The Infimum of the Energy of Unit Vector Fields on Odd-Dimensional Spheres

Abstract

We construct a one-parameter family of unit smooth vector fieldsglobally defined on the sphere \(\mathbb{S}\)2k+1 for k ≥ 2, with energyconverging to the energy of the unit radial vector field, which isdefined on the complementary of two antipodal points. So we prove thatthe infimum of the energy of globally defined unit smooth vector fieldsis $$\left( {\frac{{2k + 1}}{2} + \frac{k}{{2k - 1}}} \right){\text{ vol (}}\mathbb{S}^{2k + 1} ).$$