Some dynamical and Bundle questions arising from certain map $P:TS^{n}\to S^{n}$

Question

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 at some 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.

This post imported from StackExchange MathOverflow at 2016-11-05 15:54 (UTC), posted by SE-user Ali Taghavi

Comments

$P$ is (up to a mild re-parametrization) the restriction of the exponential map for the tangent bundle of $S^n$. So 1, yes. $H$ is basically a re-scaled length of $v$, that answers (2). (3) can similarly be answered.

This post imported from StackExchange MathOverflow at 2016-11-05 15:54 (UTC), posted by SE-user Ryan Budney

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@RyanBudney $H$ is not a re-scaled length of $v$, since the lenght of $v$ is an unbounded function but $H$ is a bounded function! Could you please elaborate your comment. I think your comments is not clear.

This post imported from StackExchange MathOverflow at 2016-11-05 15:54 (UTC), posted by SE-user Ali Taghavi

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@RyanBudney Do you mean that $X_{H}$ equal geodesic flow vector field up to a constant multiplier? Do you mean that $H$ is globally equal to $\parallel v \parallel ^2$ up to a constant?How can (3) be answered immediately?

This post imported from StackExchange MathOverflow at 2016-11-05 15:54 (UTC), posted by SE-user Ali Taghavi

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$P$ is pretty much precisely the exponential map if you re-scale the input vector by a non-linear function of its length. i.e. re-scale the input vector's length by $L \longmapsto 1/\sqrt{1+L^2}$. Exponentiate that vector, this gives you $P$. So your function $H$ is a similar non-linear rescaling of the length squared of the input vector.

This post imported from StackExchange MathOverflow at 2016-11-05 15:54 (UTC), posted by SE-user Ryan Budney

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@RyanBudney But why does this imply that $X_{H}$ has the same dynamic as the geodesic flow?Moreover, what about the third part of my question?

This post imported from StackExchange MathOverflow at 2016-11-05 15:54 (UTC), posted by SE-user Ali Taghavi

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@RyanBudney Do you mean $P(x,V)=exp(V/\sqrt{1+\parallel V \parallel^2}$? If you mean this equality, I think some thing is missing, for example it is not the case for $n=1$.

This post imported from StackExchange MathOverflow at 2016-11-05 15:54 (UTC), posted by SE-user Ali Taghavi