What are the 'closed geodesic' ?

[mhill]

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

$$l(\gamma)=\int_\gamma \sqrt{ g(\dot\gamma(t),\dot\gamma(t)) }\,dt\ ,$$

but i do not know what means 'closed' or why the geodesic of a torus would have the length (?) $$l_n =na$$ a=radius ??

 

[tiny-tim]

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

$$l(\gamma)=\int_\gamma \sqrt{ g(\dot\gamma(t),\dot\gamma(t)) }\,dt\ ,$$

but i do not know what means 'closed' or why the geodesic of a torus would have the length (?) $$l_n =na$$ a=radius ??

Hi mhill!

"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.

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 ).

 

[mhill]

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 ??

 

[tiny-tim]

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!

"a point" is rather an understatement … it intersects itself everywhere, an infinite number of times.

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.

 

[mhill]

' Selberg Trace' is related to the fact that you can express a sum $$f(E_n )$$

with $$-\nabla f(x)= E_n f(x)$$ using the 'length' of the closed geodesics

http://mathworld.wolfram.com/SelbergZetaFunction.html

 

[HallsofIvy]

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.

 

[tiny-tim]

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!

You missed the torus :

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 length (?) $$l_n =na$$ a=radius ??