Connections/geometry in Kaluza-Klein theory

[center o bass]

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!

 

[pat_connell]

The connection between the four-dimensional and five-dimensional Riemann tensors is derived in the usual way by taking derivatives of the metric. In short, we can view the extra dimension as an additional direction in which the metric can vary, and this gives rise to additional components of the Riemann tensor. There isn't really any need to understand the details of the underlying geometry in order to understand this, though if you are interested in learning more about it, then vector bundles and fiber bundles are a good place to start. In general, the extra dimension can be viewed as a kind of "hidden" space, in which the geometry of space-time is encoded. This hidden space can be explored using the mathematics of differential geometry, and this is what Kaluza-Klein theory does.