Foundation of mechanics 2nd edition and vector bundle definitions

Question

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:

1. 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.
2. 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 $U \times F$
3. This book have weaken the definition of a vector bundle requiring local charts to be only bijections, not homeomorphisms.

Definitions:

1. 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.
2. 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?

Answer 1

The first thing to say before the further explanation is about the kind of vector bundles we talk about.

There are two kinds of vector bundles: smooth and topological.

Smooth require local bundle charts to be diffeomorphisms and a projection to be smooth, meanwhile topological require charts only to be a homeomorphisms and a projection to be continuous.

The definition the author introduces is close to smooth vector bundle, but lacks smoothness of a projection. However, the zero section's projection $\pi:E\to E_0$ is smooth by theorem 1.5.4, and it is a submanifolds of $E$, so the zero section is itself a smooth vector bundle in its canonical meaning.

1. For smooth vector bundles the fact that the transition map is a diffeomorphism is derived from the fact that local bundle chart is diffeomorphism too. So, its secondary and should not necessarily be included in the definition;
2. $S$ isn't endowed with topology, so to concern the transition map a diffeomorphism (roughly speaking, we consider this map to be smooth and to preserve topological structure) we should act from topological space to another topological space;
3. As far as the base space $S$ doesn't carry any topology, we cannot consider having a homeomorphism, However, we do induce pullback topology onto $S$, so beside of chart being a bijection in the definition it happens to be a homeomorphism de facto.