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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.

MSC:

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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