Is the Geodesic Always the Shortest Path Between Two Points?

[enricfemi]

it comes from the calculus of variation that the shortest path between two points on a surface must be geodesic.
then must the geodesic connected two points be the shortest path?
if not, what about the example?
Thanks for any reply!

 

[physixlover]

it comes from the calculus of variation that the shortest path between two points on a surface must be geodesic.
then must the geodesic connected two points be the shortest path?
if not, what about the example?
Thanks for any reply!

check out the example in wikipedia
http://en.wikipedia.org/wiki/Geodesic

 

[wofsy]

it comes from the calculus of variation that the shortest path between two points on a surface must be geodesic.
then must the geodesic connected two points be the shortest path?
if not, what about the example?
Thanks for any reply!

No. On a cylinder there are infinitely many geodesics between most points. The same is true of a flat torus.

 

[enricfemi]

No. On a cylinder there are infinitely many geodesics between most points. The same is true of a flat torus.

yeah, cylinder is really a good example!

 

[wofsy]

There is an example of a geodesic on a fluted surface of negative curvature that winds almost all of the way down the surface circling around it in a helical motion then turns around and comes back! The shortest geodesic though between two adjacent points is a simple arc. I will try to look this up. It is pretty incredible.

 

[enricfemi]

it reminds me the magnetic lines of force in tokamak. they are all helical.