Skobeltsyn Institute of Nuclear Physics
Lomonosov Moscow State University

A generalization of the reference frame formalism, used in General Relativity, for arbitrary dimension is suggested. We present a convenient notation based on the complex reference frame formalism in case of an arbitrary manifold. We pay special attention to obtaining expressions for covariant derivative and curvature in case of the reference frame formalism on arbitrary manifold. Results seem to be useful in analysis of various physical theories with extra dimensions, in particular Кaluza-Кlein theories.