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Toward a classification of Killing vector fields of constant length on pseudo-Riemannian normal homogeneous spaces. (English) Zbl 1378.53082
The authors consider: a connected real reductive Lie group \(G\), endowed with a nondegenerate invariant bilinear form \(b\) on its Lie algebra \(L(G)\); a closed reductive subgroup \(H\) in \(G\) such that \(b\) is nondegenerate on \(L(H)\); the “normal” pseudo-Riemannian metric \(b^M\) induced on \(M:= {G}/ {H}\) and the Killing vector fields \(\xi^M\) on \(M\), induced by any elliptic nonzero \(\xi \in L(G)\), with constant (pseudo) length w.r.t. \(b^M\). The paper builds fundamental landmarks for the classification of the triples \((G,H,\xi)\), in the following cases: (i) When \(G\) is simple, Theorem 6.1 provides the complete classification; (ii) When \(G\) is semi-simple and \(M\) is a coset irreducible normal homogeneous space, Lemma 7.5 and Theorem 7.6 establish a partial classification (the remaining uncovered situations involve a family of pseudo-Riemannian group manifolds which contain, for example, the spaces \({(H \times H)}/ {(\operatorname{diag}(H))}\), for real simple Lie groups \(H\)).

MSC:
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C30 Differential geometry of homogeneous manifolds
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