1. Derive a geometric formula for covariant derivative of a contravariant vector via parallel transport:
2. Show that 
.
3. Derive a geometric formula for the covariant derivative of an arbitrary tensor via parallel transport.
4. Find all geodesics on the Lobachevsky plane.
5. Let G be a Lie group with biinvariant Riemannian metric. Prove that
geodesics passing through 
are exactly one-parametric subgroups, i.e.
smooth maps 
such that 
and 
for all
.
6. Describe all isometric transformations of the Lobachevsky plane
(with the metric 
, z=x+iy)
which leave
a given point p fixed. They should include ``rotations" (which actually
induce rotations
in the tangent space 
) and reflections with respect to a geodesic
passing
through p.
7. Prove that the group of all isometries of 
is a
3-dimensional Lie group
which is generated by the following transformations:
1) ``Translations": 
;
2)``Homotheties" or ``Scalings": 
;
3) ``Rotations around the point i":
4) Reflection 
.
8. Use the product rule for the covariant derivative with respect to
the tensor product
to derive the formula for 
in terms of
the curvature tensor.
9. Calculate all the components of the curvature tensor for the paraboloid
(here 
)
with the metric which is induced by the standard Euclidean metric in 
.
10. Calculate the curvature tensor on the standard sphere
where R>0.
( Hint: Use exercise 9 and symmetries.)