Math 402, Assignment 10, Spring, 2008

Due Wed. April 16 (at the beginning of class)

1. Let l' be the parallel transport of l along a line m. Let X be the midpoint of the segment of m between l and l'. Let n be another line which is transversal to l and l' and contains X.

   In class, we proved the following claim on H^2 and E^2: l' is the parallel transport of l along n. However, that proof does not work on S^2 because ASA is false on S^2. Find a symmetry of this figure. Use it to prove the claim on S^2.

   Hint: this is similar to the proofs of PTDI in the HW.
   

2. Let l' be the parallel transport of l along a line m. Let X be the midpoint of the segment of m betrween l and l'. Let n be another line which is transversal to l and l'.

   1. In class we showed that, on E^2, l' is the parallel transport of l along n. Prove that, on H^2, l' is not the parallel transport of l along n if n does not contain X.

      Hint: The argument is very similar to the argument that it is the parallel tranport on E^2. In particular, you need to break it up into cases depending of where m and n intersect.

   2. Look at your proof. Did you use either of the following facts? The sum of the interior angles of a triangle in hyperoblic space is less than 180 degrees. The sum of the interior angles of a a convex quadrilateral in hyperbolic space is less than 360 degrees.

   3. Did you use any facts which are true on H^2 space but NOT true on E^2?

3. Let l' be the parallel transport of l along a line m.

   1. In class, we showed that, on E^2, l' is equidistant to l. Prove that, on H^2, l' and l are not equidistant.

      Hint: The argument is very similar to the argument that they are equidistant on E^2. In particular, you need to find a common perpendicular.

   2. Look at your proof. Did you use either of the following facts? The sum of the interior angles of a triangle in hyperoblic space is less than 180 degrees. The sum of the interior angles of a a convex quadrilateral in hyperbolic space is less than 360 degrees.

   3. Did you use any facts which are true on H^2 space but NOT true on E^2?