The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant
Annales de l'Institut Fourier, Tome 47 (1997) no. 5, p. 1523-1539
Pour tout champ de vecteurs analytique réel tangent à une hypersurface V et ayant une singularité algébriquement isolée on introduit le déterminant jacobien relatif dans l’algèbre finiment engendrée B Ann B (h) associée à la singularité du champ de vecteurs sur V. On montre que le déterminant jacobien relatif engendre un idéal unidimensionnel qui est l’idéal minimal non trivial. À l’aide du déterminant jacobien relatif on introduit un produit bilinéaire non dégénéré dont la signature mesure la taille (avec signe) de ce point. La signature satisfait une loi de conservation. Pour les hypersurfaces de dimension paire la signature donne l’indice de Poincaré-Hopf de la restriction du champ de vecteurs à l’hypersurface.
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 B Ann 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.
@article{AIF_1997__47_5_1523_0,
     author = {G\'omez-Mont, Xavier and Marde\v si\'c, Pavao},
     title = {The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {47},
     number = {5},
     year = {1997},
     pages = {1523-1539},
     doi = {10.5802/aif.1608},
     zbl = {0891.32013},
     mrnumber = {99g:32062},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1997__47_5_1523_0}
}
Gómez-Mont, Xavier; Mardešić, Pavao. The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant. Annales de l'Institut Fourier, Tome 47 (1997) no. 5, pp. 1523-1539. doi : 10.5802/aif.1608. http://www.numdam.org/item/AIF_1997__47_5_1523_0/

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