Geometry of fibered submanifolds

Let $\theta^k$ be a differential equation given in the fibre bundle $(E,M,\rho)$. We shall regard the fibre bundle $(L,N,\pi)$ and the immersion of fibre bundles
$$ \nu\colon L\to E. $$
Let the map
$$ f\colon N\to M $$
be the immersion of manifolds, satisfayng the condition
$$ f\circ\pi=\rho\circ\nu. $$
$N_f$ shall design the immersed submanifold.
Simultaneously with the differential equation $\theta^k$ differential equation $\theta^{k+h}$ inductively defined with the help of equations
$$ \theta^{k+h}=\rho(\theta^{k+h-1}) $$
are considered, where $\rho(\theta^{k+h-1})$ designs the prolongation of the equation $\theta^{k+h-1}$.
If the immersion of the fibre bundles
$$ \nu\colon L\to E $$
is given, in the fibre bundle $J^lL$ we obtain the subset
$$ \omega_\nu^l=\nu^l(I(\theta^l))|_{E_\nu^l} $$
where $E_\nu^l$ designs the restriction $J^lE|N_f$ and
$$ \nu^l\colon E_\nu^l\to J^lL $$
is the mapping induced by immersion $\nu$.
The immersion
$$ \nu_l\colon L\to E $$
is called the $h$-deformation of immersion in respect of the differential equation $\theta^k$, if the following condition
$$ \omega_{\nu_1}^{k+h}=\omega_{\nu_2}^{k+h} $$
is satisfied.
The $h$-deformation is a relation of equivalence in the set of fibred submanifolds.
The case of the first order differential equation in involution is being studied.
It is shown that the possibility of 0-deformation is necessary and sufficient condition for the possibility of $h$-deformation of two regular fibred submanifolds.
Conclusively some applications for the geometry of homogeneous space submanifolds are regarded. A well-known theorem on the finiteness of the number of independent differential invariants of homogeneous space submanifolds is shown [3].