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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 ${g}_{ij}\left(x,y\right)$, a nonlinear connection ${N}^{i}{}_{j}\left(x,y\right)$ and two skew-symmetric tensor ${T}_{j}{}^{i}{}_{k}\left(x,y\right)$ and ${S}_{j}{}^{i}{}_{k}\left(x,y\right)$. There exists a unique $d$-connection $D{\Gamma }=\left({N}^{i}{}_{j},{L}_{j}{}^{i}{}_{k},{V}_{j}{}^{i}{}_{k}\right)$ 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 ${A}_{ijk}={\stackrel{˙}{\partial }}_{k}{g}_{ij}/2$, ${N}^{i}{}_{j}$ and ${T}_{j}{}^{i}{}_{k}$.
##### MSC:
 53B40 Finsler spaces and generalizations (areal metrics) 53A45 Vector and tensor analysis