So here I'm tasked with proving two statements about Frenet Curves. I shall enumerate them now..
I. If γ is an arclength parametrized planar curve, we can regard it as a space curve. Show that this space curve has τ ≡ 0. Also prove the converse: if a space curve has τ ≡ 0 then it lies in some plane P ⊂ R$^3$
II. Show that given functions κ : I → R , κ ( t ) > 0 for all t ∈ I , and τ : I → R both smooth, there exists a unique (up to Euclidean motion) Frenet curve in R$^3$ whose curvature and torsion are κ and τ respectively.
III. If anyone can help to classify curves of constant curvature and torsion that'd be swell!
So for question...
i. Here can I just use the definition of torsion which (for arclength parameterized curves) is $\tau$ = < T $\times$ N, N'> and show that since the 2d planar curve is spanned only by T and N (is that true? It would seem as though since T and N are more fundamental than B = T $\times$ N) that B = T $\times$ N would then be zero and that would cause the inner product and thus the torsion to be zero?
Furthermore I'm a little more lost on how I might formulate the converse of that statement, that if the torsion is zero it must lay in a plane. In that circumstance I would argue similarly that since $\tau$ is zero then either N' or T $\times$ N must be zero but I'm not sure how I'd prove it has to be T $\times$ N (thus proving theres no vector in a third orthogonal direction).
ii. I know there's some fundamental theorem of differential geometry or something of that nature that states that all space curves may be defined by their torsion and curvature (like all plane curves may be defined by their curvature) but I'm a bit lost at how to formulate this into an argument. Some help is greatly appreciated!
iii. I'm aware of the helix being a curve of constant curvature and torsion but not aware of any others.