  The aim of this paper is to open the problem of construction of a nonlinear
connection $\Gamma=(M^{(i)}_{(\alpha)\beta}, N^{(i)}_{(\alpha)j})$ on the jet
bundle of first order $J^1(T,M)$, which to be canonically produced by a
Kronecker product vertical metrical d-tensor
$G^{(\alpha)(\beta)}_{(i)(j)}=h^{\alpha\beta}g_{ij}$, possibly provided by
multi-time dependent quadratic Lagrangians coming from various branches of
theoretical physics: bosonic strings theory, electrodynamics or elasticity.
