Recent content by Korybut

Hello! I would like to be sure about my understanding of the definition provided in screenshot below 1. What is this ##\mathcal{E}(G,N)##? I know that not all extension are isomorphic so I wonder What are the elements of ##\mathcal{E}## groups? Or maybe all Es diffeomorphic to each other...

Hello! There is a proof that Grassmannian is indeed a smooth manifold provided in Nicolaescu textbook on differential geometry. Screenshots are below There are some troubles with signs in the formulas please ignore them they are not relevant. My questions are the following: 1. After (1.2.5)...

Thanks once again to everyone the notion of principal bundle is perfectly clear to me now

Thanks to everyone for help. I kinda get this formal definition I would like to summarize just in case Manifold ##P## should be designed in the way that each ##\pi^{-1}(W)## is diffeomorphic to ##W\times G##. One can act on the latter with any element of ##G## in the obvious way. However I...

Sorry, but I don't get you clarification. How LOCAL trivialization is aware of of the whole manifold ##P## since group might send this neighbourhood ##W## in general to any other domain of ##P##? Book is "Differential Geometry" by Rudolph and Schmidt

Hello there! Book provides the following definition Let ##(P,G,\Psi)## be a free Lie group action, let ##M## be a manifold and let ##\pi : P \rightarrow M## be a smooth mapping. The tuple ##(P,G,M,\Psi,\pi)## is called a principal bundle, if for every ##m\in M## there exists a local...

TL;DR Summary: Reference request Hello! Reading the book "Differential geometry of Singular Spaces and Reduction of symmetry" by J. Sniatycki https://www.cambridge.org/core/books/differential-geometry-of-singular-spaces-and-reduction-of-symmetry/7D73498C35A5975594605428DA8F9267 I found that...

Hello! I am trying to figure how one can deduce guiding center motion equation according to Hazeltine and Waelbroeck "The Framework of Plasma physics". They suggest the following: To solve equations ##\frac{d\vec{r}}{dt}=\vec{v},\;\; \frac{d\vec{v}}{dt}=\frac{e}{\epsilon...

Thanks for another clarification. I wish those lines were in the book)

Lawson and Michelson "Spin Geometry". They suggested the following ##(1,0)\rightarrow \frac{1}{2}(1\otimes 1+i\otimes i)## ##(0,1)\rightarrow \frac{1}{2}(1\otimes 1 -i \otimes i)## And I don't get how I proceed with the full proof with just that

Sorry! My Bad! Indeed! Thanks for help!

I wish someone can explain notation in this formula. What is ##T##? What does ##/<...> means?

I've seen this formula before and indeed this identification doesn't suffer from the problem I mentioned earlier however I don't get why this is isomorphism. If I consider this map ##i: \mathbb{C}\otimes \mathbb{C}\rightarrow \mathbb{C}\oplus\mathbb{C}## then I can look at element which are...

Hello! Reading book o Clifford algebra authors claim that ##\mathbb{C}\oplus\mathbb{C}\cong\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}## as algebras. Unfortunately proof is absent and provided hint is pretty misleading As vector spaces they are obviously isomorphic since ##\dim_{\mathbb{R}}...