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Let M be a surface in R^3 oriented by a unit normal vector field

U=g_{1}U_{1}+g_{2}U_{2}+g_{3}U_{3}

Then the Gauss map G:M[tex]\rightarrow[/tex][tex]\Sigma[/tex] of M sends each point p of M to the point (g_{1}(p),g_{2}(p),g_{3}(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

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# Tangent map, gauss map, and shape operator

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