Nonhomogeneous Riemannian 3-manifolds with distinct constant Ricci eigenvalues.

*(English)*Zbl 0789.53024A Riemannian manifold \((M,g)\) is said to be curvature homogeneous if, for each pair of points \(p\) and \(q\), there exists a linear isometry \(F\) between the tangent spaces \(T_ pM\) and \(T_ qM\) such that \(F^*(R_ q) = R_ p\) where \(R\) denotes the Riemannian curvature tensor. Moreover, if \((\overline{M},\overline{g})\) is a homogeneous Riemannian manifold and \(o\) a fixed point of \(\overline{M}\), then \((M,g)\) has the same curvature tensor as \((\overline{M},\overline{g})\) if, for each \(p\in M\), there exists a (linear) isometry \(F: T_ pM \to T_ oM\) such that \(F^*(\overline{R}_ o) + R_ p\). In this case, \((\overline{M},\overline{g})\) is called a homogeneous model space of \((M,g)\). Obviously, any (locally) homogeneous space is curvature homogeneous but many examples which are not locally homogeneous are now known.

For three-dimensional manifolds the curvature homogeneity is equivalent to the constancy of the Ricci eigenvalues \(\rho_ 1,\rho_ 2,\rho_ 3\). In [Nagoya Math. J. 132, 1-36 (1993; Zbl 0788.53038)] the author made a detailed study of the case \(\rho_ 1 = \rho_ 2 \neq \rho_ 3\) and obtained examples of curvature homogeneous spaces without a homogeneous model. The existence of curvature homogeneous ones with prescribed arbitrary and “distinct” Ricci eigenvalues has been proved by A. Spiro and F. Tricerri but they did not provide explicit examples. On the other hand K. Yamato found complete examples in this class but they all have a homogeneous model. In this paper the author extends the class of Yamato examples and provides noncomplete curvature homogeneous ones without any homogeneous model. His method is based on J. Milnor’s classification of three-dimensional Lie groups equipped with a left invariant metric.

For three-dimensional manifolds the curvature homogeneity is equivalent to the constancy of the Ricci eigenvalues \(\rho_ 1,\rho_ 2,\rho_ 3\). In [Nagoya Math. J. 132, 1-36 (1993; Zbl 0788.53038)] the author made a detailed study of the case \(\rho_ 1 = \rho_ 2 \neq \rho_ 3\) and obtained examples of curvature homogeneous spaces without a homogeneous model. The existence of curvature homogeneous ones with prescribed arbitrary and “distinct” Ricci eigenvalues has been proved by A. Spiro and F. Tricerri but they did not provide explicit examples. On the other hand K. Yamato found complete examples in this class but they all have a homogeneous model. In this paper the author extends the class of Yamato examples and provides noncomplete curvature homogeneous ones without any homogeneous model. His method is based on J. Milnor’s classification of three-dimensional Lie groups equipped with a left invariant metric.

Reviewer: L.Vanhecke (Heverlee)