The problem of constructive equivalence in differential geometry

Abstract

The present paper is devoted to the algorithmic construction of diffeomorphisms establishing the equivalence of geometric structures. For structures of finite type the problem reduces to integration of completely integrable distributions with a known symmetry algebra and further to integration of Maurer-Cartan forms. We develop algorithms that reduce this problem to integration of closed 1-forms and equations of Lie type that are characterized by the fact that they admit a non-linear superposition principle. As an application we consider the problem of constructive equivalence for the structures of absolute parallelism and for the transitive Lie algebras of vector fields on manifolds.