On the biharmonicity of product maps
type: Item
instance of: scholarly article
title: On the biharmonicity of product maps
zbMATH Open document ID: 1124.53026
DOI: 10.1155/IJMMS/2006/73094
author: L. Todjihounde
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2007-06-14T00:00:00Z
full work available at URL: https://eudml.org/doc/53558
review text: A harmonic (resp., biharmonic) map between Riemannian manifolds is a critical point of the energy (resp., bienergy) functional [\textit{J. Eells jun.} and \textit{J. H. Sampson}, Am. J. Math. 86, 109--160 (1964; Zbl 0122.40102)]. Let \(\phi_i :(M_i,g_i) \to (N_i,h_i)\), i=1,2, be two harmonic maps between Riemannian manifolds and let \(\alpha\), \(\beta\) be two positive functions on \(M=M_1 \times M_2\) and \(N=N_1 \times N_2\), respectively. Let \(G_{\alpha} = \pi_1*g + (\alpha \circ \pi_1)^2 \pi_2*g_2\) and similarly \(H_{\beta}\) be two warped product metrics on \(M\) and \(N\), respectively. Some necessary an sufficient conditions are given that the map \(\phi_1 \times \phi_2 : (M,G_{\alpha}) \to (N,H_{\beta})\) defined by \(\phi_1 \times \phi_2 (x,y) = (\phi_1 (x),\phi_2 (y))\) is harmonic. By [\textit{B. Chen, T. Nagano}, J. Math. Soc. Japan 36, 295--313 (1984; Zbl 0543.58015)], a Riemannian metric \(\sigma_2\) is harmonic (resp., biharmonic) w.r.t. a metric \(\sigma_1\) on a manifold \(S\) if the identity map \(i:(S,\sigma_1) \to (S,\sigma_2)\) is harmonic (resp., biharmonic). Some necessary and sufficient conditions are given here such that a warped product metric \(G_\alpha\) is harmonic (resp., nonharmonic biharmonic) w.r.t. a metric \(G\) on \(M\). Some examples are constructed.
reviewed by: Cornelia-Livia Bejan
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 5164247
zbMATH Keywords: product maps
zbMATH Keywords: biharmonic maps
zbMATH Keywords: harmonic maps
MaRDI profile type: Publication
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OpenAlex ID: W2055529107
description: scientific article; zbMATH DE number 5164247