Parallel Transport Around Triangle on Sphere: Angles Excess 180°

[Tzar]

Hey I got this problem. Any help wil be great.

Suppose you have a triangle on a sphere. Show that the amount by which a vector is rotated by a parallel transport around such a triangle equals the excess of the sum of the angles over 180 degrees.

Thanks!

 

[Tzar]

Anyone?? If you know some differential geometry, I am interested in your opinion!

 

[koushdcurryman]

Hey Tzar. What you need to use is the Gauss Bonet theorem. Once you've shown the excess angle is A/r^2 all you need to do is show that the covariant derivative is equal to this value.

 

[masudr]

Start off with a co-ordinate system. Write down the metric for your 2D space. Then calculate the covariant derivative (using the Christoffel connection), then stick it in the equation for parallel transports. Surely there's an easier way, but this method is most general.

 

[vladimir69]

doing the phys3550 assignment i take it :p