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On deflection tensor field in Lagrange geometries. (English) Zbl 0843.53014
Antonelli, P. L. (ed.) et al., Lagrange and Finsler geometry: applications to physics and biology. Proceedings of a conference. Dordrecht: Kluwer Academic Publishers. Fundam. Theor. Phys. 76, 1-14 (1996).
In the geometries based on Lagrangians such as Finsler or Lagrange geometry, the so-called deflection tensor is strongly involved. Its significance for Finsler geometry was pointed out by the reviewer [Tensor, New Ser. 17, 217-226 (1966; Zbl 0139.39604); see also Foundations of Finsler geometry and special Finsler spaces (Kaiseisha Press, Japan) (1986; Zbl 0594.53001)]. When he formulated the well-known axioms determining the Cartan connection of a Finsler space, one of the axioms requires that the deflection tensor vanishes. Let M be a smooth manifold endowed with a generalzied Lagrange metric g ij (x,y), a nonlinear connection N i j (x,y) and two skew-symmetric tensor T j i k (x,y) and S j i k (x,y). There exists a unique d-connection DΓ=(N i j ,L j i k ,V j i k ) satisfying the following four conditions: DΓ is h- and v-metrical and h- and v-torsions of DΓ are T j i k and S j i k , respectively. The deflection tensor of DΓ is defined in terms of A ijk = ˙ k g ij /2, N i j and T j i k .
MSC:
53B40Finsler spaces and generalizations (areal metrics)
53A45Vector and tensor analysis