Index of Singularities of Real Vector Fields on Singular Hypersurfaces

Journal of Singularities
volume 9 (2014), 111-121

Received 10 June 2012. Received in revised form 6 May 2013.

DOI: 10.5427/jsing.2014.9j

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Abstract:

Gómez-Mont, Seade and Verjovsky introduced an index, now called GSV-index, generalizing the Poincaré-Hopf index to complex vector fields tangent to singular hypersurfaces. The GSV-index extends to the real case. This is a survey paper on the joint research with Gómez-Mont and Giraldo about calculating the GSV-index of a real vector field X tangent to a singular hypersurface V=f^{-1}(0). The GSV-index is calculated as a combination of several terms. Each term is given as a signature of some bilinear form on a local algebra associated to f and X. Main ingredients in the proof are Gómez-Mont's formula for calculating the GSV-index on singular complex hypersurfaces and the formula of Eisenbud, Levine and Khimshiashvili for calculating the Poincaré-Hopf index of a singularity of a real vector field in R^{n+1}.

Author(s) information:

Pavao Mardešić
Université de Bourgogne
Institut de Mathématiques de Bourgogne- UMR 5584 du CNRS
UFR Sciences et Techniques
9, Avenue Alain Savary
BP 47870
21078 DIJON, France