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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.

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