JOURNAL BROWSE
Search
Advanced SearchSearch Tips
FIBRE BUNDLE MAPS AND COMPLETE SPRAYS IN FINSLERIAN SETTING
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
FIBRE BUNDLE MAPS AND COMPLETE SPRAYS IN FINSLERIAN SETTING
Crasmareanu, Mircea;
  PDF(new window)
 Abstract
A theorem of Robert Blumenthal is used here in order to obtain a sufficient condition for a function between two Finsler manifolds to be a fibre bundle map. Our study is connected with two possible constructions: 1) a Finslerian generalization of usually Kaluza-Klein theories which use Riemannian metrics, the well-known particular case of Finsler metrics, 2) a Finslerian version of reduction process from geometric mechanics. Due to a condition in the Blumenthal's result the completeness of Euler-Lagrange vector fields of Finslerian type is discussed in detail and two situations yielding completeness are given: one concerning the energy and a second related to Finslerian fundamental function. The connection of our last framework, namely a regular Lagrangian having the energy as a proper (in topological sense) function, with the celebrated Recurrence Theorem is pointed out.
 Keywords
fibre bundle;fibration;regular Lagrangian;energy;Euler-Lagrange equations;semispray;spray;Finsler fundamental function;complete vector field;proper function;
 Language
English
 Cited by
 References
1.
J. C. Alvarez Paiva and C. E. Duran, Isometric submersions of Finsler manifolds, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2409–2417

2.
M. Anastasiei, Certain generalizations of Finsler metrics, Finsler geometry (Seattle, WA, 1995), 161–169, Contemp. Math., 196, Amer. Math. Soc., Providence, RI, 1996

3.
M. Anastasiei and P. L. Antonelli, The differential geometry of Lagrangians which generate sprays, Lagrange and Finsler geometry, 15–34, Fund. Theories Phys., 76, Kluwer Acad. Publ., Dordrecht, 1996

4.
P. L. Antonelli, R. S. Ingarden, and M. Matsumoto, The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Fundamental Theories of Physics, 58. Kluwer Academic Publishers Group, Dordrecht, 1993

5.
P. L. Antonelli and D. Hrimiuc, A new class of spray-generating Lagrangians, Lagrange and Finsler geometry, 81–92, Fund. Theories Phys., 76, Kluwer Acad. Publ., Dordrecht, 1996

6.
D. Bao, S.-S. Chern, and Z. Shen, Preface for 'Finsler geometry over the reals', Finsler geometry (Seattle, WA, 1995), 3–13, Contemp. Math., 196, Amer. Math. Soc., Providence, RI, 1996

7.
D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics, 200, Springer-Verlag, New York, 2000

8.
R. G. Beil, Finsler and Kaluza-Klein gauge theories, Internat. J. Theoret. Phys. 32 (1993), no. 6, 1021–1031 crossref(new window)

9.
R. G. Beil, Comparison of unified field theories, Tensor (N.S.) 56 (1995), no. 2, 175–183

10.
A. M. Bloch, Nonholonomic Mechanics and Control, With the collaboration of J. Baillieul, P. Crouch and J. Marsden. With scientific input from P. S. Krishnaprasad, R. M. Murray and D. Zenkov. Interdisciplinary Applied Mathematics, 24. Systems and Control. Springer-Verlag, New York, 2003

11.
R. A. Blumenthal, Sprays, fibre spaces and product decompositions, Differential geometry (Santiago de Compostela, 1984), 156–161, Res. Notes in Math., 131, Pitman, Boston, MA, 1985

12.
R. L. Bryant, An introduction to Lie groups and symplectic geometry, Geometry and quantum field theory (Park City, UT, 1991), 5–181, IAS/Park City Math. Ser., 1, Amer. Math. Soc., Providence, RI, 1995

13.
M. Crasmareanu, Nonlinear connections and semisprays on tangent manifolds, Novi Sad J. Math. 33 (2003), no. 2, 11–22

14.
R. H. Cushman and L. M. Bates, Global aspects of classical integrable systems, Birkhauser Verlag, Basel, 1997

15.
D. G. Ebin, Completeness of Hamiltonian vector fields, Proc. Amer. Math. Soc. 26 (1970), 632–634 crossref(new window)

16.
W. B. Gordon, On the completeness of Hamiltonian vector fields, Proc. Amer. Math. Soc. 26 (1970), 329–331 crossref(new window)

17.
M. Hashiguchi and Y. Ichijyo, Randers spaces with rectilinear geodesics, Rep. Fac. Sci. Kagoshima Univ. No. 13 (1980), 33–40

18.
R. Hermann, A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc. Amer. Math. Soc. 11 (1960), 236–242 crossref(new window)

19.
R. Hermann, Yang-Mills, Kaluza-Klein, and the Einstein Program, With contributions by Frank Estabrook and Hugo Wahlquist. Interdisciplinary Mathematics, XIX. Math. Sci. Press, Brookline, Mass., 1978

20.
T. Kaluza, Zum Unitatsproblem der Physik, Sitzber. Preuss. Akad. Wiss. Kl. 2 (1921), 966–970

21.
Th. Kaluza, On the problem of unity in physics, Unified field theories of more than 4 dimensions (Erice, 1982), 427–433, World Sci. Publishing, Singapore, 1983

22.
Th. Kaluza, On the unification problem in physics, Translated from the German by T. Muta. An introduction to Kaluza-Klein theories (Deep River, Ont., 1983), 1–9, World Sci. Publishing, Singapore, 1984

23.
O. Klein, Quantentheorie und funf-dimensionale Relativitatstheorie, Z. Phys. 37 (1929), 89–901

24.
O. Klein, The Oskar Klein memorial lectures. Vol. 1, Lectures by C. N. Yang and S. Weinberg with translated reprints by O. Klein. Edited by Gosta Ekspong. World Scientific Publishing Co., Inc., River Edge, NJ, 1991

25.
J. E. Marsden, R. Montgomery, and T. Ratiu, Reduction, symmetry, and phases in mechanics, Mem. Amer. Math. Soc. 88 (1990), no. 436, iv+110 pp

26.
G. Meigniez, Submersions, fibrations and bundles, Trans. Amer. Math. Soc. 354 (2002), no. 9, 3771–3787

27.
R. Miron and M. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications, Fundamental Theories of Physics, 59. Kluwer Academic Publishers Group, Dordrecht, 1994

28.
R. Miron, The Geometry of Higher-Order Lagrange Spaces, Applications to mechanics and physics. Fundamental Theories of Physics, 82. Kluwer Academic Publishers Group, Dordrecht, 1997

29.
R. Miron, The Geometry of Higher-Order Finsler Spaces, With a foreword by Ruggero Maria Santilli. Hadronic Press Monographs in Mathematics. Hadronic Press, Inc., Palm Harbor, FL, 1998

30.
F. Moalla, Sur quelques theoremes globaux en geometrie finslerienne, Ann. Mat. Pura Appl. (4) 73 (1966), 319–365 crossref(new window)

31.
P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, 171. Springer-Verlag, New York, 1998

32.
J. Szilasi, A Setting for Spray and Finsler Geometry, Handbook of Finsler geometry. Vol. 1, 2, 1183–1426, Kluwer Acad. Publ., Dordrecht, 2003

33.
C. Udriste, Completeness of Finsler manifolds, Publ. Math. Debrecen 42 (1993), no. 1-2, 45–50

34.
J. G. Vargas and D. G. Torr, Of Finsler fiber bundles and the evolution of the calculus, Proceedings of the 5th Conference of Balkan Society of Geometers, 183–191, BSG Proc., 13, Geom. Balkan Press, Bucharest, 2006

35.
A. Weinstein and J. Marsden, A comparision theorem for Hamiltonian vector fields, Proc. Amer. Math. Soc. 26 (1970), 629–631 crossref(new window)