# zbMATH — the first resource for mathematics

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
Full Text: