On the singularities of quadratic forms
The authors start by pointing out that in the case of submanifolds of pseudo-Riemannian manifolds, the induced structures are not in general pseudo-Riemannian. Such is the case of compact submanifolds in \(\mathbb R^n\) equipped with the Lorentzian metric. By giving a complete classification of germs of quadratic forms, they give explicit expressions for their degeneracy locus algebraic equations. These will be used when proving, in the case of generic submanifolds of a pseudo-Riemannian manifold, that the Pfaffian forms that belong to the induced quadratic morphism extend smoothly to Pfaffian forms on the ambient manifold on the generic submanifolds. A singular connection is used to determine the geodesics relatively to the induced quadratic form on the generic manifolds. In a more general context, they analyse the dynamics of generic stratified singularities: first generic quadratic forms on a space and on its dual, then they extend it to the ambient pseudo-Riemannian manifold. They pose the problem whether there are Lagrangians that define a vector field everywhere on \(TM\) despite the fact that their Legendre transformation is singular and the projection of its integral curves gives the solution of the variational problem on \(M\). They specifically give conditions under which the Lagrangian vector is extended smoothly.
scientific article; zbMATH DE number 1463372
scientific article