Differential geometry

1. (Christoffel symbols of the first kind) If the metric g is a diagonal matrix show that for all indices a and B + a in the range 1, 2, ..., n (no summation): (a) Tana = 2gada (b) -Paas = laBa = [Baa = 2goal (c) All other lijk vanish 2. (Christoffel symbols of the second kind) If the metric g is a diagonal matrix show that for all fixed indices in the range 1, 2, ..., n (no summation): (a) Tag = [ga = ard (7 In |gaal) (b) Tg8 = - a 2gaa dra 9BB with a * B (c) All other D'; vanish