3.Consider the line element of the sphere of radius a:ds2 =a2(d02+sin20do2).The only non-vanishing Christoffel symbols arer=-sin0 cos0,=tana)Write down the metric and the inverse metric,and use the definitionTP (ngva+9uo-09m)=TPvuto reproduce the results

written above for and[You can also check that the otherChristoffel symbols vanish,for practice,but this will not be marked.b)Write down the two components of the geodesic equation.b)The geodesics of the sphere are great circles.Thinking of 0 =0 as the North pole and 0=as the South pole,find a set a solutions to the geodesic equation corresponding to meridians,andalso the solution corresponding to the equator.

Fig: 1