On the singularities of quadratic forms

The induced structures on the submanifolds in a pseudo-Riemannian manifold are not, in general, pseudo-Riemannian; also, the kernel's distributions of the induced quadratic forms do not define, in general, regular foliations. In this report, the singularities of the quadratic forms on a manifold are described in a generic context and we study their geometric and algebraic properties. Therefore, using these results, we treat the problem whether there are Lagrangians on the tangent bundle of a manifold that define a Lagrangian vector field everywhere on the tangent bundle, despite the fact that their Legendre transformation is singular, and the projection of its integral curves gives the solutions of the corresponding variational problem on the manifold. (C) 2000 Elsevier Science B.V. All rights reserved. Subj. Class.: Differential geometry 1991 MSG: 58A10; 58C27.