Hermitian harmonic maps
type: Item
instance of: scholarly article
title: Hermitian harmonic maps
zbMATH Open document ID: 0965.58020
author: Eric Loubeau
published in: Beiträge zur Algebra und Geometrie
publication date: 2001-01-09T00:00:00Z
full work available at URL: https://eudml.org/doc/226104
full work available at URL: 1.html
review text: Let \((X,\gamma)\) be a Hermitian manifold of complex dimension \(m\), equipped with the Hermitian metric \(\gamma\) and with local complex coordinates \((z^\alpha)_{\alpha= 1,\dots, m}\), and \((N,g)\) a Riemannian manifold of real dimension \(n\), with the Riemannian metric \(g\) and local real coordinates \((x^i)_{i= 1,\dots, n}\). Following \textit{J. Jost} and \textit{S. T. Yau} [Proc. Symp. Pure Math. 54, Part I, 245-280, Am. Math. Soc. (1993; Zbl 0806.58012)], a smooth map \(\phi: X\to N\) is said to be a Hermitian harmonic map if \(\text{trace}_\gamma(\nabla^N_{d\phi} d\phi)= 0\), where \(\nabla^N\) is the Lévi-Cività connection of \((N,g)\). As a parallel with (real) harmonic morphisms, in this paper the author introduces the notion of Hermitian harmonic morphisms as maps pulling local Hermitian harmonic functions back to Hermitian harmonic functions when \((N,g)\) is a Hermitian manifold, and characterizes them as Hermitian harmonic maps satisfying a geometrical condition called the biholomorphic horizontal weak conformality. The author then considers the case of maps mixing (classical) harmonicity and Hermitian harmonicity, i.e., pulling local harmonic functions back to Hermitian harmonic functions, and shows them to be precisely \(d''\)-horizontally weakly conformal Hermitian harmonic maps. Finally, the author illustrates the different notions discussed with a few examples.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 1549683
zbMATH Keywords: Hermitian harmonic map
zbMATH Keywords: Hermitian harmonic morphisms
zbMATH Keywords: Hermitian harmonic functions
zbMATH Keywords: biholomorphic horizontal weak conformality
MaRDI profile type: Publication
Recommended article: Pseudo Harmonic Morphisms
Recommended article: Harmonic morphisms between almost Hermitian manifolds
Recommended article: Q5017571
Recommended article: Hermitian harmonic maps from complete Hermitian manifolds to complete Riemannian manifolds
Recommended article: A boundary value problem for Hermitian harmonic maps and applications
Recommended article: PSEUDO HARMONIC MORPHISMS ON RIEMANNIAN POLYHEDRA
Recommended article: Q5500039
Recommended article: Harmonicity of horizontally conformal maps and spectrum of the Laplacian
Recommended article: On $p$-harmonic morphisms and conformally flat spaces
Recommended article: On pseudo-harmonic maps in conformal geometry
description: scientific article; zbMATH DE number 1549683