About pseudo-Riemannian Lichnerowicz conjecture. (English) Zbl 1335.53097
A (pseudo-)Riemannian manifold \((M,g)\) is said to have essential conformal group if there are no (pseudo-)Riemannian metrics conformal to \(g\) with respect to which \(\mathrm{Conf}(M,g)\) acts isometrically. It is a classical result obtained independently by M. Obata [J. Differ. Geom. 6, 247–258 (1971; Zbl 0236.53042)] and J. Lelong-Ferrand [Mem. Cl. Sci., Collect. Octavo, Acad. R. Belg. 39, No. 5, 3–44 (1971; Zbl 0215.50902)] that the only closed Riemannian manifold with essential conformal group is the round sphere, confirming a conjecture of Lichnerowicz. G. D’Ambra and M. Gromov [Surv. Differ. Geom., Suppl. J. Diff. Geom. 1, 19–111 (1991; Zbl 0752.57017)] asked whether the same conjecture of Lichnerowicz holds in the pseudo-Riemannian realm. More precisely, the statement of the “pseudo-Riemannian Lichnerowicz conjecture” is that a closed pseudo-Riemannian manifold with essential conformal group is conformally flat. This short and well-written note constructs counter-examples to this conjecture on \(S^1\times S^{n-1}\) with metrics of all signatures \(\geq 2\), leaving only the Lorentzian case unsettled.
Reviewer: Renato G. Bettiol (Philadelphia)
MSC:
| 53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |
| 53A30 | Conformal differential geometry (MSC2010) |
Keywords:
pseudo-Riemannian geometry; semi-Riemannian geometry; conformal geometry; essential conformal groupReferences:
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