A geometric  approach to discrete connections on principal bundles

Javier Fernández; Marcela Zuccalli

doi:10.3934/jgm.2013.5.433

Article Contents

2013, Volume 5, Issue 4: 433-444. Doi: 10.3934/jgm.2013.5.433

This issue Previous Article Regular discretizations in optimal control theory Next Article Tulczyjew triples: From statics to field theory

A geometric approach to discrete connections on principal bundles

Javier Fernández^1, and
Marcela Zuccalli^2,

1.
Instituto Balseiro, Universidad Nacional de Cuyo – C.N.E.A., Av. Bustillo 9500, San Carlos de Bariloche, R8402AGP
2.
Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, 50 y 115, La Plata, Buenos Aires, 1900

Received: April 30, 2013

Revised: September 30, 2013

Published: November 2013

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Abstract

This work revisits, from a geometric perspective, the notion of discrete connection on a principal bundle, introduced by M. Leok, J. Marsden and A. Weinstein. It provides precise definitions of discrete connection, discrete connection form and discrete horizontal lift and studies some of their basic properties and relationships. An existence result for discrete connections on principal bundles equipped with appropriate Riemannian metrics is proved.

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Mathematics Subject Classification: Primary: 53B15, 53C05; Secondary: 37J15, 70G45.

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References

[1]	H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems, in Mathematics Unlimited-2001 and Beyond, Springer, Berlin, 2001, 221-273.
[2]	_______, Lagrangian reduction by stages, Mem. Amer. Math. Soc., 152 (2001).doi: 10.1090/memo/0722.
[3]	J. Fernández, C. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems, J. Geom. Mech., 2 (2010), 69-111,doi: 10.3934/jgm.2010.2.69.
[4]	_______, Lagrangian reduction of discrete mechanical systems by stages, in preparation.
[5]	V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974.
[6]	R. Hermann, A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc. Amer. Math. Soc., 11 (1960), 236-242.doi: 10.1090/S0002-9939-1960-0112151-4.
[7]	S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I, Reprint of the 1963 original, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996.
[8]	M. Leok, Foundations of Computational Geometric Mechanics, Ph.D. thesis, California Institute of Technology, 2004.
[9]	M. Leok, J. E. Marsden and A. Weinstein, A discrete theory of connections on principal bundles, arXiv:math/0508338, 2005.
[10]	J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity, 19 (2006), 1313-1348.doi: 10.1088/0951-7715/19/6/006.
[11]	J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.doi: 10.1017/S096249290100006X.
[12]	J. Milnor, Morse Theory, Based on lecture notes by M. Spivak and R. Wells, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963.