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.