Textbooks:

1) Riemannian Geometry, by Manfredo Perdigão do Carmo. Birkhäuser, Boston, 1993.

2) General Theory of Relativity, by P.A.M.Dirac. Princeton University Press, 1996.

1. Derive the transformation rule for the Christoffel symbols under a change of curvilinear coordinates.

2. Show that is not a tensor.

3. On show that can be arbitrary functions i.e. for any choice of such functions there exists a unique affine connection with the Christoffel symbols in the canonical coordinates on .

4. Prove that for any connection with the Christoffel symbols the quantities form a tensor. (It is called the torsion tensor of the given connection.)

5. Prove that for any connection with the Christoffel symbols the quantities are the Christoffel symbols of a symmetric connection.

6. On a Riemannian manifold M with the metric prove that for any contravariant vector and the corresponding covariant vector we have i.e. the map of to defined by the metric is isometry on every tangent space provided the metric on is given by .

7. Describe geometrically the parallel transport along a parallel c of latitude on the standard unit sphere .

Hint: Consider the cone C tangent to along c and show that the parallel transport of any tangent vector along c is the same whether taken relative to or to C.

8. Calculate the Christoffel symbols on the Lobachevsky plane

(with the metric ).

9. On the Lobachevsky plane (see Problem 8 above) describe the parallel transport along the ``curve" x=t, y=1.

Hint: The transported vector rotates with constant angular velocity.

10. Show that for any Riemannian or pseudo-Riemannian metric and arbitrary tensor with there exists unique affine connection which is compatible with g and has the given torsion tensor T (see Problem 4).

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