Week 46

The topic this week has been geodesics, section 5.6.  Again this was first developed for surfaces in Euclidean 3-space, but in the end we showed that the theory can be extended to any surface which is locally isometric to some such surface, since the equations can be expressed using only  the Riemannian metric. ("Being geodesic is an intrinsic property.")  Thus it also applies to the hyperbolic plane.  Among the most important examples, it is shown that the earlier notions of lines in Euclidean, spherical and hyperbolic geometry all are geodesic,  Moreover, an important result from differential equations implies that there is always exactly one geodesic line in every direction from every point.  Hence geodesics are indeed good generalizations of the concept of lines in a geometry.

Exercises for Tuesday November 18:  5.6:  1, 4-10