In Kaluza-Klein theory one introduce an extra fifth spatial dimension, to the usual four-dimensional manifold ##M^4## in General Relativity called space-time. This extra dimension has a symmetry; i.e. it is spanned by a Killing vector and it's taken to be compact. One views this extra dimension as a little circle attached to each point, so that locally I would guess one can view it as a product manifold ##M^4 \times S^1##. Now I would really like to understand this geometry a little better.. One thing I wonder about is why one can not consider the manifold ##M^4## a hypersurface in the five-dimensional manifold? In the book "Einstein's General Theory of Relativity" by Grøn and Hervik, in the section about KK-theory, one derives a relation between the five- and four-dimensional Riemann tensors by using the Cartan equations in 5D and 4D. However then the question arises how the connection is defined in the 4-dimensional and 5-dimensional space? I've tried to find material on this, but those that I have found are very complicated and requires knowledge of vector-bundles, fiber-bundles, distributions and so on. I would think it was possible to understand it simpler terms? Thanks for any help!