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. Let be a Lorentz transformation i.e. a linear transformation which preserves the Lorentz metric . Prove that .

2. Let be a Lorentz transformation, so for any contravariant vector . Write the transformation law for any covariant vector.

3. Let be a symmetric Lorentzian inner product in (i.e. it is non-degenerate and the corresponding quadratic form is of the type +---, that is has exactly one positive and 3 negative squares). A vector x is called a time-like vector if . Prove that for any two time-like vectors x,y we have the inverse Cauchy-Schwarz inequality

4. Check that the elements of (r times V and s times , where is the dual space to V) are tensors of type r,s in the sense that they can be presented as sets of numbers associated with any choice of a basis in V with appropriate transormation law when the basis is replaced by another basis.

5. Check that is a tensor.

6. Check that and are tensors.

7. Calculate the Planck units with 10% precision.

8. Find a manifold M, , such that there exists no Lorentzian metric on M.

9. ( A construction of the Lobachevsky plane). Consider the group G of all affine orientation preserving transformations of i.e. transformations , , y>0. Write a left-invariant Riemannian metric on G, such that it coincides with the Euclidean metric at the unit element . (Here e corresponds to x=0, y=0).

( Answer: up to a positive constant factor, or in other words , .)

10. In the previous exercise denote z=x+iy, , and show that any transformation with ad-bc=1 is an isometry of G with the metric .

( Hint: Observe that .)

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