What are the 'closed geodesic' ?

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i have been reading about 'Selberg Trace formula'

i know what a Laplacian is but i do not know what is the author referring to when he talks about 'Closed Geodesic' i know what the Geodesic of a surface is

[tex] l(\gamma)=\int_\gamma \sqrt{ g(\dot\gamma(t),\dot\gamma(t)) }\,dt\ ,[/tex]

but i do not know what means 'closed' or why the geodesic of a torus would have the lenght (?) [tex] l_n =na [/tex] a=radius ??
 

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  • #2
tiny-tim
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i know what a Laplacian is but i do not know what is the author referring to when he talks about 'Closed Geodesic' i know what the Geodesic of a surface is

[tex] l(\gamma)=\int_\gamma \sqrt{ g(\dot\gamma(t),\dot\gamma(t)) }\,dt\ ,[/tex]

but i do not know what means 'closed' or why the geodesic of a torus would have the lenght (?) [tex] l_n =na [/tex] a=radius ??

Hi mhill! :smile:

"closed" simply means that the geodesic meets itself, so it has a finite length.

If the tangent of the angle to one axis is a rational multiple of the ratio of the axes, then it will meet itself. If irrational, it will go on for ever. :smile:

n is the number of turns "through the hole" before the geodesic joins up. The more turns, the longer the geodesic (though I must admit, I don't see why it's proportional :redface:).
 
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thanks tiny-tim, then you mean that a geodesic will be closed if for example x(a)=x(b) , so there is a point where the geodesic intersect itself.

and for the 'Selberg Trace' is there a pedestrian proof or a proof that a profane non-mathematician could understand ??
 
  • #4
tiny-tim
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thanks tiny-tim, then you mean that a geodesic will be closed if for example x(a)=x(b) , so there is a point where the geodesic intersect itself.

Hi mhill! :smile:

"a point" is rather an understatement … it intersects itself everywhere, an infinite number of times. :wink:
and for the 'Selberg Trace' is there a pedestrian proof or a proof that a profane non-mathematician could understand ??

Sorry … I've no idea what a Selberg trace is. :redface:
 
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HallsofIvy
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For a cylinder the closed geodesics are the circles perpendicular to the axis of the cylinder.

For a sphere the closed geodesics are the great circles.
 
  • #7
tiny-tim
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torus

For a cylinder the closed geodesics are the circles perpendicular to the axis of the cylinder.

For a sphere the closed geodesics are the great circles.
Hi HallsofIvy! :smile:

You missed the torus :wink::
i have been reading about 'Selberg Trace formula'

but i do not know what means 'closed' or why the geodesic of a torus would have the lenght (?) [tex] l_n =na [/tex] a=radius ??
 

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