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2.5 Gauss Mapping and the Third Fundamental Form

The unit normal vector (u1,u2) can be considered as a point on the unit sphere S0 centered at the origin of our space E. The resulting mapping
: X --> S0
which assigns to the point X(u1,u2) the point (u1,u2), is called the Gauss map.
Using the first fundamental form on S, we define the third fundamental form
III(V,W) = mijaibj
where V = aiXi and W = bjXj and mij = ij.
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