I just want your opinion on this one:
"1. Write two vector bundles with rank 1 on $S^1$, such that they are not isomorphic between them".
"2. Prove that $S^1\times S^1$ is parallelizable
For the first question I was thinking about these two:
- the Möebius bundle;
- $TS^1$, the vector bundle of $S^1$.
Both are vector bundle on $S^1$ with rank 1 and they are not isomorphic because $TS^1$ is trivial while the Möebius bundle it's not so.
For the second one, I was reasoning this way: I know $S^1$ is parallelizable and I know that the product of parallelizable manifolds is a parallelizable manifold, so I can conclude $S^1\times S^1$ is parallelizzable.
Are this answers correct?
Thank you so much
As mentioned in the comments, tangent bundle should be replaced with vector bundle everywhere in the comments.
Otherwise, this sounds good, you may want to show that the mobius bundle is not trivial (it is not even homeomorphic to the trivial bundle.)
A nice way to do this is to note that $\mathbb R \times S^1\setminus S^1 \times \{0\}$ is disconnected, while it's image under any homeomorphism is connected for the mobius band. (Or you could check the first stiefel whitney class, or some orientability argument.)
The second answer is perfectly fine, since $T(M \times N) \cong T(M) \times T(N)$, where the isomorphism can be seen by taking canonical projections and for each $(p,q)$ and considering $v \mapsto (d_{(p,q)} \pi_M(v),d_{(p,q)} \pi_N(v)$ which is a linear isomorphism (there is a clear inverse induced by maps $M \hookrightarrow M \times N$, $m \mapsto m \times q$, the other inclusion, and taking their sum.