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This is a problem out of Elementary Differential Geometry by Barrett O'Neill (Chapter 6 Section 3 Number 2).

Let $M$ be a flat surface in $\mathbb{R}^3$ with principal curvatures $k_1$ and $k_2$, where $k_1 = 0$ and $k_2$ is never zero. Want to show that the principal curves of $k_1$ are line segments in $\mathbb{R}^3$.

There is a hint: With $\{E_i\}$ principal and $\alpha''= \nabla_{E_1}E_1$ and use the connection equations.

So far I have concluded that $\alpha'= E_1$, $\alpha ''=E_2$. So, $$\frac{S(\alpha')\cdot \alpha'}{\alpha' \cdot \alpha'}=\frac{\alpha'' \cdot U}{\alpha' \cdot \alpha'}=\frac{E_2 \cdot E_3}{E_1 \cdot E_1}=0.$$ And $S(\alpha') \cdot \alpha' = 0$ implies $\alpha$ is asymptotic. Which means $\alpha$ is not bending away from the tangent plane. Which I suppose I already knew because $k_1$ is $0$. Also I know that the Gaussian curvature is $0$ so my surface is either a plane or trough-shaped. But $k_2$ is non-zero so $M$ cannot be a plane.

I'm not sure how any of this helps me, nor do I know how to use the hint. Also any pointers on why this is intuitively true. I have been looking at examples but cannot figure out why this has to be true.

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    $\begingroup$ Hi, Can you show how do you get $\alpha'' = E_2$? If $\alpha$ is a line you should get $\alpha ''= 0$. $\endgroup$ – user99914 Apr 21 '14 at 7:45
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You're wrong about "trough-shaped." There are cylinders, cones, and so-called tangent developables, all of which are flat, ruled surfaces.

As a hint, I'll suggest that you want to use the Mainardi-Codazzi equations.

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@Ted Shifrin May be he is right about "trough-shaped." There are cylinders, or any other non-circular trough curves dragged or extruded along a straight line generator creating a set of parallel rulings. When bending takes place curvature along the ruling is always zero, only the other principal curvature changes, product Gauss curvature remaining zero in all bent configuration mappings.

For a cone the folding lines are generators. For tangential developable folding lines are edges of regression, which are straight for helices of right circular cylinder only? Is that the error? Regards

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