A characterization of locally homogeneous Riemann manifolds of dimension 3. (English) Zbl 0738.53032

The main result is the following sufficient condition for the local homogeneity of a 3-dimensional compact connected Riemannian manifold \(M\): Assume that the eigenvalues \(\rho_ 1,\rho_ 2,\rho_ 3\) of the Ricci tensor are constant on \(M\). Suppose that \(\rho_ 1=\rho_ 2\). If \(\rho_ 1\geq0\) or \(\rho_ 3\leq0\), then \(M\) is locally homogeneous. We also give examples of non-homogeneous, complete metrics on \(\mathbb{R}^ 3\) which have distinct, constant Ricci eigenvalues. Judging from the delicate argument to prove Proposition 5.1, the author arrives at the following problem which is still unsolved: Give a compact connected Riemannian manifold of dimension 3 which is not locally homogeneous, but has constant Ricci eigenvalues.
Reviewer: K.Yamato (Nagoya)


53C20 Global Riemannian geometry, including pinching
53C30 Differential geometry of homogeneous manifolds
Full Text: DOI


[1] DOI: 10.1307/mmj/1028999604 · Zbl 0145.18602
[2] DOI: 10.1016/S0001-8708(76)80002-3 · Zbl 0341.53030
[3] Curvature homogeneous Riemannian manifolds · Zbl 0836.53029
[4] DOI: 10.14492/hokmj/1381758986 · Zbl 0266.53034
[5] DOI: 10.2748/tmj/1178241081 · Zbl 0302.53022
[6] DOI: 10.1002/cpa.3160130408 · Zbl 0171.42503
[7] Ann. Sci. École Norm. Sup 22 pp 535– (1989) · Zbl 0698.53033
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