I've been having trouble wrapping my head around the tautological line bundle and the fact that it cannot admit any nowhere vanishing smooth sections.
Let $E=\{(v,tv)\,:\,v\in\mathbb{R}P^1,~t\in\mathbb{R}\}$ be the tautological line bundle over $\mathbb{R}P^1$. We can define a covariant derivative by differentiating (where this is the usual vector valued derivative) a section of $E$ and then projecting onto the corresponding line in $\mathbb{R}^2$. Now it is pretty clear (I think this is where my mistake is!) that $\sigma:[0,\pi]\to E$ given by $$\sigma(t)=((\cos t,\sin t),(\cos t,\sin t))$$ is a smooth section of $E$. I get this by just taking the vector $(1,0)$ and parallel transporting it around $\mathbb{R}P^1$ using the connection described above. It seems like $\sigma$ is a global nowhere vanishing section of $E$. This can't exist though since $E$ is non-trivial!
Where is my mistake? How should I parallel transport in $E$? Is there a more natural connection that I can define on $E$?
What you are constructing here is the normal bundle of $S^1$ considered as a submanifold of $\mathbb{R}^2$. Remark that construct $\mathbb{R}P^1$ you take the quotient of $\mathbb{R}^2-\{0\}$ by the relation $(x,y)\simeq c(x,y)$. If you restrict that relation to $S^1$, you obtain that $v\simeq -v$, so $\sigma$ is not well defined since $\sigma(-cos(t),-sin(t))\neq \sigma((cos(t),sin(t))$.