TY - JOUR
T1 - On bounded vector fields
AU - Cima, Anna
AU - Mañosas, Francesc
AU - Villadelprat, Jordi
PY - 1999
Y1 - 1999
N2 - 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.
AB - 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.
KW - Bounded vector field
KW - Index
UR - https://www.scopus.com/pages/publications/0033145434
U2 - 10.1216/rmjm/1181071647
DO - 10.1216/rmjm/1181071647
M3 - Article
AN - SCOPUS:0033145434
SN - 0035-7596
VL - 29
SP - 473
EP - 489
JO - Rocky Mountain Journal of Mathematics
JF - Rocky Mountain Journal of Mathematics
IS - 2
ER -