I proceed studying differential theory at Foundations of Mechanics 2nd edition R. Abraham and E. Marsden, and I sometimes get confused by divergence with other sources.
Right now I got confused by a couple of points:
- Meanwhile other sources for local bundle charts $\forall(U,\varphi), (U,\psi)\in\mathcal{A}_{max}$ require only well-definedness of a map $\psi \circ \varphi^{-1}(x,v) \lvert_{\varphi(U\cap V)} =(\phi_1(x),\phi_2(x)(v))$, this book demands a map $\psi \circ \varphi^{-1}(x,v) \lvert_{\varphi(U\cap V)}$ to be a diffeomorphism as well as $\phi_2$ to be a linear isomorphism. It's of course clear that this statements has purpose of preserving local bundle structure as well as having a differentiable structure.
- A local trivialization $\varphi: U\to W\times F, U\subset S,W \subset E$ is said to act into Cartesian product with some vector space $E$ which is not in any way connected with the original space $S$, when typically it is just Cartesian product $U \times F$
- This book have weaken the definition of a vector bundle requiring local charts to be only bijections, not homeomorphisms.
Definitions:
- Local bundle chart. Let $E$ and $F$ be finite-dimensional real vector spaces, and $W \in \tau(E)$ an open neighborhood of $E$, then we call the Cartesian product $W \times F$ a local vector bundle.
- Vector bundle. A local bundle chart is a pair $(U,\varphi)$, where $U \subset S$, and $ \varphi: U \to W \times F$ is a bijection onto a local bundle $W \times F$. A vector bundle atlas on $S$ is a family of local bundle charts $\mathcal{A}=\{(U_i,\varphi_i)\}$ covering the space $S$, and $\forall(U,\varphi), (U,\psi)\in\mathcal{A}\;\psi \circ \varphi^{-1}(x,v) \lvert_{\varphi(U\cap V)} =(\phi_1(x),\phi_2(x)(v))$ is a diffeomorphism as well as $\phi_2$ a linear isomorphism.
So how can I show that these statements are equivalent to standard definitions?