Spacetimes of class two (Q1965343)

It is known that there exists an isometric immersion of the Schwarzschild spacetime as well as of the Reissner-Nordström spacetime into a semi-Euclidean space \({\mathbb{E}}^{6}_{s}\), with signature \((s, 6-s)\). It is also known that there exist no isometric immersions of these spacetimes into a \(5\)-dimensional semi-Euclidean space. We mention also that the Kerr spacetime cannot be isometrically immersed into \({\mathbb{E}}^{6}_{s}\). In the paper under review, the problem of isometric immersions of spacetimes \(M^{4}\) into \({\mathbb{E}}^{6}_{s}\) is studied. A family of necessary conditions for an isometric immersion of a spacetime \(M^{4}\) into \({\mathbb{E}}^{6}_{s}\) has been found. These results are applied to the solution of the problem of an isometric immersion of vacuum (i.e. Ricci flat) spacetimes \(M^{4}\) into \({\mathbb{E}}^{6}_{s}\). A list of open problems related to the question of an isometric immersion of a spacetime \(M^{4}\) into \({\mathbb{E}}^{6}_{s}\) is presented. For instance, it is an open question whether the Gödel spacetime can be immersed isometrically in \({\mathbb{E}}^{6}_{s}\).