Fabiano G. B. Brito

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

AbstractWe construct a one-parameter family of unit smooth vector fieldsglobally defined on the sphere

ENERGY OF GLOBAL FRAMES

\mathbb{S}

The geometry of closed hypersurfaces

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

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\left( {\frac{{2k + 1}}{2} + \frac{k}{{2k - 1}}} \right){\text{ vol (}}\mathbb{S}^{2k + 1} ).

On the volume of unit vector fields on spaces of constant sectional curvature

An integral formula for symmetric functions of curvature ofdistributions on closed constant nonnegative sectional curvature spacesis proved. The distributions under consideration are orthogonal to atotally geodesic foliation and the main theorem extends a previousresult concerning the total curvature of codimension-one foliations.