Degenerate submanifolds in semi-Riemannian geometry
Let (M,g) be a semi-Riemannian manifold of arbitrary signature and (H,g\(| H)\) be a degenerate submanifold of M, i.e. \(g| H\) is singular and has constant signature. In this paper the author extends his studies on null submanifolds of Lorentzian manifolds [ibid. 23, 33-51 (1987; Zbl 0625.53019)] to the situation described above by employing his results on degenerate manifolds [ibid. 23, 259-290 (1987; Zbl 0625.53020)]. The main problem is that there does not exist a natural projection (as in the nondegenerate case) onto the tangent space of a degenerate submanifold. Nevertheless the author succeeds in showing the usual compatibility of the induced singular semi-Riemannian structure of H with the semi-Riemannian structure of M for a certain subclass the so- called irrational stationary degenerate submanifolds. In this case Gauss- Codazzi equations are obtained. Furthermore at the end of the paper the property of total umbilicity is investigated for this type of submanifolds.
irrational stationary degenerate submanifolds
scientific article; zbMATH DE number 4050449