Natural differential operations on manifolds: an algebraic approach

Natural algebraic differential operations on geometric quantities on smooth manifolds are considered. A method for the investigation and classification of such operations is described, the method of IT-reduction. With it the investigation of natural operations reduces to the analysis of rational maps between $k$-jet spaces, which are equivariant with respect to certain algebraic groups. On the basis of the method of IT-reduction a finite generation theorem is proved: for tensor bundles $\mathscr{V},\mathscr{W}\to M$ all the natural differential operations $D\colon\Gamma(\mathscr{V})\to\Gamma(\mathscr{W})$ of degree at most $d$ can be algebraically constructed from some finite set of such operations. Conceptual proofs of known results on the classification of natural linear operations on arbitrary and symplectic manifolds are presented. A non-existence theorem is proved for natural deformation quantizations on Poisson manifolds and symplectic manifolds.
Bibliography: 21 titles.