This question was asked in my exercises of differential geometry and I am completely struck on the problem.
So, I am posting it here in hope I will get some leads:
Question: Let $M$ be a manifold of dimension n and let $ p: B \to M$ be a vector bundle with typical fiber $\mathbb{R}^k$. Let furthermore ,$ \phi_i : U_i × \mathbb{R}^k \to p^{-1} (U_i)$, $ i\in I$ be a family of local trivialization ( the $U_i$, $i\in I$ form an open cover of M) with transition functions $\phi_{ij} : U_i \cap U_j\to GL(k,\mathbb{R}$) .
(i) what is the cocycle condition?
(ii) Show that any section s of B give rises to a family of functions $s_i : U_i \to \mathbb{R}^k, i\in I$ and this family satisfies certain compatibility condition that you should specify! ( 1st define $s_i$'s).
(iii) Show that any family of functions $s_i : U_i\to \mathbb{R}^k$ that satisfies the compatibility condition given in the previous question give rises to a global section $s$ of $B$. ( 1st define s!).
Attempt: i I have done.
But for (ii) I am not able to do understand how the sections will give rise to family of functions and what could be the compatibility condition that this family satisfies and how to define $s_i$'s. Can you please help me with that?