User matgaio - MathOverflow

Reference - Asymptotic geodesics on compact surfaces without conjugate points

2013-03-20T23:43:21Z

I would like to ask about possible references on the following problem: consider a compact surface and a metric without conjugate points. Consider it's universal covering endowed whith the lifting of the metric (so there is no conjugate points on the universal covering as well).

Suppose that there exists two geodesics on the covering wich are strongly asymptotic on the future (the distance between them goes to zero as $t\rightarrow\infty$). Is there any hope of obtainning estimatives for the distance between them on the past (as $t\rightarrow -\infty$)?

I see that on the hyperbolic plane if the geodesics get closer on the future, they deviate on the past. I'm wondering if there is some similar "deviating behavior" (even if it was not monotonically increasing) in abscense of conjugate points too.

Thanks on advance.

is the geodesic flow on Hyperbolic Plane completely integrable?

2012-04-21T17:55:51Z

I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and $H:M\rightarrow\mathbb{R}$ a smooth function, its symplectic gradient is the unique field $X_H$ over $M$ satisfying

$$\textrm{d}H=\omega(X_H,\cdot)$$

and we say that the system $(M,\omega,H)$ is completely integrable is there exists $f_1,\ldots,f_{n-1}:M\rightarrow\mathbb{R}$ smooth functions Poisson commuting: ${f_i,f_j}={f_k,H}=0$, where ${f,g}=\omega(X_f,X_g)$, and with $\textrm{d}f_1,\ldots,\textrm{d}f_{n-1},\textrm{d}H$ linearly independent in a dense set of $M$.

In the cotangent bundle $T^*M$ of a manifold $M$, there exists a canonical symplectic form,

$$\omega_\textrm{can}=\sum{\textrm{d}x_i}\wedge\textrm{d}\xi_i$$

$(x_1,\ldots,x_n,\xi_1,\ldots,\xi_n)$ local coordinates of $T^\star M$. Then, if we consider a riemannian manifold $(M,g)$, we can canonically define a symplectic form on $TM$ with the bundle isomorphism $\Phi:TM\rightarrow T^\star M$ given by

$$\Phi(p,v)=(p,v^\star)$$

where $v^*(w)=g(v,w)$ is the Riesz representation of a functional. Hence we can define $\omega=\Phi^{\star}\omega_\textrm{can}$. In this way, the geodesic flow can be viewed as the flow of the symplectic gradient of the metric hamiltonian $H(p,v)=\frac{1}{2}g_p(v,v)$. Then my question is if the geodesic flow on the tangent bundle of the hyperbolic plane is completely integrable and, if yes, what is the another function beside the metric hamiltonian $H$. Any help will be appreciated.

Comment by matgaio

2013-03-24T21:07:41Z

Dear Anton, sorry for taking too long to unnacept the answer. I was really busy these days so I didn't see the changes on the status of the question. And thanks to Misha for pointing out the papers of Burns and Sullivan

Comment by matgaio

2013-03-24T21:04:30Z

I will unnacept the answer in order to let Anton delete it. I apologize everybody for accepting too early the answer.

Comment by matgaio

2013-03-21T16:49:40Z

Hi, Anton, thanks a lot for the attention. It seems that it is a counterexample for the deviation. But is that surface the universal covering of some compact surface without conjugate points? My question is more on the following sense: considering a compact surface wthout conjugate points, on its covering is it possible to occur the deviation and to estimate it? I was hoping it because we have some additional control of the metric on the covering (it is the same at the various pieces of the covering). Thanks again!

Comment by matgaio

2012-04-22T16:46:03Z

@Giuseppe, I know about the answer there because it was my question there too. I've asked there first and a friend of mine have suggested ask here, because it could be a relatively good question to discuss. Thank you for the answers. It helped me a lot.