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

DOI:
https://doi.org/10.1090/S0002-9939-04-07613-0

Published electronically:
July 22, 2004

MathSciNet review:
2084096

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider tensors on the unit sphere , where , is the standard metric and is a differentiable function on . For such tensors, we consider the problems of existence of a Riemannian metric , conformal to , such that , and the existence of such a metric that satisfies , where is the scalar curvature of . We find the restrictions on the Ricci candidate for solvability, and we construct the solutions 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 that are rotationally symmetric. Moreover, we obtain the well-known result that a tensor , , has no solution on if and only metrics homothetic to admit as a Ricci tensor. We also show that if , then equation has no solution , conformal to on , and only metrics homothetic to are solutions to this equation when . Infinitely many solutions, globally defined on , are obtained for the equation

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

**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**for constant tensors**.*, 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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
53C21,
53C50,
53C80

Retrieve articles in all journals with MSC (2000): 53C21, 53C50, 53C80

Additional Information

**Romildo Pina**

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

Email:
romildo@mat.ufg.br

**Keti Tenenblat**

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

Email:
keti@mat.unb.br

DOI:
https://doi.org/10.1090/S0002-9939-04-07613-0

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