Research Log: Week 6 |
| Monday, 6/20/05 |
| Joe and I found that the theorem about First Fundamental Forms gives that there should be a parametrization of the isometry to match a given First Fundamental Form, not that any parametrization works. This means that using the standard equation of a ruled surface does not give the identity matrix for the First Fundamental Form, and Duc's observation was right. This might mean that the First Fundamental Form is useless for our purposes. Without it, though, we don't have a method to test whether a candidate ruled surface is truely and isometry of a strip. |
| Tuesday, 6/21/05 |
We've begun looking for candidates for a counter-example to the hypothesis that any space curve is a ribbon curve. My first inclination was to look at closed curves to use an arc length argument. Joe noticed, however, that a segment of a simplified version of our tennis ball curve (without straight to straight or curve to curve parts) cannot be a ribbon curve because each planar segment of the curve must extend to a surface whose tangent plane is vertical to the plane containing the generating curve. Since the generating curve lies half in one plane and half in another perpendicular plane, any possible ribbon along the curve would have a discontinuity at the point where the planes change. |
| Wednesday, 6/22/05 |
| We realized that the example of the simplified version of our tennis ball curve is not a very interesting counterexample to the ribbon curve theorem because, while it is C1 (one time continuously differentiable), it is not C2, and since the second derivative of the curve is the normal of the ribbon surface (property of geodesics), an undefined second derivative gives an undefined normal, which gives an undefined binormal (since the binormal is the cross product of the normal and tangent). An undefined binormal gives an undefined tangent plane since the tangent plane is spanned by the tangent and the binormal. |
| Thursday, 6/23/05 |
| I was sick today. |
| Friday, 6/24/05 |
Joe noticed that we can modify our supposed counter example by considering a unit speed curve whose acceleration changes smoothly from a positive constant to zero and smoothly back to a positive constant, but still retains the quality of being contained in two perpendicular planes. This curve should be C2 by construction. I think that it is impossible to twist the strip between the two perpendicular planes. It seems that the local perpendicularity should continue even on the straightaway. Joe and I are at a loss on how to begin proving this, though. |