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Is the normal vector to a surface on the same direction as the binormal vector to a curve that lies on the surface ? To be more precise, consider a curve $\alpha(s)$ such that $a'(0) = v \in T_{p}S$ and $\alpha(0)= p$. My question is: does the normal vector to a surface at the point P correspond to the binormal vector (as in the Frenet frame) of $\alpha$ at 0?

I'm almost sure that this is true, however, I'd like to read others' opinion. Thank you.

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The answer to your question is no. It is not true that the binormal vector to an arbitrary curve on an arbitrary surface in three space is collinear with the normal vector to the surface. This is because in general, the normal vector of the curve lies in neither the tangent plane nor it is aligned with the normal vector of the surface. That is the reason for the existence of geodesic and normal curvature of a curve on a surface. If the geodesic curvature is zero (like with geodesic curves on the surface), then the normal vector of the curve is aligned with the normal vector of the surface and the binormal vector of the curve is in the tangent plane to the surface.

Curves with your property are very special and these are the asymptotic lines (curves) of the surface. If the surface is positively curved they do not exist. They exist for negatively curved surfaces and form two generically transverse families of curves, defining the so called asymptotic line parametrization of the surface.

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  • $\begingroup$ Yeah, later on the same day I realized how wrong I was thinking about the normal vector to a surface, what I said doesn't make any sense. Thanks anyway ! $\endgroup$
    – 0212user
    Nov 2 '16 at 23:12

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