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Curvature tensors and Singer invariants of four-dimensional homogeneous spaces. (English) Zbl 1020.53032
The main aim of this paper is to obtain information about the Singer invariant of 4-dimensional locally homogeneous Riemannian spaces. The approach of the authors is the following one. Consider the vector space $$\mathcal R$$ of (algebraic) curvature tensors on $$\mathbb R^4$$ with the natural action of $$O(4)$$ on it. For a Lie subalgebra $$\mathfrak {h}\subset \mathfrak {so}(4)$$ we denote $$\mathcal R^{\mathfrak h}$$ the subspace of curvature tensors invariant by $$\mathfrak {h}$$. We shall say that the curvature tensor of a 4-dimensional locally homogeneous Riemannian space $$M$$ belongs to $$\mathcal R^{\mathfrak h}$$ if there exists an orthonormal frame $$u$$ at $$p\in M$$ such that $$u^*R_p\in \mathcal R^{\mathfrak h}$$. ($$R_p$$ denotes here the curvature tensor at the point $$p$$.) The authors set themselves the following three tasks: (1) Classify the $$\mathfrak {h}$$ which are isotropy subalgebras of the action of $$\mathfrak {so}(4)$$ on $$\mathcal R$$. (2) With respect to each $$\mathfrak {h}$$ in the above, classify the homogeneous spaces whose curvature tensors belong to $$\mathcal R^{\mathfrak h}$$. (3) Compute the Singer invariants of the homogeneous spaces obtained above.
The authors were able to fulfil to a large extent their program. They found (up to conjugation) all isotropy subalgebras of the action of $$\mathfrak {so}(4)$$ on $$\mathcal R$$. (There are 8 of them.) For 5 of them they were able to realize the item (2) of the program. Moreover, they have shown that if $$\dim \mathfrak {h}\geq 2$$, a homogeneous space whose curvature tensor belongs to $$\mathcal R^{\mathfrak h}$$ is locally symmetric or locally homothetic to $$SL(2,\mathbb R)\ltimes \mathbb R^2/SO(2)$$ with the $$(SL(2,\mathbb R)\ltimes \mathbb R^2)$$-invariant metric. The Singer invariant of $$SL(2,\mathbb R)\ltimes \mathbb R^2/ SO(2)$$ is 1. The main result then states that the Singer invariant of a 4-dimensional locally homogeneous Riemannian space is at most 1.

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