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Conformally Osserman Lorentzian manifolds. (English) Zbl 1143.53336

A pseudo-Riemannian manifold \(M\) is conformally flat if the eigenvalues of the symmetric Weyl Jacobi operator, \(J_W(x)y=W(y,x)x\) (\(W(x,y)\) being the standard Weyl conformal curvature operator), has constant eigenvalues on the bundle of unit time-like (space-like) vectors. The author studies conformal Osserman Lorentzian manifolds. The main result is the following characterization: if \(M\) is a Lorentzian manifold of dimension \(n>3\), then it is a conformal Osserman manifold if and only if it is a conformally flat manifold. This result shows some rigidity of the Lorentzian signature, which is illustrated by some recent results concerning conformal Osserman manifolds in other signatures (Section 4).

MSC:

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