Solutions of Euler-Lagrange equations for self-interacting field of linear frames on product manifold of group space
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)].
scientific article; zbMATH DE number 2042732