# What are the 'closed geodesic' ?

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

## Answers and Replies

tiny-tim
Science Advisor
Homework Helper
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 lenght (?) $$l_n =na$$ 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 ).

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
Science Advisor
Homework Helper
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. HallsofIvy
Science Advisor
Homework Helper
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
Science Advisor
Homework Helper
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 lenght (?) $$l_n =na$$ a=radius ??