The setting of this question is the 2-dimensional manifold M with line element ds' = gijdX'dX'= -4(1 + y)dx dy - 4(1 - y2 )dy2, where X = {x, y} and i = 1, 2. (a) Using any suitable method, find the non-vanishing Christoffel symbol components. (b) Show that the geodesic equations can be brought into the form 2 2 X = 0, y + =0. 1 + 1+y (c) By solving the geodesic equations, or otherwise, find an improved coordinate system to show that this is a flat 2-dimensional manifold. (d) Determine the signature of the given line element.The setting of this question is the 2-dimensional manifold M with line element ds' = gijdX'dX'= -4(1 + y)dx dy - 4(1 - y2 )dy2, where X = {x, y} and i = 1, 2. (a) Using any suitable method, find the non-vanishing Christoffel symbol components. (b) Show that the geodesic equations can be brought into the form 2 2 X = 0, y + =0. 1 + 1+y (c) By solving the geodesic equations, or otherwise, find an improved coordinate system to show that this is a flat 2-dimensional manifold. (d) Determine the signature of the given line element