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Xavier Gómez-Mont; Pavao Mardešić
The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant
Annales de l'institut Fourier, 47 no. 5 (1997), p. 1523-1539, doi: 10.5802/aif.1608
Article PDF | Reviews MR 99g:32062 | Zbl 0891.32013

Résumé - Abstract

Given a real analytic vector field tangent to a hypersurface $V$ with an algebraically isolated singularity we introduce a relative Jacobian determinant in the finite dimensional algebra ${{\bf B}\over {\rm Ann}_{\bf B}(h)}$ associated with the singularity of the vector field on $V$. We show that the relative Jacobian generates a 1-dimensional non-zero minimal ideal. With its help we introduce a non-degenerate bilinear pairing, and its signature measures the size of this point with sign. The signature satisfies a law of conservation of number and for even dimensional hypersurfaces it gives a method to compute the Poincaré-Hopf index of the vector field restricted to the hypersurface.

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