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A framed f(3,-1) structure on the tangent bundle of a lagrange space

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For a tangent bundle (TM,r,M), the kernel of the differential r* of the projection r defines the vertical subbundle VTM of the bundle (TTM, rTM , TM). A supplement HTM of it is called a horizontal subbundle or a nonlinear connection on M, (R. Miron and M. Anastasiei, [5]). The direct decomposition TTM = HTM VTM gives rise to a natural almost product structure P on the manifold TM. A general method to associate to P a framed f(3, -l)-structure of any corank is pointed out. When we endow M with a regular Lagrangian L and use as the nonlinear connection that canonically induced by L, a framed f(3, -l)-structure P2 of corank 2 naturally appears on TM. This reduces to that found by us in [3] when L = F2 , for F the fundamental function of a Finsler space Fn = (M,F). Then we show that on some conditions for L the restriction of P2 to the submanifold L = 1 of TO M is an almost paracontact structure on this submanifold. The conditions taken on L hold for the -Lagrangians introduced by P.L.Antonelli and D. Hrimiuc in [2] as well as for L = F2.
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Bibliogr. 5 poz.
  • Faculty of Sciences, University of Bacău, Calea Mărăşeşti 157, Bacău, Romania
  • [1] M. Anastasiei, P. L. Antonelli, The Differential Geometry of Lagrangian which Generate Sprays, in vol. Lagrange and Finsler Geometry. Applications to Physics and Biology, edited by P.L. Antonelli and R. Miron. Kluwer Academic Publishers, FTPH 76, 1996, 15-34.
  • [2] P. L. Antonelli, D. Hrimiuc, A New Class of Spray-Generating Lagrangians, the same volume, 81-92.
  • [3] M. Gîrţu, An almost paracontact structure on the indicatrix bundle of a Finsler space, BJGA, 7(2), 2002, 43-48.
  • [4] I. Mihai, R. Roşca, L. Verstraelen, Some aspects of the differential geometry of vector fields. PADGE, Katholieke Universitiet Leuven, vol. 2, 1996.
  • [5] R. Miron, M. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications. Kluwer Academic Publishers, FTPH 59, 1994.
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