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1 Calculate the curvature tensor for the following metrics: a) ds

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1 Calculate the curvature tensor for the following metrics: a) ds

2

= a

2

(dθ

2

+ sin

2

θdφ

2

) and b) ds

2

= dv

2

− v

2

du

2

.

2 Show that, on account of symmetries, the curvature tensor R

αβγδ

(for the metric connec- tion) in an n-dimensional space has n

2

(n

2

− 1)/12 algebraically independent components.

3 Using the Bianchi identity R

αβγδ;µ

+ R

αβδµ;γ

+ R

αβµγ;δ

= 0 prove the contracted Bianchi identity ∇

α

(R

αβ

−

12

g

αβ

R) = 0 .

4 Consider two neighboring geodesics (great circles) on a 2-sphere of radius a: the equator and a geodesic parallel to the equator at φ = 0 and tilted to the north by a small angle

∆θ = b . Let w be the displacement vector between the geodesics with the same φ (hence for φ = 0 we have w = b ∂/∂θ. Using the geodesic deviation equation show that

d

2

w

θ

dφ

2

= −w

θ

, d

2

w

φ

dφ

2

= 0,

and then solve these equations for the above initial conditions.

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