# I Curvature forms and Riemannian curvatures of connections

#### martinbn

For me it is very difficult, actually impossible, to understand what you mean. And I think the problem is that you are vague and use terminology in a non-standard way. The problem is not your English, the mathematics is. Perhaps some of the notions you use are not clear to you. For example you keep saying that the global bundle is trivial. But that is confusing because globally the interesting bundles are not trivial. May be you mean that there are no global symmetries. When every sentence is like that it is frustrating to read. On the other hand these seem like interesting topics, so I can not stop myself reading.

#### RockyMarciano

For me it is very difficult, actually impossible, to understand what you mean. And I think the problem is that you are vague and use terminology in a non-standard way. The problem is not your English, the mathematics is. Perhaps some of the notions you use are not clear to you.
I empathize with what you are saying. I can assure I try, and that I'll keep trying, but I'm neither a mathematician nor a professional physicist, just a layman trying to learn, like most around here.
For example you keep saying that the global bundle is trivial. But that is confusing because globally the interesting bundles are not trivial.
Ok. this one is easy to clarify. The bundle I am referring as trivial is the principal bundle with the Minkowski space as base manifold and the compact (finite dimensional global) groups U(1), SU(2) and SU(3) as fibers (also called internal symmetries). This principal bundle is trivial among other thing because the Coleman-Mandula prevents in general any other relation between the fiber G and the base space other than the direct topological product, and that is what defines a trivial bundle. Also triviality here is demanded by the required gauge redundancy of degrees of fredom that must be recovered, in other words arbitrary rotations at each spacetime point must not affect the physics, and that's what the triviality of the bundle in this case assures.

You can also read about this bundle defined as trivial in the post #13 of the thread I linked: specifically and I quote " the trivial principal bundle $\pi (G,M)$" with M being the spacetime and G the compact global group, as explained there.

Some confusion might have arisen from the local gauge bundle related to nonabelian connections wich is nontrivial, but this is a different bundle, its base space is not Mikowski spacetime, and its fibers are the infinite dimensional local gauge groups that are also mentioned in the same post #13 that I quote above and it is the infinite dimension bundle mentioned in the notes by David Skinner that I referenced in #27, in the pages 21-22 as the space of local connections A.

If you have any further doubt about this point or any other of those you found hard to understand please don't hesitate to comment it too and I'll do my best to clarify.

On the other hand these seem like interesting topics, so I can not stop myself reading.
Same here, that's why I keep reading and asking questions in this site.

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#### Ben Niehoff

Gold Member
The only things you referenced in #32 are Wikipedia articles. Where are the notes by David Skinner?

#### RockyMarciano

The only things you referenced in #32 are Wikipedia articles. Where are the notes by David Skinner?
There is also a pointer to the thread I was commenting. The link to the notes was in #27.

Please ignore my out of line comment in #47.