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Singularities of vector fields on \({\mathbb{R}}^ 3\) determined by their first non-vanishing jet. (English) Zbl 0661.58005
The authors present a result which gives a sufficient condition for a vector field X on \({\mathbb{R}}^ 3\) to be equivalent at a singularity to the first non-vanishing jet \(j^ kX(p)\) of X at p. This condition - which only depends on the homogeneous vector field defined by \(j^ kX(p)\)- is stated in terms of the blown-up vector field \(\bar X\) (which is defined on \(S^ 2\times {\mathbb{R}})\), and essentially means that there are no saddle-connections for \(\bar X| S^ 2\times \{0\}.\)
The key tool in the proof will be a result of local normal linearization along a codimension 1 submanifold M providing a \(C^ 0\) conjugacy having a normal derivative along M equal to 1.
Reviewer: P.Bonckaert

58A20 Jets in global analysis
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
Full Text: DOI
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