R. Miron in [8] and R. Miron and Gh. Atanasiu in [5-7] studied the geometry of $Osc^kM$. Among many various problems which was solved, they introduced the adapted basis, the $d$-connection and gave its curvature theory. Here the attention on $E=Osc^3M$ will be restricted. The coefficients of the nonlinear connection, $M^{(1)},M^{(2)}$ and $M^{(3)}$ are determined in such a way that $T_{V_3}$, is orthogonal to $T_{V_0}$, $T_{V_1}$ $T_{V_2}$ with respect to the arbitrary but fixed nondegenerative metric $G$. The adapted basis constructed with such connections is unique.