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

${T}_{j}{}^{i}{}_{k}$.