2006
DOI: 10.1007/s00208-005-0692-9
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A critical radius for unit Hopf vector fields on spheres

Abstract: The volume of a unit vector field V of the sphere S n (n odd) is the volume of its image V (S n ) in the unit tangent bundle. Unit Hopf vector fields, that is, unit vector fields that are tangent to the fibre of a Hopf fibration S n → CP n−1 2 , are well known to be critical for the volume functional. Moreover, Gluck and Ziller proved that these fields achieve the minimum of the volume if n = 3 and they opened the question of whether this result would be true for all odd dimensional spheres. It was shown to be… Show more

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Cited by 24 publications
(32 citation statements)
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“…One of the intriguing points of the Gluck and Ziller problem is the dependence on dilatations of the metric [2]. The reason is that a dilatation on the base manifold produces dilatations with di¤erent rates on horizontal or vertical distributions of the tangent bundle.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…One of the intriguing points of the Gluck and Ziller problem is the dependence on dilatations of the metric [2]. The reason is that a dilatation on the base manifold produces dilatations with di¤erent rates on horizontal or vertical distributions of the tangent bundle.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…In [1] and [6] respectively, we can find a version concerning spheres of radius r for each sense of the implication of Theorem 1 mentioned in the Introduction. Recall that the study of unit vector fields on the sphere of curvature c, S n ( 1 √ c ), is equivalent to the study of fields with length 1 √ c on S n (1).…”
Section: Notationmentioning
confidence: 99%
“…Recall that the study of unit vector fields on the sphere of curvature c, S n ( 1 √ c ), is equivalent to the study of fields with length 1 √ c on S n (1). For this reason, in this issue, assumptions concerning the length of the vector field and the curvature of the manifold should not look strange, as in Theorem 4.…”
Section: Notationmentioning
confidence: 99%
“…Nevertheless, concerning the stability of Hopf vector fields as critical points of the volume it has been shown [1,7] that for m > 1 they are unstable if and only if the curvature is lower than 2m − 3. Nevertheless, concerning the stability of Hopf vector fields as critical points of the volume it has been shown [1,7] that for m > 1 they are unstable if and only if the curvature is lower than 2m − 3.…”
Section: Introductionmentioning
confidence: 99%