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Primary singularities of vector fields on surfaces
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https://doi.org/10.1007/s10711019004973Abstract
Unless another thing is stated one works in the C∞ category and manifolds have empty boundary. Let X and Y be vector fields on a manifold M. We say that Y tracks X if [Y, X] = fX for some continuous function f: M→ R. A subset K of the zero set Z(X) is an essential block for X if it is nonempty, compact, open in Z(X) and its PoincaréHopf index does not vanishes. One says that X is nonflat at p if its ∞jet at p is nontrivial. A point p of Z(X) is called a primary singularity of X if any vector field defined about p and tracking X vanishes at p. This is our main result: consider an essential block K of a vector field X defined on a surface M. Assume that X is nonflat at every point of K. Then K contains a primary singularity of X. As a consequence, if M is a compact surface with nonzero characteristic and X is nowhere flat, then there exists a primary singularity of X.
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