ON HIGHER ORDER CONTACT MANIFOLDS

Yuli Villarroel

Key words: Geometric structures on manifolds, local submanifolds,
contact theory.

1991 Math. Subject Classification: 53C15, 53B25.


Form the text.
Let $M$ be a smooth $m$-dimensional manifold and
$\quad C^{k,n}M$, $n\le m$, the manifold of contact elements
of order $k$ and dimension $n$ over M$\quad $[3]. Given an
$n$-submanifold $S\subset M$, denote by $C^k_xS$ the contact element
of order $k$, of $S$ at $x\in S$ and by $\pi ^k_0$ the canonical
projection over $M$. Let
$\tilde C^{1,n}(C^{k,n}M)\quad $ be the manifold of all contact elements
of order $1$ and dimension $n$,
of $n$-submanifolds ${\cal S}\subset C^{k,n}M$, such that the
natural projection $\pi ^{1,k}_0$ restricted to ${\cal S}$
 over M, has maximal rank [4].
 \smallskip

The manifold $\tilde C^{1,n}(C^{k,n}M)$
 is embedded in  $C^{1,n}(C^{k,n}M)$, and the natural immersion,

 $i^{1,k}:{C^{k+1,n}M\rightarrow \tilde C^{1,n}(C^{k,n}M)}$, defined by
 $i^{1,k}(C^{k+1}_xS)=C^1_{C^k_xS}C^kS$, is an embedding; where
 $C^kS\subset C^{k,n}M$ denotes
 the $n$-submanifold in $C^{k,n}M$, defined by the immersion
 $x\in S\subset M\mapsto C^k_xS\in C^{k,n}M$.
\smallskip

We prove that, there exists an involution
$\tilde \alpha $, i.e. an automorphism with
$\tilde \alpha\circ \tilde \alpha $
 equal to the identity,
 $$\tilde \alpha :\tilde C^{1,n}(C^{k,n})M\longrightarrow
 \tilde C^{1,n}(C^{k,n})M,$$
  such that, $i^{1,k}(C^{k+1,n}M)\subset \tilde C^{1,n}(C^{k,n}M)$ is
  the set of the fixed points of $\tilde \alpha $.
\smallskip

This result generalizes the following:
Let $J^k_0(\Re ^n,M),$ be
the $k$-jet bundle of maps from $\Re ^n$ into $M$,
with source at $0\in \Re ^n$ [1].
There is a
canonical involution, $\alpha$ in $\quad J^1_0(\Re ^n,J^k_0(\Re ^n,M))$
such that, $\quad J^{k+1}_0(\Re ^n,M)$ can be immersed in
$\quad J^1_0(\Re^n,J^k_0(\Re ^n,M))\quad $as the invariant set of
$\alpha \quad ([2], [5]).$
\smallskip

Using the properties of the fixed point set of $\tilde \alpha $, we
we will prove the following:
\smallskip

1)The $n$-planes $P$ in a submanifold $C^{1,n}W$,
with rank $(\pi ^k_o)_*|P=n$, contained in $C^{k+1,n}M$,
are fixed by $\tilde \alpha $.
\smallskip

2)If a submanifold $W\subset C^{k+1}M$, has at each point an $n$-plane
fixed by $\tilde \alpha $, then $W$ generates an involutive $n$-distribution
on $C^{k,n}M$. Moreover, given $X^k\in W$, there exists a submanifold
$S\subset M$, with $\pi ^k_o(X^k)=x\in S$ such that: $C^k_x=X^k$ and $C^kS$
is a solution of the distribution defined by $W$, through $X^k$.