Bulletin of the Belgian Mathematical Society - Simon Stevin

Harmonicity and minimality of vector fields and distributions on locally conformal Kähler and hyperkähler manifolds

Liviu Ornea and Lieven Vanhecke

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Abstract

We show that on any locally conformal Kähler (l.c.K.) 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 l.c.K. manifolds, we look at locally conformal hyperkähler manifolds and show that, if the Lee form is parallel (that is always 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 l.c.K. 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.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 12, Number 4 (2005), 543-555.

Dates
First available in Project Euclid: 5 December 2005

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1133793342

Digital Object Identifier
doi:10.36045/bbms/1133793342

Mathematical Reviews number (MathSciNet)
MR2205998

Zentralblatt MATH identifier
1144.53051

Subjects
Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 58E15: Application to extremal problems in several variables; Yang-Mills functionals [See also 81T13], etc. 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]

Keywords
locally conformal Kähler manifold Lee and anti-Lee field harmonic vector field and distribution minimal vector field and distribution, stability

Citation

Ornea, Liviu; Vanhecke, Lieven. Harmonicity and minimality of vector fields and distributions on locally conformal Kähler and hyperkähler manifolds. Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 4, 543--555. doi:10.36045/bbms/1133793342. https://projecteuclid.org/euclid.bbms/1133793342


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