# Shape operator and The Gauss tangent map

1. May 10, 2010

Let M be a surface in R3 oriented by a unit normal vector field
U=g1U1+g2U2+g3U3
Then, the Gauss Map G: M to E, of M sends each point p of M to the point (g1(p),g2(p),g3(p)) of the unit sphere E.
Show that the shape operator of M is (minus) the tangent map of its Gauss map: If S and G are both derived from U, the S(v) and -G*(v) are parallel for every tangent vector v to M.

We know by definition that S(v)=-$$\nabla$$vU=v[g1]U1+v[g2]U2+v[g3]U3
Next, I know that I must start with the definition of the tangent map and transform it into something similar to the definition for S(v).

I'm confused on where to begin with the second part of this problem. Any help would be greatly appreciated!