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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 $E$?

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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 $\sigma(-cos(t),-sin(t))\neq \sigma((cos(t),sin(t))$.

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  • $\begingroup$ Ahh, I see my mistake. Do you mind elaborating how I would do parallel transport in the tautological line bundle? $\endgroup$ – pomegranate Mar 22 '18 at 13:49
  • $\begingroup$ Is parallel transport not defined globally in this line bundle? It seems like I should be able to take $\sigma((1,0))$ and parallel transport it around a closed curve in $\mathbb{R}P^1$. I think it should flip sign if I do this -- which means that it needs to be zero somewhere along its path. I'm not sure how to construct such a parallel transport though since it seems to be changing magnitude (which of course is fine that it changes magnitude), but it seems by differentiating and projecting that such a change in magnitude will lead to a non-zero covariant derivative $\endgroup$ – pomegranate Mar 22 '18 at 13:58
  • $\begingroup$ To define a parallel transport you need a connection, what is the connection that you are using here ? And the parallel transport map under a path $c:I\rightarrow M$ is a map $F_{c(0)}\rightarrow F_{c(1)}$. $\endgroup$ – Tsemo Aristide Mar 22 '18 at 14:02
  • $\begingroup$ So I'm having a bit of trouble actually writing down a connection -- but I think this should work. We can write $\mathbb{R}P^1=\mathbb{R}/\mathbb{Z}$ and so we can parameterize it by $\theta\in[0,1]$. Then we can define the covariant derivative of a section $\sigma\in\Gamma(E)$ by $\nabla_{\partial_{\theta}}\sigma:=d\sigma/d\theta$. I think this works out and is well defined since we can identify section of $E$ with anti-periodic functions on $\mathbb{R}$. Is there a more natural connection to consider here? Thank you so much, I really appreciate it. $\endgroup$ – pomegranate Mar 22 '18 at 14:11
  • $\begingroup$ Thanks so much! I figured it out :) My main confusion is that I was thinking that a section over the pullback bundle $c^*(E)$ was the same as a section of $E$. Clearly, there are smooth non-vanishing sections of the pullback bundle since it is a vector bundle over a contractible space... On the other hand, no such section can exist in $E$ -- which is exactly what you pointed out! My bad. Please let me know if I made a mistake in this comment $\endgroup$ – pomegranate Mar 22 '18 at 21:34

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