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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 \(D\Gamma= (N^i{}_j, L_j{}^i {}_k, V_j{}^i {}_k)\) satisfying the following four conditions: \(D\Gamma\) is \(h\)- and \(v\)-metrical and \(h\)- and \(v\)-torsions of \(D\Gamma\) are \(T_j{}^i{}_k\) and \(S_j{}^i{}_k\), respectively. The deflection tensor of \(D\Gamma\) is defined in terms of \(A_{ijk}= \dot \partial_k g_{ij} /2\), \(N^i{}_j\) and \(T_j{}^i{}_k\).
For the entire collection see [Zbl 0833.00033].

MSC:

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53A45 Differential geometric aspects in vector and tensor analysis
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