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Let $S$ be a closed surface embedded in $\mathbb{R}^3$, let's say of genus zero. I seek examples of $S$ with the following property: If one selects a random any point $p$ on $S$, and a random direction $u$ tangent to $S$ at $p$, then the geodesic issuing from $p$ in direction $u$ is a closed geodesic with positive probability. [Question modified to reflect @alvarezpaiva's comment.]

I know this is trivially true for Zoll surfaces, on which every geodesic is closed; see figure below. But are there non-Zoll $S$ where geodesics are prevalent enough to yield a positive probability (perhaps $1$)? Or, to be more explicit:

Q. If, for every $p \in S$, closed geodesics issuing from $p$ occur with probablility $1$ among all tangent directions $u$, does that imply that every such geodesic is closed?


           ZollDiscr
           (Zoll Surface: Image from Polthier&Schmies via this MO question)

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Well, just take a metric on the two-sphere that agrees with the standard metric in a neighborhood of an equator. How about asking that at every point $p$ there is a positive probability that a geodesic through $p$ be closed? –  alvarezpaiva Feb 20 at 12:44
    
@alvarezpaiva: Ah, good point! I have so modified the question. –  Joseph O'Rourke Feb 20 at 12:48
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By the way, your "sub-question" asking for surfaces where with probability one a unit vector is the initial condition of a closed geodesic looks very interesting. I wonder if there is anything besides surfaces all of whose geodesics are closed. –  alvarezpaiva Feb 20 at 14:20
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If you substitute dense for probability one, it will not be the same problem. I think that if you take a metric of revolution on the sphere the initial conditions for closed geodesics may be dense in the unit tangent bundle (the invariant tori with rational rotation numbers are foliated by periodic orbits). –  alvarezpaiva Feb 21 at 21:55
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It's related to something I worked on, but for Finsler metrics. It turns out that there are continuous Finsler metrics on the torus, for example, that are smooth almost everywhere and for which with probability one every unit momentum is the initial condition of a closed geodesic of length 1. This has neat applications to the geometry of numbers. See this question: mathoverflow.net/questions/132818/… –  alvarezpaiva Feb 21 at 22:56

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