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Zbl 0891.32013
Gómez-Mont, Xavier; Mardešić, Pavao
The index of a vector field tangent to a hypersurface and the signature of the relative Jacobian determinant.
(English)
[J] Ann. Inst. Fourier 47, No.5, 1523-1539 (1997). ISSN 0373-0956; ISSN 1777-5310/e

Let $\delta= \sum^n_{i=0} h_i{\partial \over \partial x_i}$ be a real analytic vector field on $\bbfR^{n+1}$ with an algebraically isolated singularity at 0 and $B= A_{\bbfR^{n +1}, 0}/(h_0, \dots, h_n)$, $A_{\bbfR^{n+1}, 0}$, the ring of germs of real analytic functions on $\bbfR^{n+1}$ at 0.\par The product in the algebra $B$ and a linear map $\ell: B\to \bbfR$ with $\ell(\text {det} ({\partial h_i \over \partial x_j})) >0$ defines a non-degenerate bilinear form $\langle,\rangle: B\times B\to \bbfR$. The index of $\delta$ is the signature of this bilinear form. \par Let $V\subseteq \bbfR^{n+1}$ be a hypersurface with an isolated singularity at 0 defined by $f=0$ such that $\delta$ is tangent to $V$, i.e. $\delta(f) \in(f)$.\par 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 $\ell:B/ \text {Ann} (h)\to \bbfR$ such that $\ell(J_f (\delta)) >0$.\par 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.\par 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 $\delta$.
[G.Pfister (Kaiserslautern)]
MSC 2000:
*32S05 Local singularities (analytic spaces)
32S25 (Hyper-) Surface singularities (analytic spaces)
58C25 Differentiable maps on manifolds (global analysis)
58K99 None of the above, but in this section
37G99 Bifurcation theory
13H10 Special types of local rings

Keywords: index of a vector field; singularity of a vector field; hypersurface singularities; relative Jacobian determinant

Cited in: Zbl 1175.58015

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