Does there exist a fibre preserving diffeomorphism from $T S^1$ onto $S^1 \times \mathbb R\ $?

Question

Does there exist a fibre preserving diffeomorphism $\phi : TS^1 \longrightarrow S^1 \times \mathbb R\ $?

(Here $TS^1$ denotes the tangent bundle of $S^1$)

Any help would be warmly appreciated. Thanks for your time.

Answer 1

Yes, it exists. Taking the angular coordinate $\theta$ gives you a globally defined vector field $\frac{\partial}{\partial\theta}$ on $S^1$ (even though the coordinate function itself is defined up to a point).

Any tangent vector $v$ at $p\in S^1$ is proportional to this vector field at $p$, $v=v^\theta \frac{\partial}{\partial\theta}|_p$. The isomorphism you are looking for is

$$ (p,v_p)\in TM \mapsto (p,v_p^\theta)\in S^1\times\mathbb{R} $$

There are other ways of seeing that $S^1$ has trivial tangent bundle (in the jargon, $S^1$ is parallelizable). $S^1$ is a Lie group, and the tangent bundle of every $m$-dimensional Lie group is isomorphic to $G\times \mathbb{R}^m$.