The index of a vector field tangent to a hypersurface and the signature of the relative Jacobian determinant
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 \(\delta\).
scientific article; zbMATH DE number 1092703
scientific article