Five-dimensional general relativity and Kaluza-Klein theory

We consider 5D spaces which admit the most symmetric 3D subspaces. 5D vacuum Einstein equations are constructed and 5D analog of the mass function is found. The corresponding conservation law leads to 5D analog of Birkhoff's theorem. Hence the cylinder condition is dynamically implemented for the considered spaces. For some obtained metrics a period of space with respect to the fifth coordinate was found. The problem of the dynamical degrees of freedom of the fields system obtained in the process of dimensional reduction is discussed, and the problem of their interpretation is considered. One can think that the parametrization of the scalar field and 4D metric leading to the conformally invariant 4D theory for interacting gravitational and scalar fields is most natural and adequate.