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

Gîrţu M.

Demonstratio Mathematica

2004

Vol. 37, nr 4

955--961

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.