Curvature homogeneous spaces with a solvable Lie group as homogeneous model
Following I. M. Singer, a Riemannian manifold \((M,g)\) is said to be curvature homogeneous if, for every two points \(p,q\in M\), there exists a linear isometry \(F: T_ p M\to T_ q M\) such that \(F^* R_ q=R_ p\), where \(R\) denotes the Riemannian curvature tensor of \((M,g)\). Of course, any (locally) homogeneous space is curvature homogeneous. Further, a Riemannian manifold \((M,g)\) is said to have the same curvature tensor as a homogeneous space \((\widetilde M,\widetilde g)\) if, for every pair of points \(m\in M\) and \(\widetilde m\in\widetilde M\), there exists a linear isometry \(F: T_ m M\to T_{\widetilde m}\widetilde M\) such that \(F^*\widetilde R_{\widetilde m}=R_ m\). Also in this case \((M,g)\) is curvature homogeneous and \((\widetilde M,\widetilde g)\) is called a model space for \((M,g)\). The main task is to construct examples of non- homogeneous curvature homogeneous manifolds, to determine their classification and to study their geometry. An extensive study of curvature homogeneous spaces with a symmetric model has been made by several authors and, in the generic case, their classification is now known [see \textit{E. Boeckx}, \textit{O. Kowalski} and \textit{L. Vanhecke}, Nonhomogeneous relatives of symmetric spaces, to appear in Differ. Geom. Appl., for more details and further references]. We note that all these spaces are semi-symmetric manifolds. Some examples of curvature homogeneous spaces with a non-symmetric model space are already known, for examples in the class of isoparametric hypersurfaces in spheres. The main purpose of this paper is to construct new examples and to study their geometry and isometry classes. They are obtained by deforming a flat right invariant metric on a Lie group which is a semi- direct product of \(\mathbb{R}^ p\) and \(\mathbb{R}^ q\). They are all non-compact in contrast to the isoparametric examples. These last ones are the only known compact examples and the search for other ones seems to be a difficult problem. The second purpose is to treat a remarkable 4-dimensional curvature homogeneous space discovered by \textit{K. Tsukada} [Tôhoku Math. J., II. Ser. 40, 221-244 (1988; Zbl 0651.53037)] and to show that this example has no homogeneous model at all. Moreover, one treats some 3-dimensional examples discovered by \textit{K. Yamato} [Nagoya Math. J. 123, 77-90 (1991; Zbl 0738.53032)]. They all have a homogeneous model. Recently, O. Kowalski found other examples of 3-dimensional curvature homogeneous spaces without a homogeneous model.
scientific article; zbMATH DE number 98459