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Math Help - differential geometry

  1. #1
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    differential geometry

    How to show the normal spherical image x is never constant?
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  2. #2
    Member HappyJoe's Avatar
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    I assume your curve is parametrized by arc-lenght?

    If so: Suppose the unit normal vector N was constant. This normal vector is always perpendicular to the tangent vector T, so the vector T must be in the plane perpendicular to N. But if T is containing entirely in a plane, then the normal unit vector will also be in this plane (being the derivative of T), a contradiction.
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