Geometry of fibered submanifolds

R. V. Vosylius

Itogi Nauki i Tekhniki. Seriya Problemy Geometrii. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 5 (1974), pp. 201-237

Résumé

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].