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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)

MSC:

53C20 Global Riemannian geometry, including pinching
53C30 Differential geometry of homogeneous manifolds
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