on the right of this post you can see 4 (approximately) geodesics in the mobius strip, one is transversal to the core curve and the others (three) begin at that transversal… One almost can see that all geodesics in the surface have curvature but without 3d torsion… would it be a theorem?
Rendered with Mathematica-6.
let us describe how the euclidean duality is carried to the tangent bundle of a surface.
We know how euclidean duality is: For each euclidean space, , the coordinated (rectangular) functions: are linear, so their derivatives, , are equal themselves i.e the gradients obey , they are dubbed and they satisfy duality:
Now, if is a parameterization of the surface, , and is a “measure” then, doing calculus in the surface means do calculus to . Let .
Let us name the coordinated (rectangular) functions on .
So by the chain rule we have: that is, in terms of gradients:
So for the functions we get and by the same rule just above
where evaluating at the basis of give
but since it is known that the generate , then both: generate , the co-tangent space at ,… wanna see a picture?