Using special coordinate transformations we introduce
subbundles and complementary subbundles of a vector bundle. The new
results come from the fact that these bundles are considered together.
In the former investigations as in Dragomir [5], Miron [7], Oproiu [8]
the subbundle of the vector bundle was defined. In the relations
between their tangent spaces the unit normal vectors of the subbundle
were involved. Here, they are substituted by the tangent vectors of the
complementary subbundle. The coordinates in the vector bundle $\xi=(E,\pi,M)$ are $(x^i,y^a)$ in
the subbundle $\tilde\xi$ are $(u^\alpha,v^A)$ and in the complementary
subbundle $\tilde{\tilde\xi}$ are $(\bar u^{\bar\alpha},\bar v^{\bar A})$.
We need six types of indices. With respect to the special coordinate
transformations (given by (1.1), (2.1) and (2.5)) the nonlinear
connections $N_i^a(x,y)$, $N_{\alpha}^A(u,v)$ and
$N_{\bar\alpha}^{\bar A}(\bar u,\bar v)$ are given. Using them, the
adapted bases $B=\{\delta_i,\partial_a\}$ and $\widehat B
=\{\delta_\alpha,\partial_A,\delta_{\bar\alpha},\partial_{\bar A}\}$
of $T(E)$ are constructed. The generalized connection
$\nabla: T(E)\otimes T(E)\to T(E)$ in the basis $B$ has $2^3$ types and
in the basis $\widehat B$ $4^3$ types of connection coefficients. The
relations between these coefficients are given. These formulae are very
general and have nice special cases. When the second fundamental forms of the subbundle and complementary
subbundle are equal to zero, i.e. when the so called induced nonlinear
connections $N_\alpha^A$ and $N_{\bar\alpha}^{\bar A}$ are used, then
these relations are simpler ((3.4)$'$--(3.7)$'$). In this case we
obtain that Miron's $d$-connection defined in $T(E)$ induces also
$d$-connection in the tangent space of the subbundle, in $T(\widetilde E)$.
