Exercises 4.1 Conformal metrics on a Riemann surface Let us consider the metric on a Riemann surface ds2 = F(y)(dx2 + dy2)

1. Compute the Christoel symbols ????k ij .

2. Prove that the geodesic equations are given by the ODEs (4.47) and

(4.48).

3. Compute the Riemann curvature Rp ijk.

4. Compute the Ricci curvature Rij .

5. Compute the scalar curvature R.