The index of a vector field tangent to a hypersurface and the signature of the relative Jacobian determinant
type: Item
instance of: scholarly article
title: The index of a vector field tangent to a hypersurface and the signature of the relative Jacobian determinant
zbMATH Open document ID: 0891.32013
author: X. Gómez-Mont
author: Pavao Mardešić
publication date: 1997-12-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75272
review text: Let \(\delta= \sum^n_{i=0} h_i{\partial \over \partial x_i}\) be a real analytic vector field on \(\mathbb{R}^{n+1}\) with an algebraically isolated singularity at 0 and \(B= A_{\mathbb{R}^{n +1}, 0}/(h_0, \dots, h_n)\), \(A_{\mathbb{R}^{n+1}, 0}\), the ring of germs of real analytic functions on \(\mathbb{R}^{n+1}\) at 0. The product in the algebra \(B\) and a linear map \(\ell: B\to \mathbb{R}\) with \(\ell(\text{det} ({\partial h_i \over \partial x_j})) >0\) defines a non-degenerate bilinear form \(\langle,\rangle: B\times B\to \mathbb{R}\). The index of \(\delta\) is the signature of this bilinear form. Let \(V\subseteq \mathbb{R}^{n+1}\) be a hypersurface with an isolated singularity at 0 defined by \(f=0\) such that \(\delta\) is tangent to \(V\), i.e. \(\delta(f) \in(f)\). Let \(h= {\delta(f) \over f}\) and define \(J_f(\delta) ={\text{det} ({\partial h_i \over \partial x_j}) \over h} \in B/ \text{Ann} (h)\) the relative Jacobian determinant of \(\delta\). Choose a linear map \(\ell:B/ \text{Ann} (h)\to \mathbb{R}\) such that \(\ell(J_f (\delta)) >0\). The product in \(B/\text{Ann} (h)\) together with \(\ell\) defines a binlinear form on \(B/ \text{Ann} (h)\), let \(\text{Sgn}_{f,0} (\delta)\) be the signature of this bilinear form. It is proved that the function \(\text{Sgn}_{f,0}\) satisfies the law of conservation of number: \[ \text{Sgn}_{f,0} (\delta)= \text{Sgn}_{f,0} (\delta_t) +\sum_{x\in V \setminus \{0\}\atop \delta_t(x) =0}\text{Index}_{V,x} (\delta_t|V) \] for \(x\) close to 0 and \(\delta_t\) tangent to \(V\) and close to \(\delta\).
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 32S25
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 1092703
zbMATH Keywords: index of a vector field
zbMATH Keywords: singularity of a vector field
zbMATH Keywords: hypersurface singularities
zbMATH Keywords: relative Jacobian determinant
reviewed by: Gerhard Pfister
MaRDI profile type: Publication
OpenAlex ID: W2319874468
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DOI: 10.5802/AIF.1608
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 1092703