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Curvature homogeneous metrics of cohomogeneity one on vector spaces. (English) Zbl 0911.53029

Let \((M,g)\) be an \(n\)-dimensional Riemannian manifold with Riemannian curvature tensor \(R\). Then \((M,g)\) is said to be curvature homogeneous if, for every pair of points \(p,q\) of \(M\), there exists a linear isometry \(F:T_p M\to T_qM\) such that \(F^*R_q =R_p\). Locally homogeneous spaces are trivial examples but there exist a lot of examples which are not locally homogeneous. We refer to [E. Boeckx, O. Kowalski and L. Vanhecke, ‘Riemannian manifolds of conullity two’ (World Scientific, Singapore) (1996; Zbl 0904.53006)] for a survey. K. Tsukada provided a particularly interesting example in [TĂ´hoku Math. J., II. Ser. 40, 221-244 (1988; Zbl 0651.53037)]. It is a cohomogeneity one hypersurface of the 5-dimensional hyperbolic space \(H^5(-1)\).
The main aim of the author is to classify curvature homogeneous manifolds among the cohomogeneity one manifolds A first step towards this goal is the classification of curvature homogeneous \(G\)-invariant metrics \(g\) on a real vector space \(V\) where \(G\subset SO(n)\) is a compact, connected Lie group acting transitively on the standard unit sphere \(S^{n-1}\) in \(V=\mathbb{R}^n\).
In this paper, he proves that any curvature homogeneous \(G\)-invariant metric on \(V\setminus \{0\}\) that admits an extension to a smooth, complete metric \(g\) on \(V\) is Einstein and homogeneous (in fact, locally isometric to a rank one symmetric space). The explicit expression of these metrics \(g\) are given, up to a \(G\)-diffeomorphism.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30 Differential geometry of homogeneous manifolds
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