Time to fess up. The answer to the question below is "No." Furthermore, there are a number of irrelevant conditions. So let me ask a simpler and straightforward version of the question: ----------------------------------------- Let A = { p in R^2 : 1 <= ||p|| <= 2 }. PUZZLE: Prove there is no C^2 homeomorphism H : S^1 x [1,2] -> A such that if L(r) denotes the length of the curve H(S^1, r) in A, then L(r) is independent of r. ----------------------------------------- --Dan << Can you find a smooth (C^oo) function f : D -> [0,1] (where D is the closed unit disk in R^2) such that 1) f is onto. 2) The unit circle bd(D) is finv(1), and 3) For 0 < c <= 1, finv(c) is a smooth curve with length = 2pi. where finv(c) denotes the inverse image of c.

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_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele