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A note on Osserman Lorentzian manifolds. (English) Zbl 0865.53018

Let \(M\) be a Lorentzian manifold of dimension \(m\geq 3\). We say that \(M\) is spacelike \((\varepsilon= +1)\) or timelike \((\varepsilon= -1)\) Osserman at a point \(p\in M\) if the eigenvalues of \({\mathcal R}_X: Y\to R(X, Y)X\) are constant on \(\{X\in T_p (M): |X|^2= \varepsilon\}\). We show that if \(M\) is spacelike Osserman at \(p\), then \(M\) has constant sectional curvature at \(p\); similarly if \(M\) is timelike Osserman at \(p\), then \(M\) has constant sectional curvature at \(p\). The reverse implications are immediate. The timelike case and 4-dimensional spacelike case where first studied by E. García-Río, D. N. Kupeli, and M. E. Vázquez-Abal [Differ. Geom. Appl. 7, No. 1, 85-100 (1997)]; we use a different approach from these authors.
Reviewer: P.Gilkey (Eugene)

MSC:

53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics