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 | 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.**A. Barnes,*Space times of embedding class one in general relativity*, General Relativity and Gravitation**5**(1974), no. 2, 147–162. MR**0411526****2.**Brinkmann, H.W.,*Einstein spaces which are mapped conformally on each other*, Math. Ann. 94, (1925), 119-145.**3.**Jian Guo Cao and Dennis DeTurck,*Prescribing Ricci curvature on open surfaces*, Hokkaido Math. J.**20**(1991), no. 2, 265–278. MR**1114407**, 10.14492/hokmj/1381413843**4.**Jian Guo Cao and Dennis DeTurck,*The Ricci curvature equation with rotational symmetry*, Amer. J. Math.**116**(1994), no. 2, 219–241. MR**1269604**, 10.2307/2374929**5.**M. Cahen and J. Leroy,*Exact solutions of Einstein-Maxwell equations*, J. Math. Mech.**16**(1966), 501–508. MR**0204127****6.**Dennis M. DeTurck,*Existence of metrics with prescribed Ricci curvature: local theory*, Invent. Math.**65**(1981/82), no. 1, 179–207. MR**636886**, 10.1007/BF01389010**7.**Dennis M. DeTurck,*Metrics with prescribed Ricci curvature*, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 525–537. MR**645758****8.**Dennis M. DeTurck,*The Cauchy problem for Lorentz metrics with prescribed Ricci curvature*, Compositio Math.**48**(1983), no. 3, 327–349. MR**700744****9.**Dennis M. DeTurck and Norihito Koiso,*Uniqueness and nonexistence of metrics with prescribed Ricci curvature*, Ann. Inst. H. Poincaré Anal. Non Linéaire**1**(1984), no. 5, 351–359 (English, with French summary). MR**779873****10.**Richard Hamilton,*The Ricci curvature equation*, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), Math. Sci. Res. Inst. Publ., vol. 2, Springer, New York, 1984, pp. 47–72. MR**765228**, 10.1007/978-1-4612-1110-5_4**11.**D. Kramer, H. Stephani, E. Herlt, and M. MacCallum,*Exact solutions of Einstein’s field equations*, Cambridge University Press, Cambridge-New York, 1980. Edited by Ernst Schmutzer; Cambridge Monographs on Mathematical Physics. MR**614593****12.**Wolfgang Kühnel,*Conformal transformations between Einstein spaces*, Conformal geometry (Bonn, 1985/1986) Aspects Math., E12, Vieweg, Braunschweig, 1988, pp. 105–146. MR**979791****13.**John M. Lee and Thomas H. Parker,*The Yamabe problem*, Bull. Amer. Math. Soc. (N.S.)**17**(1987), no. 1, 37–91. MR**888880**, 10.1090/S0273-0979-1987-15514-5**14.**R. G. McLenaghan, N. Tariq, and B. O. J. Tupper,*Conformally flat solutions of the Einstein-Maxwell equations for null electromagnetic fields*, J. Mathematical Phys.**16**(1975), 829–831. MR**0376089****15.**Romildo Pina and Keti Tenenblat,*Conformal metrics and Ricci tensors in the pseudo-Euclidean space*, Proc. Amer. Math. Soc.**129**(2001), no. 4, 1149–1160 (electronic). MR**1814152**, 10.1090/S0002-9939-00-05817-2**16.**Romildo Pina and Keti Tenenblat,*On metrics satifying equation 𝑅ᵢⱼ-\frac12𝐾𝑔ᵢⱼ=𝑇ᵢⱼ for constant tensors 𝑇*, J. Geom. Phys.**40**(2002), no. 3-4, 379–383. MR**1866996**, 10.1016/S0393-0440(01)00043-2**17.**Stephani, H.,*Konform flache Gravitationsfelder*, Comm. in Math. Phys. 5, (1967) 337-342.**18.**Xingwang Xu,*Prescribing a Ricci tensor in a conformal class of Riemannian metrics*, Proc. Amer. Math. Soc.**115**(1992), no. 2, 455–459. MR**1093607**, 10.1090/S0002-9939-1992-1093607-8**19.**Shing Tung Yau (ed.),*Seminar on Differential Geometry*, Annals of Mathematics Studies, vol. 102, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. Papers presented at seminars held during the academic year 1979–1980. MR**645728**

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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