Consider the line element of the sphere of radius a: ds² = a² (do² + sin 20 20 do²). The only non-vanishing Christoffel symbols are го =-sin cos 0, ΓΦ θα =1ø 60 = 1 tan a) Write down the metric and the inverse metric, and use the definition (+9µv) =гºvp гр μν 1 Оро to reproduce the results written above for r ΦΟ and Fo 06 [You can also check that the other Christoffel symbols vanish, for practice, but this will not be marked.] b) Write down the two components of the geodesic equation. c) The geodesics of the sphere are great circles. Thinking of 0 = 0 as the North pole and 0 = π as the South pole, find a set a solutions to the geodesic equation corresponding to meridians, and also the solution corresponding to the equator.

Consider the line element of the sphere of radius a: ds² = a² (do² + sin 20 20 do²). The only non-vanishing Christoffel symbols are го =-sin cos 0, ΓΦ θα =1ø 60 = 1 tan a) Write down the metric and the inverse metric, and use the definition (+9µv) =гºvp гр μν 1 Оро to reproduce the results written above for r ΦΟ and Fo 06 [You can also check that the other Christoffel symbols vanish, for practice, but this will not be marked.] b) Write down the two components of the geodesic equation. c) The geodesics of the sphere are great circles. Thinking of 0 = 0 as the North pole and 0 = π as the South pole, find a set a solutions to the geodesic equation corresponding to meridians, and also the solution corresponding to the equator.