Conformally flat semi-Riemannian manifolds with commuting curvature and Ricci operators. (English) Zbl 1055.53054

The author studies conformally flat, simply connected, complete semi-Riemannian manifolds \((M,g)\) of any dimension \(n= 4\) and any signature \(q\) which satisfy the condition \(R(X,Y)Q= 0\), where \(R\) denotes the curvature tensor and \(Q\) the Ricci operator. The main theorem is a classification result: Such an \((M,g)\) is (i) of constant curvature or (ii) a product of manifolds of constant curvature of one-dimensional manifolds or (iii) a complex sphere or (iv) such that \(Q^2= 0\). Case (iv), which never occurs in proper Riemannian geometry is studied in detail.


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