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I just did exercise 7 from Manfredo's section 3.2, which says that if the mean curvature is zero at a non-planar point then the asymptotic directions are perpendicular.

My question is: Is the converse true? If the asymptotic directions are perpendicular then the mean curvature is zero? I tried to prove this and I couldn't, but I don't find a counter example either.

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  • $\begingroup$ How did you try to prove it? Edit your post to include this. $\endgroup$
    – Ted Shifrin
    Nov 20 at 20:25
  • $\begingroup$ I already got it, thank you $\endgroup$
    – Leandro Santos
    Nov 21 at 13:57

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