Geometrization of jet bundles (Q2266490)

This paper is a continuation of the author's previous paper [Jap. J. Math., New Ser. 8, 109-176 (1982; Zbl 0548.58002)] dealing with the case of higher order canonical differential systems \((J^ k(M,N,p),C^ k)\) on fibred manifold (M,N,p) of fibre dimension (rank) \(m\geq 2\). First, the author considers the canonical systems \((J(M,n),C)\) on Grassmann bundles J(M,n) of the tangent n-planes in the tangent bundle of the \((m+n)\)- dimensional manifold M. He obtains characterizations by graded algebras. Next, the author defines inductively the higher order prolongations \((J^ k(M,n),C^ k)\) of \((J(M,n),C)\) and introduces the notion of a contact manifold of order k, (K,D) of order k and bidegree, (n,m) as a manifold K with a differential system D such that (1) there exists a family \(\{D^ 1,...,D^ k\}\) of differential systems on K such that \(D^ k=D\) and \(D^ r=\partial D^{r+1}\); \(r=1,...,k-1\), i.e. the r-th derived sheaf \(\partial^ rD\) of D defines a differential system \(\partial^ rD=D^{k-r},\) \(r=1,...,k-1.\) (2) \(D^ 1\) is a differential system of codimension m; (3) there exists a completely integrable subbundle F of \(D^ 1\) of codimension n such that \(F\supset Ch(D^ 1);\) (4) The Cauchy-Cartan characteristic system \(Ch(D^ r)\) of \(D^ r\) is a subbundle of \(D^{r+1}\) of codimension n, for \(r=1,...,k-1\); (5) \(Ch(D^ k)(x)=0,\) \(x\in K;\) (6) \(Ch(D^{k-1})=D^ k\cap F,\) \(k\geq 2\); (7) \(\dim K=m\times (\sum^{k}_{r=1}{}_ nH_ r)+m+n\) where \(_ nH_ r=\left( \begin{matrix} n+r-1\\ r\end{matrix} \right).\) The main result is: Let D be a differential system on a manifold K. Then (K,D) is a contact manifold of order k of bidegree (n,m) if and only if it is locally isomorphic with \((J^ k(M,N,p),C^ k)\) where dim N\(=n\), dim M\(=m+n\).