On the index of a vector field at an isolated singularity

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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.