defined on $R^n$ and conformal to the Euclidean metric, for certain bounded scalar curvature functions that vanish at infinity.

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  • 1. Barnes, A., Space-times of embedding class one in general relativity, General Relativity and Gravitation, 5, (1974), 147-161. MR 53:15259
  • 2. Brinkmann, H.W., Einstein spaces which are mapped conformally on each other, Math. Ann. 94, (1925), 119-145.
  • 3. Cao, J.; DeTurck, D., Prescribing Ricci curvature on open surfaces, Hokkaido Math. J. 20 (1991), 265-278. MR 92k:53068
  • 4. -, The Ricci curvature equation with rotational symmetry, American Journal of Mathematics 116 (1994), 219-241. MR 94m:53052
  • 5. Cahen, M. and Leroy, J. Exact solutions of the Einstein-Maxwell equations, J. Math. Mech. 16, (1966), 501-508. MR 34:3973
  • 6. DeTurck, D., Existence of metrics with prescribed Ricci curvature: Local theory, Invent. Math. 65 (1981), 179-207. MR 83b:53019
  • 7. -, Metrics with prescribed Ricci curvature, Seminar on Differential Geometry, Ann. of Math. Stud. Vol. 102, (S. T. Yau, ed.), Princeton University Press, (1982), 525-537. MR 83e:53014
  • 8. -, The Cauchy problem for Lorentz metrics with prescribed Ricci curvature, Compositio Math. 48 (1983), 327-349. MR 85c:53041
  • 9. DeTurck, D.; Koiso, N., Uniqueness and non-existence of metrics with prescribed Ricci curvature, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 351-359. MR 86i:53022
  • 10. Hamilton, R.S., The Ricci curvature equation, Seminar on nonlinear partial differential equations, Publ. Math. Sci. Res. Inst. 2, (1984), 47-72. MR 86b:53040
  • 11. Kramer, D., Stephani, H., MacCallun, M.A.H. and Herlt, E., Exact solutions of Einstein's field equations, Cambridge University Press, 1980. MR 82h:83002
  • 12. Kühnel, W., Conformal transformations between Einstein spaces, Conformal geometry (Bonn, 1985/1986), 105-146, Aspects Math., E12, Vieweg, Braunschweig, 1988. MR 90b:53055
  • 13. Lee, J.M.; Parker, T.H., The Yamabe problem, Bulletin (New Series) of the Amer. Math. Soc. 17, (1987), 37-91. MR 88f:53001
  • 14. McLenaghan, R.G., Tarig, N. and Tupper, B.O.J. Conformally flat solutions of the Einstein-Maxwell equations for the null electromagnetic fields, J. Math. Phys. 16, (1975), 829-831. MR 51:12275
  • 15. Pina, R., Tenenblat, K., Conformal metrics and Ricci tensors in the pseudo-euclidean space, Proc. Amer. Math. Soc. 129, (2001), 1149-1160. MR 2001k:53137
  • 16. Pina, R., Tenenblat, K., On metrics satisfying equation $R_{ij}-K g_{ij}/2 =T_{ij}$ for constant tensors $T$., Journal of Geometry and Physics 40, (2002) 379-383. MR 2002g:53125
  • 17. Stephani, H., Konform flache Gravitationsfelder, Comm. in Math. Phys. 5, (1967) 337-342.
  • 18. Xu, X., Prescribing a Ricci tensor in a conformal class of Riemannian metrics, Proceedings of the American Mathematical Society Vol. 115, (1992), 455-459. MR 92i:53036
  • 19. Yau, S.T., Ed., Seminar on Differential Geometry, Princeton University Press, 1982. MR 83a:53002

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Additional Information

Romildo Pina
Affiliation: IME, Universidade Federal de Goi\a’as, 74001-970 Goi\a^{a}nia, GO, Brazil

Keti Tenenblat
Affiliation: Departamento de Matem\a’atica, Universidade de Bras\a’ılia, 70910-900, Bras\a’ılia, DF, Brazil

Keywords: Ricci tensor, conformal metric, scalar curvature
Received by editor(s): May 30, 2002
Received by editor(s) in revised form: May 14, 2003
Published electronically: July 22, 2004
Additional Notes: The first author was partially supported by FUNAPE/UFG and PROCAD
The second author was partially supported by CNPq, PRONEX and PROCAD
Communicated by: Wolfgang Ziller
Article copyright: © Copyright 2004 American Mathematical Society

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