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Tangent map, gauss map, and shape operator

  1. May 9, 2010 #1
    Can anyone help me with this problem??
    Let M be a surface in R^3 oriented by a unit normal vector field
    U=g1U1+g2U2+g3U3
    Then the Gauss map G:M[tex]\rightarrow[/tex][tex]\Sigma[/tex] of M sends each point p of M to the point (g1(p),g2(p),g3(p)) of the unit sphere [tex]\Sigma[/tex].
    Show that the shape operator of M is (minus) the tangent map of its Gauss Map: If S and G:M[tex]\rightarrow[/tex][tex]\Sigma[/tex] are both derived from U, then S(v) and -G*(v) are parallel for every tangent vector v to M.
    Any help is appreciated. Thanks
     
  2. jcsd
  3. May 10, 2010 #2

    lavinia

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    Science Advisor

    the normal map is the same as the Gauss map. Its derivative is the tangent map. the shape operator by definition is the negative of this - I think.
     
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