Index of singularities of real vector fields on singular hypersurfaces

DOI10.5427/JSING.2014.9JzbMATH Open1305.58021arXiv1301.1781OpenAlexW2333110655MaRDI QIDQ2879068

Pavao Mardešić

Publication date: 8 September 2014

Published in: Journal of Singularitiesl (Search for Journal in Brave)

Abstract: G'omez-Mont, Seade and Verjovsky introduced an index, now called GSV-index, generalizing the Poincar'e-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 G'omez-Mont and Giraldo about calculating the GSV-index $In{d}_{V_{p}m,0}(X)$ of a real vector field $X$ tangent to a singular hypersurface $V=f^{-1}(0)$. The index $In{d}_{V_{pm,0}}(X)$ 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 G'omez-Mont's formula for calculating the GSV-index on singular complex hypersurfaces and the formula of Eisenbud, Levine and Khimshiashvili for calculating the Poincar'e-Hopf index of a singularity of a real vector field in $R^{n+1}$

Full work available at URL: https://arxiv.org/abs/1301.1781

zbMATH Keywords

real hypersurfaceshessianalgebraically isolated singularitieshamiltonian vector fieldsPoincaré-Hopf index, GSV-index of real vector fieldssignature of bilinear forms

Mathematics Subject Classification ID

Singularities of vector fields, topological aspects (58K45) Singularities of holomorphic vector fields and foliations (32S65) Topological invariants on manifolds (58K65) Topological properties of mappings on manifolds (58K15)

Related Items (6)

The index of a vector field tangent to a hypersurface and the signature of the relative Jacobian determinant ⋮ An index formula of singularities for vector fields on manifolds with boundary and its application ⋮ Indices of vector fields and 1-forms ⋮ Some index classes of singularity of vector fields on manifolds with boundary ⋮ On the index formula for singular surfaces ⋮ Index sums of isolated singular points of positive vector fields

This page was built for publication: Index of singularities of real vector fields on singular hypersurfaces