Variational sequences on finite order jet spaces.

*(English)*Zbl 0813.58014
Janyška, Josef (ed.) et al., Differential geometry and its applications. International conference, Brno, Czechoslovakia, 27 Aug. - 2 Sept. 1989. Singapore: World Scientific. 236-254 (1990).

The variational sequence of order \(r\) is introduced in the following way. Let \(Y\) be a fibered manifold with base \(X\) and \(J^{r}Y\) the \(r\)-jet prolongation of \(Y\); further \(\Omega^ k_ r\) denotes the sheaves of \(k\)-forms over \(J^ rY\) while \(\Omega^ k_{r(c)}\) stands for their subsheaves of contact. We set \(\theta^ k_ r = d\Omega^{k - 1}_{r(c)} + \Omega^ k_{r(c)}\) and consider the quotient sheaves \(\Omega^ k_ r/ \theta^ k_ r\) jointly with the quotient morphisms \(E_ k : \Omega^ k_ r/\theta^ k_ r \to \Omega^{k + 1}_ r/\theta^{k + 1}_ r\) induced by the exterior derivatives \(d_ k : \Omega^ k_ r \to \Omega^{k + 1}_ r\).

The quotient sequence: \[ 0 \to R \to \Omega^ 0_ r \to \Omega^ 1_ r/\theta^ 1_ r \to \Omega^ 2_ r/\theta^ 2_ r \to \cdots \] is called the variational sequence of order \(r\) over \(Y\). One of the main theorems of the paper states that such a sequence is an acyclic resolution of the constant sheaf \(R\) over \(Y\). Explicit expression for \(E_{n + 1}\) is provided and it is shown that \(E_ n\) is the Euler- Lagrange mapping in the sense introduced earlier by the same author.

The paper reads well and constitutes a valuable contribution to the calculus of variations on manifolds.

For the entire collection see [Zbl 0777.00040].

The quotient sequence: \[ 0 \to R \to \Omega^ 0_ r \to \Omega^ 1_ r/\theta^ 1_ r \to \Omega^ 2_ r/\theta^ 2_ r \to \cdots \] is called the variational sequence of order \(r\) over \(Y\). One of the main theorems of the paper states that such a sequence is an acyclic resolution of the constant sheaf \(R\) over \(Y\). Explicit expression for \(E_{n + 1}\) is provided and it is shown that \(E_ n\) is the Euler- Lagrange mapping in the sense introduced earlier by the same author.

The paper reads well and constitutes a valuable contribution to the calculus of variations on manifolds.

For the entire collection see [Zbl 0777.00040].

Reviewer: J.J.Telega (Warszawa)

##### MSC:

58E30 | Variational principles in infinite-dimensional spaces |