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Mar14-12, 11:16 AM
P: 7
Thank you all for your replies!
The amusingly-named Hairy Ball Theorem says that the tangent bundle of the 2-sphere has no nonvanishing sections. In other words, it's impossible to assign a nonzero vector (say, the one you want to associate with "up") to every point on the sphere in a smooth (or even continuous) way.
I didn't want it to sound as if I am ignoring the fact that the sphere is not parallelizable ( aka the HBT). My intuition was to start with a pseudo "up" direction and try to keep your "eyes" in that direction while moving along the curve from two points in such a way that we don't cross any "bad points" (where the parallel vector field frame vanishes - I forgot the fancy and yet semantically incorrect names of the tools that Diff Geom operates with: section, bundle, fibre, sheaf, etc. and I must re-read a Riemannian Geometry book). As I remember, the thing with parallelizable manifolds is that you can define an universal (kind of moving) frame for the reunion of the tangent spaces (may this be the tangent bundle or whatever the correct name for this object is)..
For a manifold in Euclidean space, a constant speed curve is a geodesic if its acceleration is normal to the manifold. Intuitively, the geodesic does not wiggle along the surface. A vector field is parallel if its change in direction is normal to the surface. I'd like to see some examples for non-trivial surfaces, say a surface of revolution of constant negative curvature.
One way to find geodesics is to slice the surface with a knife that is perpendicular to the surface. On a sphere this produces a great circle. What about on a symmetrically shaped torus? What about on a cylinder?
Nice explanation, very natural. It makes sense because if the acceleration were to have any "parallel" components, it would "shift" the advancing direction, hence rendering that curve not "the straightest" path of them all, i.e. no longer a geodesic. As for cutting the surface with a perpendicular knife, it's dubbed the "greedy algorithm" to approximate geodesics on manifolds (probably not producing genuine geodesics..) - how can I prove or disprove this?
Again, thank you..