Harmonic maps on contact metric manifolds
type: Item
instance of: scholarly article
title: Harmonic maps on contact metric manifolds
zbMATH Open document ID: 0851.53018
author: Anna Maria Pastore
author: S. Ianuş
published in: Annales Mathématiques Blaise Pascal
publication date: 1996-11-10T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/79139
review text: The authors initiate a study of harmonic maps between almost contact metric manifolds. Let \(M(\varphi, \xi, \eta, g)\) and \(M'(\varphi', \xi', \eta', g')\) be two almost contact metric manifolds [cf. \textit{D. E. Blair}, Contact manifolds in Riemannian geometry, Lect. Notes Math. 509, Springer (1976; Zbl 0319.53026)] and let \(f : M \to M'\) be a smooth map. \(f\) is said to be \(+\varphi\)-holomorphic if \(f_* \circ \varphi = \varphi \circ f_*\), and \(-\varphi\)-holomorphic if \(f_* \circ \varphi = - \varphi \circ f_*\). By an analogy to the Kählerian case, \(f\) (here, it is sufficient for \(M'\) to be Riemannian) is called \(\varphi\)-pluriharmonic if its second fundamental form \(\alpha\) satisfies the condition (*) \(\alpha (X,Y) + \alpha(\varphi X, \varphi Y) = 0\) for any vector fields \(X\), \(Y\) on \(M\), and \(f\) is called \(D\)-pluriharmonic if (*) holds for any \(X,Y \in D = \text{ker }\eta\). It is proved that if \(M\) and \(M'\) are contact metric, then any \(\pm \varphi\)-holomorphic map \(f\) is harmonic. Any \(\varphi\)-pluriharmonic map \(f\) is harmonic. When \(M\), \(M'\) are Sasakian manifolds, then a \(\pm \varphi\)-pluriharmonic map \(f\) is \(D\)-pluriharmonic; and \(f\) is \(\varphi\)-pluriharmonic if and only if it is isometric. Harmonic maps on globally \(\varphi\)-symmetric Sasakian manifolds of constant \(\varphi\)-sectional curvature are also studied; furthermore, certain examples on principal circle bundles are discussed.
reviewed by: Zbigniew Olszak
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 871700
zbMATH Keywords: almost contact metric manifold
zbMATH Keywords: \(\varphi\)-holomorphic map
zbMATH Keywords: harmonic map
zbMATH Keywords: \(\varphi\)-harmonic map
MaRDI profile type: Publication
OpenAlex ID: W2058950227
cites work: Geometry of manifolds with structural group \(\mathcal U(n)\times\mathcal O(s)\)
cites work: Contact manifolds in Riemannian geometry
cites work: On contact manifolds
cites work: Q3999385
cites work: Q3662316
cites work: Another Report on Harmonic Maps
cites work: Harmonic Maps and Minimal Immersions with Symmetries (AM-130)
cites work: Globally framed f-manifolds
cites work: Some global properties of contact structures
cites work: Q4354733
cites work: Q3273675
cites work: Q5589060
cites work: Q4046729
cites work: Q5618076
cites work: Geometry seminar ''Luigi Bianchi'' II - 1984. Lecture given at the Scuola Normale Superiore, Pisa, Italy, 1984
cites work: On fiberings of almost contact manifolds
cites work: On Pluriharmonicity of Stable Harmonic Maps
cites work: On contact metric manifolds
cites work: Q3700395
cites work: Sasakian \(\varphi\)-symmetric spaces
cites work: Some transformations on manifolds with almost contact and contact metric structures
DOI: 10.5802/AMBP.46
Recommended article: Q4718367
Recommended article: Q4595921
Recommended article: Harmonicity on maps between almost contact metric manifolds
Recommended article: Q4650721
Recommended article: Generalized harmonic maps on normal almost contact manifolds
Recommended article: Almost contact metric manifolds whose Reeb vector field is a harmonic section
Recommended article: Q5461092
Recommended article: Harmonic almost contact structures
Recommended article: Some examples of pluriharmonic and \(\varphi\)-pluriharmonic maps
Recommended article: Q4685245
description: scientific article; zbMATH DE number 871700