The Gluck and Ziller problem with the Euclidean metric. (English. French, English summaries) Zbl 1073.53081

Borrelli, Vincent

Seminar on spectral theory and geometry. 2003–2004. St. Martin d’Hères: Université de Grenoble I, Institut Fourier. Séminaire de Théorie Spectrale et Géométrie 22, 83-92 (2004).

Let \(S^{2m+1}\) be the standard round sphere of radius \(r\) and \(T^1S^{2m+1}\) its unit tangent bundle. H. Gluck and W. Ziller [Comment. Math. Helv. 61, 177–192 (1986; Zbl 0605.53022)] set the problem of finding unit vector fields \(V: S^{2m+1} \to T^1S^{2m+1}\) whose image \(V(S^{2m+1})\) has minimal volume. This problem depends on the choice of a metric on \(T^1S^{2m+1}\) (in the literature, it is usually the Sasaki metric). The author considers the tangent bundle endowed with the metric obtained by pulling back the Euclidean metric of \(\mathbb R^{2m+2}\) by the embedding \(T^1S^{2m+1} \to \mathbb R^{2m+2}: v_x \mapsto (x,v)\). He shows that Hopf vector fields (i.e., unit vector fields tangent to the fibre of any Hopf fibration) are critical for the volume functional with respect to the above metric. By a result of H. Gluck and W. Ziller (loc. cit.) these critical points are stable for 3-dimensional spheres and the author shows that for spheres of dimension \(n\geq 5\) they are stable if \(r \leq 1/\sqrt{n-3}\).
For the entire collection see [Zbl 1054.35001].

Reviewer: Sergio Console (Torino)

MSC:

53C43	Differential geometric aspects of harmonic maps
58E15	Variational problems concerning extremal problems in several variables; Yang-Mills functionals

Keywords:

unit tangent bundle; minimum volume; Hopf fibration

Citations:

Zbl 0605.53022