Solutions of Euler-Lagrange equations for self-interacting field of linear frames on product manifold of group space
type: Item
instance of: scholarly article
title: Solutions of Euler-Lagrange equations for self-interacting field of linear frames on product manifold of group space
zbMATH Open document ID: 1048.58011
DOI: 10.1016/S0034-4877(03)90005-X
author: Piotr Godlewski
published in: Reports on Mathematical Physics
publication date: 2004-02-15T00:00:00Z
review text: With an anholonomic basis \((E_A)\) on a manifold \(N\) are associated: the structure functions \(\gamma^A_{BC}\), a pseudo-Riemannian metric \(\gamma_{AB}= \gamma^C_{AD}\gamma^D_{BC}\), a \((2,1)\)-tensor density and the Euler-Lagrange equations \[ H^{BC}_A= \sqrt{|\text{det}[\gamma_{AB}]} (\gamma^{BD}\gamma^C_{DA}- \gamma^{CD}\gamma^B_{DA}),\quad E_C H^{BC}_A+ \gamma^D_{DC} H^{BC}_A= 0. \] The situation is studied on \(N= M\times G\), \(G\) is a semisimple Lie group acting freely and transitively on a manifold \(M\), with \(\dim G= \dim M\). Two bases \((X_a,\Xi_a)\) and \((\text{Ad}^b_a X_b,\Xi_a)\), where \((\Xi_a)\) is a left-invariant basis for the Lie algebra of \(G\), \((X_a)\) the corresponding operators of \(G\) on \(M\) and \(\gamma^A_{BC}\) are determined by the structure constants \(C^a_{bc}\) of \(G\) are constructed as solutions of the Euler-Lagrange equations. See also [\textit{P. Godlewski}, ibid. 35, 77--99 (1995; Zbl 0891.53071), ibid. 38, 29--44 (1996; Zbl 0886.58117), ibid. 40, 71--90 (1997; Zbl 0891.53057)].
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 53C07
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 70S05
zbMATH DE Number: 2042732
zbMATH Keywords: Euler-Lagrange equations
zbMATH Keywords: anholonomic basis
zbMATH Keywords: representation of semisimple Lie group
reviewed by: Maido Rahula
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2042732