Euclidean and Non Euclidean Geometry

Problem 5.1 What Is Straight in a Hyperbolic Plane?

a. On a hyperbolic plane, consider the curves that run radially across each annular strip. Argue that these curves are intrinsically straight. Also, show that any two of them are asymptotic, in the sense that they converge toward each other but do not intersect.

• The straight lines as defined in the course are the same as the , that is, the shortest curves between two points.

• We shall say that two geodesics which converge in this way are asymptotic geodesics. Note that there are no geodesics (straight lines) on the plane which are asymptotic.

• The reason straight lines in the disc model do not appear to be straight is that the hyperbolic plane is negatively curved (which means, roughly speaking, that the neighborhood of every point is something like a Pringle crisp) so that it cannot be drawn in the Euclidean plane without distortion. This situation is rather like the impossibility of drawing a flat map of the world without distortion (this time because the sphere is positively curved).

b. Find other geodesics on your physical hyperbolic surface. Use the properties of

straightness (such as symmetries) you talked about in Problems 1.1, 2.1, and 4.1 Try holding two points between the index finger and thumb on your two hands. Now pull gently, a geodesic segment (with its reflection symmetry should appear between the two points. Also (if your surface is durable enough), try folding the surface along a geodesic. Also, use a ribbon to test for geodesics.

c. What properties do you notice for geodesics on a hyperbolic plane? How are they the same as geodesics on the plane or spheres, and how are they different from geodesics on the plane and spheres?

• Every pair of points is joined by a unique geodesic. • Every geodesic segment has a geodesic perpendicular bisector.

• Every angle (between two geodesics) has a geodesic angle bisector. • Two geodesics intersect no more than once.

A geodesic, the shortest distance between any two points on a sphere, is an arc of the great circle through the two points. The formula for determining a sphere’s surface area is 4πr2; its volume is determined by (4/3)πr3.

• On the sphere, the geodesics are great circles (like the equator).

• The geodesics in a space depend on the Riemannian metric, which affects the notions of distance and acceleration.

• Geodesics preserve a direction on a surface and have many other interesting properties. The normal vector to any point of a geodesic arc lies along the normal to a surface at that point

• Furthermore, no matter how badly a sphere is distorted, there exist an infinite number of closed geodesics on it.