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We get stuck on this homework problem:

Prove that if the principal curvatures of a surface $M \subset \mathbb{R}^{3}$ are constant, then $M$ is either part of a plane, a sphere, or a circular cylinder. In the case $k_{1} \neq k_{2}$ assume that there is a principal frame field on all of $M .$ Show that the $k_{1}$ -curves are parallel lines and that the $k_{2}$ -curves are circles.

We have finished proving the first part of the hint that $k_1$-curves are parallel lines, but we could not find a good way to prove $k_2$-curves are circles.

Xiao Hong
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    This looks remarkably like one of the exercises in my differential geometry text. The suggestion for the second part is to do the preceding exercise: When you have one constant principal curvature and no umbilic points, then you get a circular tube about a curve. You're using the language of frame fields; are you working with differential forms and moving frames? – Ted Shifrin Nov 01 '22 at 18:24
  • @Xiao Hong: By translation we get tubes/cylinders in one direction and parallel lines in the other. By rotation, circles define spheres lying in both directions. When curvatures are zero we have planes in either case. – Narasimham Nov 06 '22 at 19:06

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