Harmonic and minimal unit vector fields on the symmetric spaces \(G_2\) and \(G_2/\mathrm{SO}(4)\). (English) Zbl 1279.53060

Verhóczki, László

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 51, No. 1, 101-109 (2012).

Given a compact orientable Riemannian manifold \(N\) which admits smooth unit vector fields, one can define the energy functional \(\text{E}(\xi)\) and the volume functional \(\text{Vol}(\xi)\) for a smooth unit vector field \(\xi\) on \(N\). A unit vector field \(\xi\) is called harmonic if it is a critical point of the energy functional \(\text{E}\) and is called a minimal unit vector field if it is a critical point of the volume functional \(\text{Vol}\). There exists a criterion for the harmonic unit vector field (or minimal unit vector field), e.g., stated on page 103 in this paper. Moreover, a unit vector field on a manifold which is not compact or not orientable is called harmonic or minimal if it satisfies the respective criterion.
The main result of the paper shows that the radial unit vector field on the Riemannian symmetric spaces \(G_2\) (or \(G_2/\mathrm{SO}(4)\)) associated to isometric actions of \(\mathrm{SU}(3)\times \mathrm{SU}(3)\) (or \(\mathrm{SU}(3)\)) is harmonic and minimal. Since the singular orbits of these actions are not reflective, a previous result about harmonicity and minimality in [J. Berndt et al., Ill. J. Math. 47, No. 4, 1273–1286 (2003; Zbl 1045.53036)] does not apply in this case. Instead the author proves the conclusion by checking the criteria through detailed calculations with convenient coordinates.

Reviewer: Jun Yu (Princeton)

MSC:

53C43	Differential geometric aspects of harmonic maps
53C35	Differential geometry of symmetric spaces
57S15	Compact Lie groups of differentiable transformations
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

Riemannian symmetric space; harmonic unit vector field; minimal unit vector field; isometric action

Citations:

Zbl 1045.53036

References:

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