Some examples of vector fields on the 3-sphere
Annales de l'Institut Fourier, Volume 20 (1970) no. 2, p. 1-20
Let S 3 denote the set of points with modulus one in euclidean 4-space R 4  ; and let Γ 0 1 (S 3 ) denote the space of nonsingular vector fields on S 3 with the C 1 topology. Under what conditions are two elements from Γ 0 1 (S 3 ) homotopic ? There are several examples of nonsingular vector fields on S 3 . However, they are all homotopic to the tangent fields of the fibrations of S 3 due to H. Hopf (there are two such classes).We construct some new examples of vector fields which can be classified geometrically. Each of these examples has a finite number of closed integral curves. There is one denumerable class of examples which have exactly one closed integral curve and there is a denumerable class of examples which have exactly two closed integral curves. Among the latter, there are examples of all homotopy classes.
Soit S 3 l’ensemble des points de module un de l’espace numérique réel R 4 de dimension quatre ; et soit Γ 0 1 (S 3 ) l’espace des champs de vecteurs non singuliers définis sur S 3 avec la topologie C 1 . Quelles conditions sont suffisantes pour que deux éléments dans Γ 0 1 (S 3 ) soient homotopes ? Il y a plusieurs exemples de champs de vecteurs non singuliers définis sur S 3 . Cependant ils sont tous homotopes aux champs de vecteurs tangents aux fibrés de H. Hopf (il y a deux telles classes).Nous construirons des exemples nouveaux de champs qui admettent une classification géométrique. Ces exemples ont un nombre fini de courbes intégrales fermées. Nous obtenons une classe dénombrable de champs qui ont seulement une courbe intégrale fermée, et une classe dénombrable de champs qui ont exactement deux courbes intégrales fermées. Parmi celles-ci, il y a des exemples de toutes les classes d’homotopie.
@article{AIF_1970__20_2_1_0,
     author = {Wilson, F. Wesley},
     title = {Some examples of vector fields on the 3-sphere},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {20},
     number = {2},
     year = {1970},
     pages = {1-20},
     doi = {10.5802/aif.349},
     zbl = {0195.25403},
     mrnumber = {44 \#3340},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1970__20_2_1_0}
}
Wilson, F. Wesley. Some examples of vector fields on the 3-sphere. Annales de l'Institut Fourier, Volume 20 (1970) no. 2, pp. 1-20. doi : 10.5802/aif.349. http://www.numdam.org/item/AIF_1970__20_2_1_0/

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