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No.

Consider the example of a right circular cylinder as the surface in question and let $C$ be a "generator" (i.e., a line on the cylinder parallel to the axis). Then the geodesic curvature is everywhere zero, but if epsilon is greater than half the circumference there is nor no epsilon-tubular neighborhood.

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What if

No.

Consider the surface is example of a right circular cylinder of very small radius as the surface in question and let $C$ be a "generator" (i.e., a line on the curve cylinder parallel to the axis). Then the geodesic curvature is a generator?everywhere zero, but if epsilon is greater than half the circumference there is nor epsilon-tubular neighborhood.

1

What if the surface is a right circular cylinder of very small radius and the curve is a generator?