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Let $S : x^2+y^2-z^2 = 1 $ be the hyperboloid of one sheet. We know that there exists a closed geodesic, namely the unit circle $\{(\cos\theta, \sin\theta,0) , \theta \in [0,2\pi]\}$, since it is the locus of fixed points of the reflection along the $xy$ plane . The claim of uniqueness should follow from Clairaut's relation, however I can't see why that is. In fact, the problem can be solved in a more general case; A surface of revolution with negative gaussian curvature has at most one simple closed geodesic. Can anyone enlighten me on how this follows from Clairaut's relation?

samu
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    Clairaut will tell you more — namely, that every other geodesic must be unbounded. But what you’re looking for is an immediate consequence of Gauss-Bonnet. – Ted Shifrin Feb 25 '22 at 23:39
  • @TedShifrin Could you please expand a bit on why Clairaut tells us they are unbounded? – samu Feb 25 '22 at 23:53
  • In the notation I use, $r\cos\phi$ is constant along geodesics. This tells you that as $r$ increases, $\phi$ must also increase. So $r$ cannot be bounded above (because if there's a maximum $r$ we would have $\phi=0$, and if the geodesic spirals to a parallel $r=r_0$, likewise $\phi\to 0$). – Ted Shifrin Feb 26 '22 at 00:08
  • That’s minimum, not maximum! – Ted Shifrin Feb 26 '22 at 01:00
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    A complete surface of negative curvature with cyclic fundamental group indeed has at most one simple closed geodesic. – Moishe Kohan Nov 23 '23 at 16:02

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Let $ r_i, \psi_{i}, r_{min}$ be initial radius, initial angle of arc filament to meridian and minimum radius at waist.

The arc tracing is decided entirely compelled by Clairaut's Law .

There are three cases.

  1. If $ \psi = \pi/2 ,~ ~r= r_{min} $ it is stuck at $r_{min}= r_i$ at the start keeping always to the circumference and it is the lone closed geodesic. The other two following cases are open geodesics:

  2. The case $ \psi> \sin^{-1}\dfrac{r_{min}}{r_{i}}$ the filament arc comes to a point $ r> r_{min} $ where it makes $\psi = \pi/2$ and retraces its path as open geodesic to only one side, else the angle $ \psi $ would goes imaginary.

  3. The case $ \psi< \sin^{-1}\dfrac{r_{min}}{r_{i}}$ the arc filament goes past the waist to infinite distances parallel to symmetry axis as an open geodesic, all radii and $\psi $ are real.

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Geodesics on a 1-sheet hyperboloid of revolution

Narasimham
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