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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.

Gauge theory of dislocational electromagnetic field in earthquake preparation zone

Nagahama H.

Acta Geophysica Polonica

2001

Vol. 49, nr 4

437-448

From the view point of the continuum theory of defects, we consider how the generated electromagnetic field can be related to the deformation field due to dislocations and disclinations. Based on the theory of connections in the higher-order space (Kawaguchi space), the Finsler deformation theory of ferromagnetic substances is introduced and the relation between this Finsler theory and the gauge theory of dislocations and disclinations is pointed out clearly. Moreover, the various preferred directions as internal degrees of freedom of each geomaterial point (e.g., polarization, spin moment, directors) of crystal materials (geomaterials) are discussed and the "exciting" state of these various preferred directions is regarded as the electromagnetic field radiation from an earthquake preparation zone.