This question was stated during a lesson where I asked my professor if every two dimensional smooth manifold arises as a tangent bundle of some one dimensional smooth manifold. My professor said that something more strong happens since for example $\mathbb{P}^2(\mathbb{R})$ can't be a smooth fiber bundle of any one dimensional smooth manifold.
I already know that if such smooth fiber bundle exists then the base space must be $S^1$ since every smooth one dimensional manifold is diffeomorphic to $\mathbb{R}$ or $S^1,$ but being $\mathbb{P}^2(\mathbb{R})$ compact implies that the base space must be compact, hence diffeomorphic to $S^1.$
Restating my problem, I want to prove:
There is no smooth fiber bundle with base space $S^1$ and total space $\mathbb{P}^2(\mathbb{R}).$
I've proved this using algebraic topology (concretely at the level of fundamental groups) but my professor said that he prefers a solution with the tools of differential geometry, something like studying the possible fibers and so on. At this point I don't know how to proceed.
I hope someone could help me on this.