On the structure of tensor fields given on manifolds

In the present work, we consider the metric questions of diffeomorphic mappings of the manifolds.We find such bases on tangential spaces of manifolds corresponding to given mappings, which allow investigate their metric properties, independent of connection coefficients, by using of strictly analysis' methods.For simplicity of our considerations, we suffice with consideration of manifolds defined by finite number of maps.When our consideration is restricted with neighborhoods of some points, we shall suffice with one map defining the manifold.