Undergrad Connection Types: Affine vs Non-Affine

[kent davidge]

Based on what will you choose a connection to be affine or non-affine?

It seems to me that it's always more easy to work with affine connections, and I've seen only them being used in General Relativity. Are non-affine connections forbidden in the theory?

 

[Matterwave]

The "affine" part of the connection refers to the type of data that is being connected. In that case, the tangent spaces (which are affine spaces) at different points on a manifold are connected and thus you get an "affine connection". You can have non-affine connections which identify non-affine spaces with each other - e.g. if you had a more general fiber bundle than the tangent bundle, you might have a non-affine connection that connects fibers with each other in that structure. I don't have too much experience in general fiber bundles, so my knowledge kind of ends here. The connections used in GR is affine connections because the spaces that are important in GR is the base manifold and the tangent bundle (and cotangent bundle). If you want to explore more general connections and structures, you might want to ask in the differential geometry forum.

 

[fresh_42]

... you might want to ask in the differential geometry forum.

... or read https://www.physicsforums.com/threa...reoprojected-fibers-look-close-can-be.960769/
with an interesting link: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.684.168&rep=rep1&type=pdf.