Extended retraction maps: a seed of geometric integrators

The classical notion of retraction map used to approximate geodesics is extended and rigorously defined to become a powerful tool to construct geometric integrators. Using the geometry of the tangent and cotangent bundles, we are able to tangently and cotangent lift such a map so that these lifts inherit the same properties as the original one and they continue to be extended retraction maps. In particular, the cotangent lift of this new notion of retraction map is a natural symplectomorphism, what plays a key role for constructing geometric integrators and symplectic methods. As a result, a wide range of numerical methods are recovered and canonically constructed by using different extended retraction maps, as well as some operations with Lagrangian submanifolds.