| (1) |
Read chapter 5 (Jacobi Fields) and 7 (Hadamard’s Theorem) from Do Carmo. |
| (2) |
Show that on a non-compact complete Riemannian manifold  there always exists at least one ray starting from any point, i.e. a unit-speed geodesic  with  for all  . |
| (3) |
Let be a Riemannian manifold, and let  , be a geodesic segment with no self-intersections such that  is not conjugate to  for all  . Show that  is locally minimizing, i.e. there exists a neighborhood  of the image of such that any curve in between and  has length  , and equal length only if it agrees with up to parameterization. Give an example in which is NOT globally minimizing. Comment: We will see later that the converse is true. That is, a geodesic is no longer locally minimizing after its first conjugate point. |
| (4) |
DoCarmo Chapter 7, problems 9 and 10 (completeness of hyperbolic space).
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