Shape operator and The Gauss tangent map

[badgers14]

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!

 

[lippyka]

The tangent map of the Gauss Map G is defined as the directional derivative of G at a point p in the direction of a tangent vector v to M. That is, G*(v) = \nabla_vg(p) = \nabla_v(g1(p),g2(p),g3(p))Now, as g1(p),g2(p),g3(p) are all functions of the normal vector U, we can writeG*(v) = \nabla_v(U) = (v[g1]U1+v[g2]U2+v[g3]U3)Comparing this with the definition for S(v), we see that G*(v)=-S(v). Thus, the shape operator of M is (minus) the tangent map of its Gauss map, and S(v) and -G*(v) are parallel for every tangent vector v to M.