| (1) |
Read chapters 0 and 1 of DoCarmo. |
| (2) |
Prove that the following definitions of “orientable” are equivalent:
| (a) |
There exists an atlas (a system of coordinate charts) for  such that the derivative at each point of each coordinate interchange function has positive determinant. |
| (b) |
There exists an orientation (equivalence class of basis) of each  which varies smoothly in  , i.e. for each  there exists a neighborhood  and ordered vector fields  on which at each point of agrees with the orientation. |
| (c) |
For every closed path  and every collection  of vector fields along  which form a basis of each  , the orientation of  agrees with the orientation of  .
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| (3) |
(DoCarmo exercise 0.9) Let  be a group acting properly discontinuously on a smooth manifold . Prove that  is orientable iff there exists an orientation of that is preserved by all the diffeomorphism of . Use this to show that the Klein bottle and the Moebius band are non-orientable, and that  is orientable iff  is odd. |
| (4) |
True or False: every isometric embedding  is linear. |
| (5) |
Let  and  be Riemannian manifolds. Prove that the distance function associated to the product metric on  is given by:
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| (6) |
Let be a simply connected Riemannian manifold, a group of isometries acting properly discontinoulsy on , and  the associated Riemannian covering. Show that every isometry of lifts to an isometry  of with  , and conversely that every such gives rise to an isometry of . Conclude that  , where  denotes the normalizer of in  . Use this to compute the isometry group of . Is it connected? |
| (7) |
Suppose that  both act properly discontinously. Prove that  and  are isometric iff  and  are conjugate in . |
| (8) |
If  acts properly discontinuously on  , prove that  is diffeomorphic to either a cylinder, Moebius band, torus, or Klein bottle. You may use without proof the fact that every isometry of is either (1) a rotation about some point, (2) a translation or (3) a glide reflection, i.e. a reflection about some line composed with a translation in the direction of the same line. |
| (9) |
Classify up to isometry all cylinders, Mobius bands, tori, and Klein bottles of the form , where acts properly discontinously. We will see later that this classifies all 2-dimensional manifolds with zero curvature.
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