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##.(adsbygoogle = window.adsbygoogle || []).push({});

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!

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Connections/geometry in Kaluza-Klein theory

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads - Connections geometry Kaluza | Date |
---|---|

I Connections on principal bundles | Jan 22, 2018 |

A Is the Berry connection a Levi-Civita connection? | Jan 1, 2018 |

A Confusion on notion of connection & covariant derivative | Mar 14, 2016 |

Existence of book on connection projective geometry and perspective? | Mar 1, 2011 |

Connections in differential geometry | Nov 10, 2007 |

**Physics Forums - The Fusion of Science and Community**