I could not find any other question here related to this. If I have missed out, then this could be voted as a duplicate(Sorry if it is!).
I was just trying to figure out the tangent space to the projective space and intuitively I thought that as the space is itself obtained by identifying antipodal points on the sphere, one could look at the tangent space to this quotient space as quotient of the tangent spaces to the sphere at the antipodal points.
In other words, if $p \in \mathbb{R}P^n$ is obtained by identifying $x,-x \in \mathbb{S}^n$, then does $T_p(\mathbb{R}P^n)$ stem out of identifying the spaces $T_x(\mathbb{S}^n)$ and $T_{-x}(\mathbb{S}^n)$??
Is this "quotient of tangent spaces" being the same as the tangent space of the quotient space true in general??
On a related note, I came across a problem instructing one to show that if
$$\pi : \mathbb{S}^n \to \mathbb{R}P^n$$ is the quotient map with $\pi(x) = p$, then there is an isomorphism $$\pi_*|_x :T_x(\mathbb{S}^n) \to T_p(\mathbb{R}P^n)$$.
How does one show this?? I can show that $\pi$ is smooth, but not sure whether it is a diffeomorphism, whence the above result would follow ??
I hope I have not missed out on something obvious here. Thanks for your patience.
Best Answer
I would suggest you to read the Example starting on the end of p. 10 here http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf
Also in Milnors "Characteristic Classes" he computes the tangent space of the projective space in Chapter 2 or 3 I think. I strongly suggest you this one!
To answer your other question: Since the antipodal action is smooth, you get a smooth covering space to the quotient $S^n \to RP^n$, hence a local diffeomorphism. In particular induced maps on tangent spaces are isomorphisms.