Define the map $$P:TS^{n}\to S^{n} \;\;\;\text{by}\;\; P((x,v))=\frac{x+v}{\parallel x+v \parallel}$$ where $$TS^{n}=\{(x,v)\in S^{n} \ \times \mathbb{R}^{n+1}\mid v \perp x \}$$

This map is used in the book of Alain Hatcher, Algebraic topology, to give a proof for the fact that every vector field on even spheres must vanish on at least one point of the sphere.

Question:

1. Does $P$ define a (nontrivial) fiber bundle?

2. Define the Hamiltonian function $H:TS^{n} \to \mathbb{R}$ with $H(x,v)={\parallel P(x,v)-x \parallel}^{2} $ where the latter norm is the standard Euclidean norm on $\mathbb{R}^{n+1}$. What can be said about the dynamical behavior of the corresponding Hamiltonian vector field $X_{H}$? Are there any periodic orbit?
3. Assume that $V$ is a vector field on the sphere. To $V$, we associate a self map $f(x)=P(x,V(x))$ on the sphere. Are there some relations between the continuous dynamics of $V$ and the discrete dynamics of $f$. Note that $V$ and $f$ have the same fixed points.