Tensor invariants in numerical geometric integration of ODEs

We consider geometric numerical integration (GI) of ordinary differential equations (ODEs). We propose that in GI one needs concepts which are both geometric and algebraic. In this paper we start from an algebraic point of view: we introduce tensor invariants attached to an ODE as well as to an integrator. The notion of “sharing a tensor invariant” generalizes the well known notion of conserving a symplectic structure by an integrator. Several examples are given.