On the index of a vector field at an isolated singularity

1997/10/07 by Wolfgang Ebeling, Ebeling, Wolfgang, S. M. Gusein‐Zade +2
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #alg-geom #math.AG

paper · pdf · doi:10.48550/arxiv.alg-geom/9710008

AMS-LaTeX, 11 p. with 1 fig.; remarks, definition, and references added to Section 1

openalex publication_date 1997/10/07 · arxiv created 1998/03/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider manifolds with isolated singularities, i.e., topological spaces which are manifolds (say, C^∞--) outside discrete subsets (sets of singular points). For (germs of) manifolds with, so called, cone--like singularities, a notion of the index of an isolated singular point of a vector field is introduced. There is given a formula for the index of a gradient vector field on a (real) isolated complete intersection singularity. The formula is in terms of signatures of certain quadratic forms on the corresponding spaces of thimbles.

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