# Foundation of mechanics 2nd edition and vector bundle definitions

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?

• I think the definition should be included in the question if possible to avoid referencing a book others may not have. Jul 31 at 2:57
• When you refer to a book, you should mention the author(s). Jul 31 at 9:41
• Concerning question 3 see math.stackexchange.com/q/4710773. Jul 31 at 9:53
• Updated the question according to the comments Jul 31 at 12:58

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.
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.