1. Prove that in 4-dimensional case the cyclic relation
is independent of the antisymmetry and symmetry relations
i.e. there exists a tensor of type (0,4) in 
such that it satisfies
(2) but not (1).
2. Prove that in 2-dimensional case
where 
is the scalar curvature.
3. Show that the curvature tensor of the Schwarzschild metric does not vanish.
4. Find the mass of a black hole if its horizon sphere has the size
of an atom i.e.
the radius about 
cm.
5. Prove that the Schwarzschild solution indeed has a singularity (which can not be corrected by changing coordinates).
6. Consider a particle moving in the Schwarzschild gravitation field. Prove that its trajectory is a plane curve.
7. Consider a satellite rotating around a black hole of the given
mass m (or given
Schwarzschild radius 2m) along a circular orbit of a radius R>2m
(outside the horizon sphere). Find
the period of the rotation by a distant observer clock and also by the clock in
the satellite. What happens with these periods as 
?
8. Let M be a Riemannian manifold. Define canonical coordinates
near 
by transfering arbitrary linear orthonormal coordinates in 
near 0 to
a neighborhood of p in M via the geodesic exponential map
.
Prove that in these coordinates 
,
,
for any 
and sufficiently small 
,
and
9. Let G be a compact Lie group with a biinvariant Riemannian metric, X,Y,Z left-invariant vector fields on G. Prove that
and
Hint. Use the fact that one-parametric subgroups and their left translations are geodesics.
10. Let G be a compact Lie group with a biinvariant Riemannian metric, X,Y,Z,W left-invariant vector fields on G. Prove that
and
