Bi-differential calculi, bi-Hamiltonian systems and conformal Killing tensors (Q2703487)

A simple bi-differential calculus on a manifold \(M\) is defined by a pair \((d,\delta)\) of derivations of degree \(1\) on the exterior algebra of differential forms on \(M\) satisfying the relations \(d^2 = \delta^2 = 0\) and \(d \delta + \delta d =0\). A gauged bi-differential calculus is roughly the same thing, except that we must replace each derivation by an operator of the form \(d + A\) where \(A\) is a square matrix of differential \(1\)-forms. NEWLINENEWLINENEWLINEIn section 2, the authors expose elements of simple bi-differential calculus in the following case: \(d\) is the ordinary exterior derivative and \(\delta f := R^* df\), where \(R\) is a tensor of type \((1,1)\) on \(M\). The case where \(R\) is compatible with a Poisson or a symplectic structure is studied ; conditions to obtain a Poisson-Nijenhuis structure on \(M\) are given. Section 3 is devoted to the study of existence of a simple bi-differential calculus (i.e., the existence of the tensor \(R\)) when \(M\) is a cotangent bundle equiped with a hamiltonian (of mechanical type). NEWLINENEWLINENEWLINEIn section 4, gauged bi-differential calculus is studied in relation with bi-hamiltonian structures. Section 5 is an extension of section 3 from the point of view of gauged bi-differential calculus. NEWLINENEWLINENEWLINESection 6 contains applications to previous results of Ibort et al. [\textit{A. Ibort, F. Magri} and \textit{G. Marmo}, J. Geom. Phys. 33, 210-228 (2000; Zbl 0952.37028)] and \textit{S. Benenti} [Rend. Semin. Mat., Torino 50, 315-341 (1992; Zbl 0796.53017)].