- I would like to know how does one imagine/write-down the tangent bundle of the real/complex projective space. Is there something simplifying that happens especially for $\mathbb{RP}^1$? Isn't there some relationship of this tangent bundle to spheres and Möbius bands?
I can write down transition functions on these but that doesn't answer "what" is the tangent bundle.
- I vaguely know that Möbius band can be imagined as the anti-podal identification of the normal bundle on the circle. What is the exact statement and how to see that?
Since projective spaces are easier to think of as quotients of simpler spaces like spheres I am motivated to ask the following question,
- In general if a group action on a manifold is such that the quotient space is again a manifold then the same action will also do a quotient of the tangent bundle. Now is the quotient of the tangent bundle (will that be again a vector bundle?) anyway related to the tangent bundle of the quotient space?
Best Answer
$\mathbf P^1(\mathbb R)$ is a closed curve, so its tangent bundle is trivial. There is not much to be imagined, really, in that case! In general, what kind of answer do you expect for your «what is the tangent bundle of $\mathbf P^n$?». It is the tangent bundle of $\mathbf P^n$...
Regarding your last question: if a group $G$ acts properly discontinuously and differentiably on a manifold $M$, then $M/G$ is a manifold, $G$ acts on $TM$ in a natural way (also properly discontinuously, and morover linearly on fibers), and $(TM)/G$ is "the same thing" as $T(M/G)$.