In the framework of jet spaces endowed with a non-linear connection, the special curves of these spaces (h-paths, v-paths, stationary curves and geodesics) which extend the corresponding notions from Riemannian geometry are characterized. The main geometric objects and the paths are described and, in the case when the vertical metric is independent of fiber coordinates, the first two variations of energy and the extended Jacobi field equations are derived.
@article{bwmeta1.element.doi-10_2478_BF02475960,
author = {Vladimir Balan and Nicoleta Voicu},
title = {Distinguished geodesics and jacobi fields on first order jet spaces},
journal = {Open Mathematics},
volume = {2},
year = {2004},
pages = {516-526},
zbl = {1064.58002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02475960}
}
Vladimir Balan; Nicoleta Voicu. Distinguished geodesics and jacobi fields on first order jet spaces. Open Mathematics, Tome 2 (2004) pp. 516-526. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02475960/
[1] M. Anastasiei, I. Bucâtaru: “A notable submersion in higher order geometry”, BJGA Vol. 1, (1996), pp. 1–9. | Zbl 0903.53047
[2] M. Anastasiei, I. Bucâtaru: “Jacobi fields in generalized Lagrange spaces”, Rev. Roum. Math. Pures Appl., Vol. 42, (1997), pp. 9–10, 689–695. | Zbl 0928.53035
[3] V. Balan: “Lorentz-type equations in first-order jet spaces endowed with nonlinear connection” Proceedings of The First French-Romanian Colloquium of Numerical Physics, October 30–31, 2000, Bucharest, Romania, Geometry Balkan Press, (2002), pp. 105–114.
[4] V. Balan: “Notable curves in geometrized J 1 (T,M) jet framework”, BJGA, Vol. 8, (2003), pp. 1–10. | Zbl 1062.58005
[5] V. Balan: “Synge-Beil and Riemann-Jacobi jet structures with applications to physics”, Jour. of Math. and Math. Sci, Hindawi Publ. Corp., Vol. 27, (2003), pp. 1693–1702. | Zbl 1026.58002
[6] V. Balan: “Variational problems in the geometrized first-order jet framework”, Proc. Int. Workshop on Global Analysis, April 15–17, (2004), Ankara, Turkey, [to appear]. | Zbl 1117.58002
[7] V. Balan, N. Voicu: “Note on geodesics in distinguished jet framework”, Homagial volume in honor of Prof. K. Teleman, Univ. of Bucharest Editors, Bucharest 2004, [to appear]. | Zbl 1088.58501
[8] D. Bao, S.-S. Chern, Z. Shen: An Introduction to Riemann-Finsler Geometry, Springer-Verlag, 2000. | Zbl 0954.53001
[9] I. Comic: “Horizontal and vertical geodesics in the Riemannian space”. Mat. Vestnik, Vol. 42, (1990), pp. 3–4, 139–153.
[10] B.T.M. Hassan: “Sprays ans Jacobi fields in Finsler geometry”, An. Univ. Timişoara, Ser. Şt. Mat., Vol. XIX, (1981), pp. 129–139. | Zbl 0498.53051
[11] S. Kobayashi, K. Nomizu: Foundations of Differential Geometry I, II, Interscience Publishers, New York, 1963, 1969.
[12] J. Milnor: Morse Theory, Ann. of Math. Stud., Princeton Univ. Press, 1963.
[13] R. Miron: The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Acad. Publishers, 1994. | Zbl 0831.53001
[14] R. Miron, M. Anastasiei: The Geometry of Vector Bundles. Theory and Applications, Kluwer, Dordrecht, 1994.
[15] R. Miron, M. Tatoiu-Radivoiovici: “A Lagrangian theory of electromagnetism”, Rep. Math. Phys., Vol. 27, (1989), pp. 49–84. http://dx.doi.org/10.1016/0034-4877(89)90035-9 | Zbl 0709.53021
[16] M. Neagu: “The geometry of autonomous metrical multi-time Lagrange space of electrodynamics”, International Journal of Mathematics and Mathematical Sciences, Hindawi Publishing Corporation, (2001). http://xxx.lanl.gov/ abs/ math.DG/0010091, (2000).
[17] M. Neagu: “Generalized metrical multi-time Lagrangian geometry of physical fields”. http://xxx.lanl.gov/abs/math.DG/0011003, (2000).
[18] M. Neagu, C. Udrişte: “The geometry of metrical multi-time Lagrange spaces”, http://xxx.lanl.gov/abs/math.DG/0009071, (2000). | Zbl 1106.53019
[19] D.J. Saunders: The Geometry of Jet Bundles, Cambridge University Press, 1989. | Zbl 0665.58002
[20] Z. Shen: Differental Geometry of Sprays and Finsler Spaces, Kluwer Acad. Publishers, 2001.
[21] P.C. Stavrinos, H. Kawaguchi: “Deviation of geodesics in the gravitational field of Finslerian Space-Time”, Memoirs of Shonan Inst. of Technol., Vol. 27, (1993), pp. 35–40 | Zbl 0805.53022
[22] N. Voicu: “On metrical linear connections with torsion in Riemannian geometry”, An. Şt. Univ. “Al.I.Cuza”, Iaşi, [submitted]. | Zbl 1096.53011
[23] N. Voicu: “The Exponential Map on the Second Order Tangent Bundle”, Studia Mathematica, University of Cluj, [submitted]. | Zbl 1112.53047