Totally geodesic and parallel hypersurfaces of Gödel-type spacetimes (Q6123198)

A submanifold \(M\) of a pseudo-Riemannian manifold is said to be parallel if its second fundamental form \(h\) (and hence, all the extrinsic invariants derived from it) is covariantly constant. Parallel submanifolds extend in a natural way the notion of totally geodesic submanifolds, for which \(h = 0\). In this paper, the authors classify parallel and totally geodesic hypersurfaces within the wide class of Gödel-type spacetimes, i.e. Lorentzian 4-manifolds of signature \((1,3)\) with metric \[ (dt+H(r)d\phi)^2-dr^2-D(r)^2d\phi^2-dz^2, \] where \(t\) is the time variable and \((r,\phi,z)\) are cylindrical coordinates.