Summary
For the bundle of linear frames LM of a manifold M, diffeomorphism invariance on the vertically adapted linear frame bundle L π (LM) and its infinitesimal counterpart, invariance under the natural representation of vector fields of M, are analyzed. Furthermore, the structure of the vector-field-invariant Lagrangians on L π (LM) is determined.
Dedicated in honour of Professor Marisa Menéndez.
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References
Baez, J.C.: Diffeomorphism-invariant generalized measures on the space of connections modulo gauge transformations. In: Proc. of the Conference on Quantum Topology, Manhattan, KS 1993, pp. 21–43. World Sci. Pub., River Edge (1994)
Bleecker, D.: Gauge Theory and Variational Principles. Addison-Wesley, Reading (1981)
Fulp, R.O., Lawson, J.K., Norris, L.K.: Generalized symplectic geometry as a covering theory for the Hamiltonian theories of classical particles and fields. J.Geom. Phys. 20, 195–206 (1996)
Gotay, M.: A multisymplectic framework for classical field theory and the calculus of variations. II. Space + time decomposition. In: Francaviglia, M. (ed.) Mechanics, Analysis and Geometry: 200 Years after Lagrange, pp. 203–235. North Holland, Amsterdam (1991)
Gotay, M.J., Isenberg, J.A., Marsden, J.E.: Momentum maps and classical fields, I: Covariant field theory. arXiv.org preprint Physics/9801019 (1997)
Gronwald, F.: Metric-affine gauge theory of gravity. I. Fundamental structure and field equations. Int. J. Mod. Phys. D 6, 263–303 (1997)
Kobayashi, S.: Transformation Groups in Differential Geometry. Springer, Heidelberg (1995)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 1. Wiley, New York (1963)
Lawson, J.K.: A frame bundle generalization of multisymplectic geometries. Rep. Math. Phys. 45, 183–205 (2000)
Lawson, J.K.: A frame bundle generalization of multisymplectic momentum mappings. Rep. Math. Phys. 53, 19–37 (2004)
de León, M., McLean, M., Norris, L.K., Rey-Roca, A.C., Salgado, M.: Geometric structures in field theory. arXiv.org preprint math-ph/0208036 (2002)
Luo, Y., Shao, M.X., Zhu, Z.Y.: Diffeomorphism invariance of geometric descriptions of Palatini and Ashtekar gravity. Phys. Lett. B 419, 37–39 (1998)
McLean, M., Norris, L.K.: Covariant field theory on frame bundles of fibered manifolds. J. Math. Phys. 41, 6808–6823 (2000)
Muñoz Masqué, J., Rosado, M.E.: Invariant variational problems on linear frame bundles. J. Phys. A 35, 2013–2036 (2002)
Norris, L.K.: Generalized symplectic geometry on the frame bundle of a manifold. In: Proc. Symp. Pure Math., vol. 54(2), pp. 435–465 (1993)
Norris, L.K.: n-symplectic algebra of observables in covariant Lagrangian field theory. J. Math. Phys. 42, 4827–4845 (2001)
Piguet, O.: Ghost equations and diffeomorphism-invariant theories. Class Quantum Grav. 17, 3799–3806 (2000)
Saunders, D.J.: The Geometry of Jet Bundles. Cambridge University Press, Cambridge (1989)
Sławianowski, J.J.: Field of linear frames as a fundamental self-interacting system. Rep. Math. Phys. 22, 323–371 (1985)
Sławianowski, J.J.: GL(n,R) as a candidate for fundamental fymmetry in Field Theory. Nuovo. Cimento. B 11(106), 645–668 (1991)
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Lawson, J.K., Rosado-María, M.E. (2011). Invariant Lagrangians on the Vertically Adapted Linear Frame Bundle. In: Pardo, L., Balakrishnan, N., Gil, M.Á. (eds) Modern Mathematical Tools and Techniques in Capturing Complexity. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20853-9_6
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