Quote by Cinquero
Could you please try to give an example of a calculation? I would like to get some feeling of how to handle all up to the mapping from a Lie algebra element to a Lie group element via parallel transport along a loop. Additionally, I'd like to see what a projection from the manifold into the tangent space looks like in practice.
Just to make things more clear: what exactly is the nature of this manifold you are talking of? Is there any physical interpretation?

hello Cinquero, have you by any chance looked at the beginning treatment of LQG in Rovelli and Upadhya's paper? This was my introduction to the subject back in 2003. Several of us at Physics Forum were reading that paper back then.
It is short (on the order of 10 pages) and shows how a number of things are calculated. If you are interested in learning LQG, then I could review the paper myself, and read some of it with you.
If you do not already have Rovelli/Upadhya and would like the link, please let me know. the date at arxiv is about 1998.