# Some examples of vector fields on the 3-sphere

Annales de l'institut Fourier (1970)

- Volume: 20, Issue: 2, page 1-20
- ISSN: 0373-0956

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topWilson, F. Wesley. "Some examples of vector fields on the 3-sphere." Annales de l'institut Fourier 20.2 (1970): 1-20. <http://eudml.org/doc/74014>.

@article{Wilson1970,

abstract = {Let $S^3$ denote the set of points with modulus one in euclidean 4-space $R^4$ ; and let $\Gamma ^1_0(S^3)$ denote the space of nonsingular vector fields on $S^3$ with the $C^1$ topology. Under what conditions are two elements from $\Gamma ^1_0(S^3)$ 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.},

author = {Wilson, F. Wesley},

journal = {Annales de l'institut Fourier},

keywords = {topology},

language = {eng},

number = {2},

pages = {1-20},

publisher = {Association des Annales de l'Institut Fourier},

title = {Some examples of vector fields on the 3-sphere},

url = {http://eudml.org/doc/74014},

volume = {20},

year = {1970},

}

TY - JOUR

AU - Wilson, F. Wesley

TI - Some examples of vector fields on the 3-sphere

JO - Annales de l'institut Fourier

PY - 1970

PB - Association des Annales de l'Institut Fourier

VL - 20

IS - 2

SP - 1

EP - 20

AB - Let $S^3$ denote the set of points with modulus one in euclidean 4-space $R^4$ ; and let $\Gamma ^1_0(S^3)$ denote the space of nonsingular vector fields on $S^3$ with the $C^1$ topology. Under what conditions are two elements from $\Gamma ^1_0(S^3)$ 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.

LA - eng

KW - topology

UR - http://eudml.org/doc/74014

ER -

## References

top- [1] H. HOPF, Uber die abbildungen von sphären auf sphären neidrigerer dimension. Fund. Math., 25 (1935), 427-440. Zbl0012.31902JFM61.0622.04
- [2] V. NEMYTSKII and V. STEPANOV, Qualitative Theory of Differential Equations. Princeton, 1963. Zbl0089.29502
- [3] G. REEB, Variétés Feuilletées. Actualités Sci. Ind., No. 1183 (Publ. Inst. Math. Univ. Strasbourg, 11) Hermann and Cie, Paris, 1952.
- [4] N. STEENROD, The Topology of Fiber Bundles. Princeton, 1951. Zbl0054.07103
- [5] F. W. WILSON, Smoothing derivatives of functions and applications, Trans. A.M.S. 139 (1969), 413-428. Zbl0175.20203MR40 #4974
- [6] F. W. WILSON, The structure of the level surfaces of a Lyapunov function, J. Diff. Eq. 3 (1967), 323-329. Zbl0152.28701MR37 #6964
- [7] F. W. WILSON, Elliptic flows are trajectory equivalent, to appear, Am. J. Math. Zbl0209.39503

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