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Homework Statement
Let γ : I → R3 be an arclength parametrized curve whose image lies in the 2-sphere S2 , i.e. ||γ(t)||2 = 1 for all t ∈ I. Consider the “moving basis” {T, γ × T, γ} where T = γ'.
(i) Writing the moving basis as a 3 × 3 matrix F := (T, γ × T, γ) (where we think of T and etc. as column vectors) show that F : I → SO(3, R);
(ii) Define the curvature κ :=< T' , γ × T > in analogy to plane curves (noting that γ × T is normal to T and tangent to the 2-sphere–so γ × T takes the role of JT for plane curves). Calculate A := F-1F' and show that A: I → so(3, R) takes values in skew-symmetric matrices so(3, R), depends only on κ, and has a very special form (which one).
(iii) Calculate the curvature κ for the circles C in S2 obtained by slicing S2 by the planes z = c with 0 ≤ c < 1, and calculate ∫κ (contour integral)
(iv) Let κ: I → R be a smooth function. Show that there is a curve γ : I → S 2 whose curvature is κ. Hint : build the matrix function A out of κ (see (ii)) and consider the linear matrix ODE F' = F A; use Problem 1 to solve; take the 3rd column of F as a candidate for γ...
(v) The curve constructed in (iii) is unique up to a rotation of S 2 . (The group of rotations SO(3, R) of R 3 preserve S 2–why?)
Homework Equations
The Attempt at a Solution
Anyone who can address any part of the problem please do so, doesn't need to be the first (or consecutive) parts. Thanks!
I am beginning at part i) and so I basically made my 3d column vectors as follows γ = (x(t), y(t), z(t))τ, γ' = (x'(t), y'(t), z'(t))τ and then for the middle column {γ x γ'} -- I know they're orthogonal to both γ and γ' (hence the basis) but can I basically just write the column vector down defined as follows (γ x γ') = (χ(t), ψ(t), ξ(t))τ with χ orthogonal to x and x' and so on...?
Moving forward I imagine to show that this matrix is part of SO(3, R) we need to show it has the properties of the special orthogonal group, which are; that a matrix in this group multiplied by its transpose equals 1 and these matrices have determinant 1. Correct?
For ii) it looks like we just need to calculate the inner product between γ'' and (γ x γ') which shouldn't be too hard but I'm not sure what my professor is getting at with the JT thing for plane curves -- I'm unfamiliar with that but if I recall J was a 2x2 matrix (when in ℝ2) that was something like diag(-1, 1) and rotated the following matrix by 90 degrees? Hence it was the normal vector sort of? Anyways, what exactly does it mean to say that a matrix (A in this case) only takes values in skew-sym matrices -- is it asking to show that A is a skew symmetric matrix? And if so, how would I prove that?
I think that's enough for now, I will post again if I make progress and need more help. Please feel free to answer questions about parts iii), iv) and v) as well.Hoping for an interesting and lively discussion here! Good evening to you all..