A note on Fischer-Marsden’s conjecture. (English) Zbl 0867.53035

A Riemannian metric \(g\) with Levi Civita connection \(D_g\) and Ricci tensor \(Ric_g\) is called singular if the equation \[ D_gd(v)=vRic_g+(\Delta v)g \] admits a nontrivial solution \(v\). This equation comes from a variational problem which also motivates the definition. The above PDE system is overdetermined. It was proved by J.-P. Bourguignon [Compositio Math. 30, 1-41 (1975; Zbl 0301.58015)] and A. E. Fischer and J. E. Marsden [Proc. Symp. Pure Math. 27, 219-263 (1975; Zbl 0314.53031)] that a singular metric is either Ricci-flat or has positive constant scalar curvature. This led Fischer-Marsden to conjecture that a singular metric must be Einstein. An equivalent statement (based on Obata’s analytical characterization of round spheres) is if \(g\) is singular on \(M\) with positive scalar curvature, then \((M,g)\) is a round sphere. Counterexamples were found independently by O. Kobayashi [J. Math. Soc. Japan 34, 665-675 (1982; Zbl 0495.53038)] and J. Lafontaine [J. Math. Pures Appl. 62, 63-72 (1983; Zbl 0513.53046)]. All their examples contain a totally geodesic \((n-1)\)-sphere where \((M^n,g)\) is conformally flat.
The present note shows that this is not accidental, at least in dimension \(3\). Precisely, it is proved that if \(g\) is a singular metric with positive scalar curvature on a \(3\)-dimensional closed manifold \(M\), then \((M,g)\) contains a totally geodesic \(2\)-sphere. The proof is based on observing that the equation defining a singular metric is the one satisfied by a static perfect fluid in general relativity.


53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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[1] M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differential Geometry 3 (1969), 379 – 392. · Zbl 0194.53103
[2] Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. · Zbl 0613.53001
[3] Jean-Pierre Bourguignon, Une stratification de l’espace des structures riemanniennes, Compositio Math. 30 (1975), 1 – 41 (French). · Zbl 0301.58015
[4] Gary L. Bunting and A. K. M. Masood-ul-Alam, Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time, Gen. Relativity Gravitation 19 (1987), no. 2, 147 – 154. · Zbl 0615.53055
[5] John A. Flueck and James F. Korsh, A generalized approach to maximum likelihood paired comparison ranking, Ann. Statist. 3 (1975), no. 4, 846 – 861. · Zbl 0325.62020
[6] W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164 (1967), 1776-1779.
[7] Osamu Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Japan 34 (1982), no. 4, 665 – 675. · Zbl 0486.53034
[8] H. P. Künzle, On the spherical symmetry of a static perfect fluid, Comm. Math. Phys. 20 (1971), 85 – 100.
[9] Jacques Lafontaine, Sur la géométrie d’une généralisation de l’équation différentielle d’Obata, J. Math. Pures Appl. (9) 62 (1983), no. 1, 63 – 72 (French). · Zbl 0513.53046
[10] L. Lindblom, Some properties of static general relativistic stellar models, J. Math. Phys. 21, No. 6 (1980), 1455-1459.
[11] A. K. M. Masood-ul-Alam, On spherical symmetry of static perfect fluid spacetimes and the positive-mass theorem, Classical Quantum Gravity 4 (1987), no. 3, 625 – 633. · Zbl 0613.53038
[12] Morio Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333 – 340. · Zbl 0115.39302
[13] D. C. Robinson, A simple proof of the generalization of Israel’s Theorem, General Relativity and Gravitation 8, No. 8 (1977), 695-698. · Zbl 0429.53043
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