Oriented Bundle over $S^1$

Question

Is it true that oriented bundle over $S^1$ is always trivial bundle? For example take $S^2$ and let $\gamma: S^1\to S^2$ is a great circle. As $TS^2$ is orientable, Then is it true that $\gamma^*TS^2\equiv S^1\times \mathbb R^2$.

Answer 1

Let $E\to S^1$ be an $SL(n)$ Bundle. Recall the construction how to classify vector bundle on $S^n$, $E\to S^1$ depends on the homotopy class of $S^0\to SL(n)$. Since $SL(n)$ is connected, $S^0\to SL(n)$ is trivial homotopic, which means $E$ is trivial.