Harmonicity and minimality of vector fields and distributions on locally conformal Kähler and hyperkähler manifolds. (English) Zbl 1144.53051

Ornea, Liviu; Vanhecke, Lieven

Bull. Belg. Math. Soc. - Simon Stevin 12, No. 4, 543-555 (2005).

Authors’ abstract: We show that on any locally conformal Kähler manifold \((M,J, g)\) with parallel Lee form the unit anti-Lee vector field is harmonic and minimal and determines a harmonic map into the unit tangent bundle. Moreover, the canonical distribution locally generated by the Lee and anti-Lee vector fields is also harmonic and minimal when seen as a map from \((M, g)\) with values in the Grassmannian \(G^{or}_2(M)\) endowed with the Sasaki metric. As a particular case of locally conformal Kähler manifolds, we look at locally conformal hyper-Kähler manifolds and show that, if the Lee form is parallel (that is the case if the manifold is compact), the naturally associated three- and four-dimensional distributions are harmonic and minimal when regarded as maps with values in appropriate Grassmannians. As for locally conformal Kähler manifolds without parallel Lee form, we consider the Tricerri metric on an Inoue surface and prove that the unit Lee and anti-Lee vector fields are harmonic and minimal and the canonical distribution is critical for the energy functional, but when seen as a map with values in \(G^{or}_2(M)\) it is minimal, but not harmonic.

Reviewer: Witold Mozgawa (Lublin)

MSC:

53C20	Global Riemannian geometry, including pinching
58E15	Variational problems concerning extremal problems in several variables; Yang-Mills functionals
53C55	Global differential geometry of Hermitian and Kählerian manifolds

Keywords:

locally conformal Kähler manifold; Lee field; anti-Lee field; harmonic vector field; harmonic distribution; minimal vector field; minimal distribution; stability