| (1) |
Read about vector bundles in chapters 2-3 of “Characteristic Classes” by Milnor-Stasheff. Read about connections in vector bundles in chapter 2 of “Differential Geometric Structures” by Poor. Read about the Levi-Civita connection in chapter 2 of Do Carmo.
| (2) |
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| (a) |
If  and  are connections in the same vector bundle and  are positive real numbers which sum to one, prove that  is a connection as well. |
| (b) |
Use a partitian of unity argument to prove that every vector bundle admits a connection.
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| (3) |
Let  be a vector bundle with projection map  . Suppose that  and  are two connection in . If  is a vector field on  and  is a section of the bundle, define:
| (a) |
Prove that  for all vector fields on , all sections  of the bundle, and all real-valued functions  on . |
| (b) |
Prove that the value of  at  depends only on the values of and  at  , and that  is linear in each variable. |
| (c) |
If and are both compatible with a fixed Euclidean structure on , prove that is skew-symmetric, ie,  . |
| (d) |
Suppose that is an orientable rank-2 vector bundle. Prove that connections on which are compatible with a fixed Euclidean structure are in one-to-one correspondence with real-valued one-forms on . HINT: Any connection, , can be written in terms of a fixed connection, , as  , for some with the above properties.
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| (4) |
If a Riemannian manifold  admits a coordinate chart over which each Christoffel symbol  is zero, prove that each  must be constant, and therefore must be locally isometric to  on this coordinate chart. |
| (5) |
(Do Carmo exercise 2.7) Let  be the unit sphere,  an arbitrary parallell of latitude on  , and a tangent vector to at a point of . Describe geometrically the parallel transport of along . HINT: consider the cone  tangent to along and show that parallel transport of along is the same, whether taken relative to or to .
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