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Nilpotent spacelike Jordan Osserman pseudo-Riemannian manifolds. (English) Zbl 1053.53018
Slovák, Jan (ed.) et al., The proceedings of the 23th winter school “Geometry and physics”, Srní, Czech Republic, January 18–25, 2003. Palermo: Circolo Matemàtico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 72, 99-105 (2004).
The authors prove that there exist pseudo-Riemannian manifolds of signature $$(s,2s)$$ for any $$s\geq 2$$ which are space-like Jordan Osserman nilpotent of order 3 and which are not time-like Jordan Osserman. To prove this theorem they first construct a family of algebraic curvature tensors $$R$$ on a vector space $$V$$ of signature $$(s,2s)$$ which are space-like Jordan Osserman nilpotent of order 3 and which are not time-like Jordan Osserman. Next, the authors realize this family geometrically. Their construction shows that in fact there are many such examples. It is worth noticing that these examples and techniques are quite new.
For the entire collection see [Zbl 1034.53002].

##### MSC:
 53B30 Local differential geometry of Lorentz metrics, indefinite metrics 53B20 Local Riemannian geometry
##### Keywords:
Jacobi operator; Osserman conjecture