Some examples of vector fields on the 3-sphere

F. Wesley Wilson

Annales de l'institut Fourier (1970)

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

Abstract

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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.

How to cite

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Wilson, 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

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

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