Reduction of bihamiltonian manifolds and construction of the recursion operator
Since Magri's work, the study of completely integrable systems from a geometrical point of view is based on differentiable manifolds \(M\) with compatible Poisson tensors \(P\) and \(Q\) (i.e. such that their Schouten bracket equals zero). When \(Q\) is invertible, the field of endomorphisms \(h = PQ^{-1}\) is called a ``recursion'' operator. The study of the integrability of a system begins with the investigation of the possibility of reduction of the system to a Magri manifold with a recursion operator. In this paper a sufficient condition for such a reduction is obtained: if \((M,P,Q)\) is a Magri manifold such that \(P(\text{Ker }Q)\subset Q(\text{Ker }P)\), then exists a procedure of reduction which enables the construction of a recursion operator on certain quotient manifolds. This procedure is applied in the paper to the example of the heat equation.
scientific article; zbMATH DE number 70175