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A question related to frame bundle of a vector bundle
I would probably reformulate the text you copied by saying first that a trivialization of $E$ over $U$ induces a trivialization of $\text{Hom}(\mathbb R^k_M,E)$ over $U$. This has values in $U\times\...
- 20k
1
vote
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A question on the manifold $ \{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\} $.
This is not true.
There's an action of $SO(3)$ on such pairs $(n, m)$ which for $O\in SO(3)$ sends the pair $(n, m)$ to $(On, Om)$. The action is transitive. You've correctly identified that the ...
- 2,599
1
vote
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Differential of a map $\Gamma : M \times N \to P$
$\Gamma_y$ is the composition $$X \stackrel{i_y}\longrightarrow X \times Y \stackrel{\Gamma}\longrightarrow Z$$ where the first map is $i_y(x) = (x,y)$.
Then, with the identification $T_{(x,y)}(X \...
- 4,512
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