Harmonicity of sections of sphere bundles

Abstract

We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold (M, ·, · ) equipped with the Sasaki metric and discuss the characterising condition for critical points. Furthermore, we provide a useful method for computing the tension field in some particular situations. Such a method is shown to be adequate for many tensor fields defined on manifolds M equipped with a G-structure compatible with ·, · . This leads to the construction of several new examples of differential forms which are harmonic sections or determine a harmonic map from (M, ·, · ) into its sphere bundle.

Harmonic almost contact structures via the intrinsic torsion

“…(11) . So if an almost contact metric structure is of type C 10 ⊕ C 11 such that ∇η • ϕ = ∇ ζ F , then dF = 0.…”

“…To be more precise, the component of dF in C a 10,11 is given by (dF ) (10,11) = −η ∧ 2(∇η) (10) • ϕ + ϕ(∇ ζ F ) . (11) .…”

“…In particular, an almost contact metric structure of type C 5 ⊕ · · · ⊕ C 10 ⊕ C 12 is harmonic if and only if ∇ * ∇ζ = ∇ζ 2 ζ, that is, the characteristic vector field ζ is a harmonic unit vector field (see [11,25] for this notion).…”

“…Therefore, (7) 2 + (2 + 1 2n−1 ) ξ σ1 (8) 2 + (1 + 1 2n−1 ) ξ σ1(9) 2 + (2 − 1 2n−1 ) ξ σ1(10) 2 + ξ σ1 (12) 2 + 1 2 2(∇η) (10) • ϕ − (∇ ζ F ) (11) 2 + e i (∇ ei F ) (4) − (∇ ζ η) (12) • ϕ 2 .…”

“…Since for n > 1 all the coefficients are again positive, we obtain (7) = ξ σ1 (8) = ξ σ1 (9) = ξ σ1(10) = ξ σ1 (11) = ξ σ1 (12) = 0.…”