Problem 10.1 Parallel Transport on the PlaneShow that if l and h are lines on the plane such that they are parallel transports along a transversal l, then they are parallel transports along any transversal. Prove this using any assumptions you find necessary. Make as few assumptions as you can, and make them as simple as possible. Be sure to state your assumptions clearly.a. What part of your proof does not work on a sphere or on a hyperbolic plane?SuggestionsThis problem is by no means as trivial as it at first may appear. In order to prove this theorem, you will have to assume something — there are many possible assumptions* so use your imagination. But at the same time, try not to assume any more than is necessary. If you are having trouble deciding what to assume, try to solve the problem in a way that seems natural to you and then see what develops while making explicit any assumptions you are using.On spheres and hyperbolic planes, try the same construction and proof you used for the plane. What happens? You should find that your proof does not work on these surfaces. So what is it about your proof (on a sphere and hyperbolic plane) that creates difficulties?Problem 10.1 emphasizes the differences between parallelism on the plane and parallelism on spheres and hyperbolic planes. On the plane, non-intersecting lines exist, and one can “parallel transport” everywhere. Yet, as was seen in Problems 8.2 and 8.3, on spheres and hyperbolic planes two lines are cut at congruent angles if and only if the transversal line goes through the center of symmetry formed by the two lines. That is, on spheres and hyperbolic planes two lines are parallel transports only when they can be parallel transported through the center of symmetry formed by them. Be sure to draw a picture locating the center of symmetry and the transversal. On spheres and hyperbolic planes it is impossible to slide the transversal along two parallel transported lines while keeping both angles constant (something you can do on the plane). In Figures 10.1, the line r’ is a parallel transport of line r along line /, but it is not a parallel transport of r along /’.We will now divide parallel postulates into three groups: those involving mostly parallel transport, those involving mostly equidistance, and those involving mostly intersecting or non-intersecting lines. This division is useful even though it is rough and unlikely to fit every conceivable parallel postulate.b. Which of these three groups is the most appropriate for your assumption from Problem 10.1? We will call the assumption you made in Problem 10.1 “your parallel postulate.’’’’
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