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Solutions of Euler-Lagrange equations for self-interacting field of linear frames on product manifold of group space. (English) Zbl 1048.58011

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 [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)].

MSC:

58E30 Variational principles in infinite-dimensional spaces
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C80 Applications of global differential geometry to the sciences
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
22E46 Semisimple Lie groups and their representations
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
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References:

[1] Godlewski, P., Rep. Math. Phys., 35, 77 (1995)
[2] Godlewski, P., Rep. Math. Phys., 38, 29 (1996)
[3] Godlewski, P., Rep. Math. Phys., 40, 71 (1997)
[4] Kobayashi, S.; Nomizu, K., Foundations of Differential Geometry (1963), Interscience Publishers: Interscience Publishers New York London · Zbl 0119.37502
[5] Slawianowski, J. J., Rep. Math. Phys., 22, 323 (1985)
[6] Slawianowski, J. J., Rep. Math. Phys., 23, 177 (1986)
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