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The infimum of the energy of unit vector fields on odd-dimensional spheres. (English) Zbl 1031.53090

Let \(M\) be an oriented, closed, \(n\)-dimensional Riemannian manifold and \(T^1M\) be the unit tangent bundle of \(M\) considered as a closed Riemannian manifold with the Sasaki metric. The energy of a unit smooth vector field \(v\) on \(M\) is \(E(v)= {1\over 2}\int_M(n+\|\nabla\|^2)dM\), where \(dM\) is the volume form on \(M\). The authors construct a one-parameter family of unit smooth vector fields globally defined on the sphere \(\mathbb{S}^{2k +1}\) for \(k\geq 2\), with energy converging to the energy of the unit radial vector field defined on the complement of two antipodal points. The infimum of the energy of these vector fields is \(({2k+1\over 2}+ {k\over 2k-1}) \text{vol} (\mathbb{S}^{2k+1})\).
Reviewer: V.Oproiu (Iaşi)

MSC:

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