Some subtleties of Riemannian geometry

英文摘要：

It is shown, that the conventional presentation of the Maxwell equations for the electromagnetic field in the Riemannian space-time appears to be problematic. The reason of hesitations is the fact, that a solution of the Maxwell equations in the space-time of Minkowski do not turn into solution of the Maxwell equations in the Riemannian space-time after replacement of Minkowskian world function $σ_{M}$ by the world function $σ_R$ of the Riemannian space-time in the solution.