difference between canonical line bundle on $\mathbb RP^n$ and bundle obtained from normal bundle on sphere

Question

How can I, possibly geometrically, differentiate following two bundles?

One is obtained from normal bundle on sphere $S^n$ with identifications $(x,tx)\sim (-x,-tx)$, and another one is canonical line bundle on real projective space $\mathbb RP^n$.

Answer 1

First way: The outward unit normal vector field on $S^n$, $(x,x)$, descends to a global section of your bundle since the identification $(x,tx)\sim(-x,-tx)$ does precisely what it needs to make $(x,x)$ well-defined here for $[x]\in\mathbb{R}P^n$. So this is the trivial line bundle. On the other hand, the tautological bundle on $\mathbb{R}P^n$ is nontrivial and does not admit a nonvanishing section.

Second way: the zero section disconnects your bundle (because you have an "inside" and "outside" if you use the $e^tx$ identification with $\mathbb{R}^{n+1}-\{0\}$ and the identification $\sim$ preserves this distinction) but doesn't do that for the tautological line bundle.

Third way: Compute the Stiefel-Whitney class, $w_1(\text{tautological})\neq 0$, but your bundle has $w_1=0$.