Primary singularities of vector fields on surfaces

  • M. W. Hirsch
  • F. J. TurielEmail author
Original Paper


Unless another thing is stated one works in the \(C^\infty \) 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 \(f:M\rightarrow \mathbb {R}\). A subset K of the zero set \({\mathsf {Z}} (X)\) is an essential block for X if it is non-empty, compact, open in \({\mathsf {Z}}(X)\) and its Poincaré-Hopf index does not vanishes. One says that X is non-flat at p if its \(\infty \)-jet at p is non-trivial. A point p of \({\mathsf {Z}}(X)\) 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 non-flat at every point of K. Then K contains a primary singularity of X. As a consequence, if M is a compact surface with non-zero characteristic and X is nowhere flat, then there exists a primary singularity of X.


Vector field Singularity Zero set Essential block Surface 

Mathematics Subject Classification

20F16 58J20 37F75 37O25 54H25 



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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wisconsin at Madison and University of California at BerkeleyCross PlainsUSA
  2. 2.Geometría y Topología, Facultad de CienciasMálagaSpain

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