jimbo007
Oct22-04, 02:28 AM
hi,
i am trying to show that the amount by which a vector is rotated by parallel transport around a triangle whose sides are arcs of great circles equals the excess of the sum of the angles over 180 degrees.
this is what i have found out so far
call the angles of the triangle (assuming locally flat space) a, b and c.
then
a+b+c=\pi +\frac{1}{R^2} Area(T)
where Area(T) is the area of the triangle.
this is as far as i can get, i have looked up a few places on parallel transport but the notation used to explain it is very nasty...which is a bit over kill for which i believe to be a much easier problem than it looks.
pls help
i am trying to show that the amount by which a vector is rotated by parallel transport around a triangle whose sides are arcs of great circles equals the excess of the sum of the angles over 180 degrees.
this is what i have found out so far
call the angles of the triangle (assuming locally flat space) a, b and c.
then
a+b+c=\pi +\frac{1}{R^2} Area(T)
where Area(T) is the area of the triangle.
this is as far as i can get, i have looked up a few places on parallel transport but the notation used to explain it is very nasty...which is a bit over kill for which i believe to be a much easier problem than it looks.
pls help