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A two point calibration on an Sp(1) bundle over the three-sphere. (English) Zbl 1052.53041

Let \(\mathbb H\) be the algebra of quaternions, \(S^3:=\{q\in \mathbb{H}\mid \| q\| =1\}\) the unit sphere endowed with the canonical round metric induced by \(\mathbb{R}^4\), viewed as a Lie group with the bi-invariant metric of constant sectional curvature one, \((\mathbb{H},\rho)\) the finite dimensional orthogonal real representation of \(S^3\) given by \(\rho(q)u:=u\bar{q}\) (quaternionic product), and \(\pi:E=S^3\times\mathbb{H}\to S^3\) the trivial vector bundle. For \(v\in\mathbb{H}\), let \(L_v,R_v:S^3\to E\), \(L_v(p):=(p,v)\), \(R_v(p):=(p,\rho(p^{-1})v)\), \(p\in S^3\). The sections \(L_v\) (resp. \(R_v\)) are called left (resp. right) invariant. There exists a unique connection \(\nabla\) on \((E,\pi, S^3)\) such that \(\nabla_ZL_v=L_{\frac{1}{2}d\rho(Z)v}\) for all \(v\in \mathbb{H}\) and all left invariant vector fields \(Z\) on \(S^3\) and consider the canonical Sasaki metric on \(E\) induced by \(\nabla\).
The author proves that the left and right invariant unit sections have minimum volume among all unit sections of \(\pi:E\to S^3\) in their homology classes, and that they are unique by this property. The proof uses a single calibration which calibrates both the left and right invariant sections. The author’s result is the analogue of a theorem by H. Gluck and W. Ziller [Comment. Math. Helv. 61, 177–192 (1986; Zbl 0605.53022)], stating that Hopf vector fields on \(S^3\) have minimum volume among all unit vector fields. Indeed, Hopf vector fields are precisely those with unit length which are left or right invariant, and \(TS^3\) is a trivial vector bundle with a connection induced by the adjoint representation.

MSC:

53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C38 Calibrations and calibrated geometries
53C43 Differential geometric aspects of harmonic maps

Citations:

Zbl 0605.53022
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