Tensor bundles are vector bundles

Question

I would like to prove that every tensor bundle is a vector bundle. First some definitions.

A bundle of $(k,l)$-tensors on $M$ is defined as as their disjoint union, $T^k_lM:=\bigsqcup_{p\in M} T^k_l(T_pM)$

A vector bundle $(E,M,\pi)$ contains a the total space $E$ and the base $M$, both smooth manifolds with a surjective map $\pi:E\to M$ such that

1. Every fiber $E_p:=\pi^{-1}(p)$ is a vector space.
2. For every $p\in M$ exist a neighborhood $U$ of $p$ and a diffeomorphism $\phi: \pi(U)^{-1}\to U\times\mathbb{R}^k$ (local trivilisation of E) s.t. $P\circ\phi(\pi^{-1}(U))=U$, where $P$ is the projection onto the first factor.
3. Restriction of $\phi$ to a fiber, $\phi:E_p\to \{p\}\times\mathbb{R}^k$, is a linear isomorphism

Define the map $\pi:T^k_lM\to M$ by $T^k_l(T_pM)\mapsto p$. in a neighborhood $U$ of $p\in M$ $F\in T^k_l(T_pM)$ can be written as

$F=F^{j_1,\dots,j_l}_{i_1,\dots,i_k}\,\,\partial/\partial x^1\otimes\dots\otimes\partial/\partial x^1\otimes dx^i\otimes\dots\otimes dx^n$

Then define the map $\phi:\pi^{-1}(U)\to U\times\mathbb{R}^{n^{k+l}}$ by $F\mapsto(p,F^{\,j_1,\dots,j_l}_{i_1,\dots,i_k})$. It is clear that $\pi^{-1}(p)=T^k_l(T_pM)$ is a vector space and that $P\circ\phi(\pi^{-1}(U))=U$. How can I show that $\phi$ is a diffeomorphism and restricted to a fiber a linear isomorphism.

Thanks for your help.

Answer 1

First of all, you can’t prove that it is a diffeomorphism, since your tensorbundle doesn’t have any smooth structure. The fact, that these maps should be diffeomorphism, is exactly the smooth structure. The $\phi$ is a chart, and you need to prove that the change of these charts is smooth, to prove that you have an $C^\infty$-atlas.

Here are the steps you need to understand: Check that $\phi$ is bijective. Prove that the change of charts $\phi_1\circ \phi_2^{-1}$ is smooth. Test if $\phi\big|_{\pi^{-1}(p)}(aF+bG)= a\phi\big|_{\pi^{-1}(p)}(F)+b\phi\big|_{\pi^{-1}(p)}(G)$. Then, to see that $\phi$ restricted to a fiber is an isomorphism, you can check that it is injective and the dimensions of the domain and codomain coincide.