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I understood everything up to the last part ("It follows that the gaussian curvature $K = K(s, v)$ of the tube is given by..."). Why? What am I missing here? I know we can just compute it from the second fundamental form, but I want to understand what was done here.

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It seems you are reading Do Carmo's book on Differential Geometry, aren't you?

If so, have a look at the discussion at page 166. There you see the equation $$ dN(w_1) \wedge dN(w_2) = K w_1 \wedge w_2,$$ for a basis $\{w_1, w_2\}$ of a tangent space at a point. This is exactly what you need.

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  • $\begingroup$ That solves it. Thanks! $\endgroup$ – Matheus Andrade Mar 15 at 23:38

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