On the singularities of quadratic forms
type: Item
instance of: scholarly article
title: On the singularities of quadratic forms
zbMATH Open document ID: 0959.58003
DOI: 10.1016/S0393-0440(99)00042-X
author: Dimitris Pliakis
author: Spyros Pnevmatikos
published in: Journal of Geometry and Physics
publication date: 2001-04-23T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58A10
Mathematics Subject Classification ID: 58K05
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 53C17
zbMATH DE Number: 1463372
zbMATH Keywords: quadratic forms
zbMATH Keywords: singularities
zbMATH Keywords: variational problem
reviewed by: L. Del Riego
MaRDI profile type: Publication
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full work available at URL: s0393-0440(99)00042-x
OpenAlex ID: W2087056289
description: scientific article; zbMATH DE number 1463372