On bounded vector fields

Anna Cima, Francesc Mañosas, Jordi Villadelprat

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Abstract

We introduce the notion of a strongly bounded vector field, which is closely related to the usual notion of a bounded vector field, and we prove that any C1 strongly bounded vector field in rn with finitely many critical points satisfies that the sum of the indices of the vector field at these critical points is equal to (–1)n. In the planar case we improve this result since we prove it for C1 bounded vector fields. Moreover, when n ≥ 3, we present examples of C∞ bounded vector fields in rn, being obviously not strongly bounded, not satisfying that the sum of the indices at the critical points is (-1)n. © 1999 Rocky Mountain Mathematics Consortium.
Original languageEnglish
Pages (from-to)473-489
JournalRocky Mountain Journal of Mathematics
Volume29
Issue number2
DOIs
Publication statusPublished - 1 Jan 1999

Keywords

  • Bounded vector field
  • Index

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