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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].

53B40Finsler spaces and generalizations (areal metrics)
53A45Vector and tensor analysis