Degenerate manifolds (Q579634)

The author studies the geometry of manifolds M equipped with a singular semi-Riemannian metric g of constant type. A Koszul derivative \(\nabla: (\Gamma TM)^ 2\to \Gamma TM\) is introduced for (M,g) by requiring that for g(\(\nabla,\cdot,z)\) the common equations for a symmetric covariant derivative are valid for all \(z\in \Gamma TM\) and that in addition \(\nabla\) be metric with respect to g. An existence theorem for Koszul derivatives is established. Furthermore torsion curvature and sectional curvature functions are introduced for this derivative, and their properties are compared with those in the Riemannian case. Special attention is given to the case of constant sectional curvature. In the second part a quotient construction [see also the author's paper, ibid., 33-51 (1987; Zbl 0625.53019)] is used to associate a semi- Riemannian vector bundle over M to (M,g). Then the relation between its canonical semi-Riemannian connection and the Koszul derivative introduced above is investigated. Also some relation to the induced semi-Riemannian connection on a non-degenerate submanifold of maximal dimension is observed. Finally the author studies fibrations of manifolds of the above type over semi-Riemannian manifolds. In particular he shows that every singular semi-Riemannian manifold locally is a warped product.