R. V. Vosylius, “Geometry of fibered submanifolds”, Tr. Geom. Sem., 5, VINITI, Moscow, 1974, 201

Abstract: Let

θ^{k}

$\theta^k$ be a differential equation given in the fibre bundle

(E, M, ρ)

$(E,M,\rho)$ . We shall regard the fibre bundle

(L, N, π)

$(L,N,\pi)$ and the immersion of fibre bundles

ν : L \to E .

$\nu\colon L\to E.$
Let the map

f : N \to M

$f\colon N\to M$
be the immersion of manifolds, satisfayng the condition

f \circ π = ρ \circ ν .

$f\circ\pi=\rho\circ\nu.$

N_{f}

$N_f$ shall design the immersed submanifold.
Simultaneously with the differential equation

θ^{k}

$\theta^k$ differential equation

θ^{k + h}

$\theta^{k+h}$ inductively defined with the help of equations

θ^{k + h} = ρ (θ^{k + h - 1})

$\theta^{k+h}=\rho(\theta^{k+h-1})$
are considered, where

ρ (θ^{k + h - 1})

$\rho(\theta^{k+h-1})$ designs the prolongation of the equation

θ^{k + h - 1}

$\theta^{k+h-1}$ .
If the immersion of the fibre bundles

ν : L \to E

$\nu\colon L\to E$
is given, in the fibre bundle

J^{l} L

$J^lL$ we obtain the subset

ω_{ν}^{l} = ν^{l} (I (θ^{l})) |_{E_{ν}^{l}}

$\omega_\nu^l=\nu^l(I(\theta^l))|_{E_\nu^l}$
where

E_{ν}^{l}

$E_\nu^l$ designs the restriction

J^{l} E | N_{f}

$J^lE|N_f$ and

ν^{l} : E_{ν}^{l} \to J^{l} L

$\nu^l\colon E_\nu^l\to J^lL$
is the mapping induced by immersion

ν

$\nu$ .
The immersion

ν_{l} : L \to E

$\nu_l\colon L\to E$
is called the

h

$h$ -deformation of immersion in respect of the differential equation

θ^{k}

$\theta^k$ , if the following condition

ω_{ν_{1}}^{k + h} = ω_{ν_{2}}^{k + h}

$\omega_{\nu_1}^{k+h}=\omega_{\nu_2}^{k+h}$
is satisfied.
The

h

$h$ -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

$h$ -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].

Citation in format AMSBIB

\Bibitem{Vos74}

\by R.~V.~Vosylius

\paper Geometry of fibered submanifolds

\serial Tr. Geom. Sem.

\yr 1974

\vol 5

\pages 201--237

\publ VINITI

\publaddr Moscow

\mathnet{http://mi.mathnet.ru/intg57}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=431309}

\zmath{https://zbmath.org/?q=an:0301.53032}