2 Show that Γ

1 Calculate the Christoel symbols for the Euclidean metric ds

= dx

+ dy

in polar coordinates.

2 Show that Γ

= 1 2g

∂g

∂x

, where g = det(g

) . Using that show that for a vector eld v

and a scalar eld Φ we have

∇

v

= 1

p|g| ∂

(p|g|v

), ∇

∇

Φ = 1

p|g| ∂

hp

|g| g

∂

Φ i

3 Show that the anely parametrized geodesic equation is the Euler-Lagrange equation for the action

S[x

(s)] = Z

g

(x) dx

ds

dx

ds ds .

4 Find all timelike geodesics on the Lorentzian manifold M = {(t, x) ∈ R

× R} with the metric ds

= t

(−dt

+ dx

) .

5 Consider the 2-sphere with the metric ds

= dθ

+ sin

θ dφ

.

(a) Write down the geodesic equation and show that the lines of constant φ are geodesics but the lines of constant θ are not geodesics unless θ = π/2.

(b) A vector v

is parallely transported from a point (θ

, φ

) to a point (θ

, φ

) along the

curve (θ

, φ

+ λ) . Find v

(λ) .