December  2013, 5(4): 433-444. doi: 10.3934/jgm.2013.5.433

A geometric approach to discrete connections on principal bundles

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, Argentina

Received  May 2013 Revised  October 2013 Published  December 2013

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.
Citation: Javier Fernández, Marcela Zuccalli. A geometric approach to discrete connections on principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 433-444. doi: 10.3934/jgm.2013.5.433
References:
[1]

H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems,, in Mathematics Unlimited-2001 and Beyond, (2001), 221.   Google Scholar

[2]

_______, Lagrangian reduction by stages,, Mem. Amer. Math. Soc., 152 (2001).  doi: 10.1090/memo/0722.  Google Scholar

[3]

J. Fernández, C. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems,, J. Geom. Mech., 2 (2010), 69.  doi: 10.3934/jgm.2010.2.69.  Google Scholar

[4]

_______, Lagrangian reduction of discrete mechanical systems by stages,, in preparation., ().   Google Scholar

[5]

V. Guillemin and A. Pollack, Differential Topology,, Prentice-Hall, (1974).   Google Scholar

[6]

R. Hermann, A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle,, Proc. Amer. Math. Soc., 11 (1960), 236.  doi: 10.1090/S0002-9939-1960-0112151-4.  Google Scholar

[7]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I,, Reprint of the 1963 original, (1963).   Google Scholar

[8]

M. Leok, Foundations of Computational Geometric Mechanics,, Ph.D. thesis, (2004).   Google Scholar

[9]

M. Leok, J. E. Marsden and A. Weinstein, A discrete theory of connections on principal bundles,, , (2005).   Google Scholar

[10]

J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids,, Nonlinearity, 19 (2006), 1313.  doi: 10.1088/0951-7715/19/6/006.  Google Scholar

[11]

J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numer., 10 (2001), 357.  doi: 10.1017/S096249290100006X.  Google Scholar

[12]

J. Milnor, Morse Theory,, Based on lecture notes by M. Spivak and R. Wells, (1963).   Google Scholar

show all references

References:
[1]

H. Cendra, J. E. Marsden and T. S. Ratiu, Geometric mechanics, Lagrangian reduction, and nonholonomic systems,, in Mathematics Unlimited-2001 and Beyond, (2001), 221.   Google Scholar

[2]

_______, Lagrangian reduction by stages,, Mem. Amer. Math. Soc., 152 (2001).  doi: 10.1090/memo/0722.  Google Scholar

[3]

J. Fernández, C. Tori and M. Zuccalli, Lagrangian reduction of nonholonomic discrete mechanical systems,, J. Geom. Mech., 2 (2010), 69.  doi: 10.3934/jgm.2010.2.69.  Google Scholar

[4]

_______, Lagrangian reduction of discrete mechanical systems by stages,, in preparation., ().   Google Scholar

[5]

V. Guillemin and A. Pollack, Differential Topology,, Prentice-Hall, (1974).   Google Scholar

[6]

R. Hermann, A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle,, Proc. Amer. Math. Soc., 11 (1960), 236.  doi: 10.1090/S0002-9939-1960-0112151-4.  Google Scholar

[7]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol. I,, Reprint of the 1963 original, (1963).   Google Scholar

[8]

M. Leok, Foundations of Computational Geometric Mechanics,, Ph.D. thesis, (2004).   Google Scholar

[9]

M. Leok, J. E. Marsden and A. Weinstein, A discrete theory of connections on principal bundles,, , (2005).   Google Scholar

[10]

J. C. Marrero, D. Martín de Diego and E. Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids,, Nonlinearity, 19 (2006), 1313.  doi: 10.1088/0951-7715/19/6/006.  Google Scholar

[11]

J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numer., 10 (2001), 357.  doi: 10.1017/S096249290100006X.  Google Scholar

[12]

J. Milnor, Morse Theory,, Based on lecture notes by M. Spivak and R. Wells, (1963).   Google Scholar

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