Recent content by Bballer152

Well, even though my reply is close to a year late, I thought you should know that I've read this response and I love it, THANK YOU. It all makes sense now :)

Okay, I think I've got it, now let's see if I can put it into words (and thanks, by the way, for the patience/hints). We use the notation of the first post above. Namely, \phi_i:U_i\times G\rightarrow \pi^{-1}(U_i) is the local trivialization such that \phi_i(p,g)=\sigma_i(p)g with the usual...

In particular, here is precisely what I don't understand. The argument of \mathrm{d}_Pg_i(\sigma_{i*}X) is a tangent vector on P, which means that the thing that is acting on it should have some kind of "1-form" quality to it, which is good because the thing acting on it is the exterior...

Well I understand how this gives a Lie-algebra valued one-form on G itself, but I don't see how it does on the full bundle P. For example, on the next page, Nakahara says that \mathrm{d}_Pg_i(\sigma_{i*}X)=0 \mathrm{\ since\ } g\equiv e \mathrm{\ along\ } \sigma_{i*}X, and again I have no clue...

How so? I don't see how that expression specifies an action sending tangent vectors to Lie-algebra elements...

Hi All, I'm extremely confused by what's going on in section 10.1.3, pg 377 of the 2nd edition of Nakahara, in regards to his notation for lie algebra-valued one forms. We let \{U_i\} be an open covering of a smooth manifold M and let \sigma_i be a local section of M into the...

Yes, that splitting is exactly what's going on, but that should still mean that dim_ℂT_pM^+=dim_ℂT_pM^-=\frac{1}{2}dim_ℂT_pM^ℂ=dim_ℂM as opposed to dim_ℂT_pM^+=dim_ℂT_pM^-=\frac{1}{2}dim_ℂT_pM^ℂ=\frac{1}{2}dim_ℂM, right? (The only difference is in the last equality). The +'s and -'s correspond...

Can someone please confirm that there is a typo in Nakahara's Geometry, Topology, and Physics, on page 319 (the last line of the page) in the line following equation 8.27. There is a string of equalities, all of which make sense except the last one. I believe there should be no (1/2) in front...