(a) Consider a 2-dimensional manifold M with the following line element
For which values of z is this line element well defined.
(b) Find the non-vanishing Christoffel symbols
(c) Obtain the geodesic equations parameterised by l.
(d) Solve the geodesic equations and suggest an improved coordinate system. What is the metric in the new coordinates? What lines describ the geodesic geometrically?
(e) What can you say about the Riemann curvature tensor, the Ricci tensor and the Ricci scalar of this manifold.
The Attempt at a Solution
(a) The line element is well defined for all values of y and z other then z=0
(b) gzz,z= -2/z3
The only non vanishing christoffel symbol is,
(c) The geodesic equations are given by,