the Gauss map
/ GOWSS map /
At every point of a smooth surface there is one direction that sticks straight out, perpendicular to the surface: the unit normal vector. Now do a simple but profound thing. At each point, take that little outward-pointing unit arrow and move it, keeping its direction, so its tail sits at the origin. Its head then lands somewhere on the unit sphere. The Gauss map is this assignment: it sends each point of the surface to the point on the unit sphere that records 'which way the surface faces here.'
Formally, write the unit normal as N(p) for each point p of a (regular, oriented) surface S. Since |N| = 1, N(p) is itself a point of the unit sphere S^2. The Gauss map is N : S -> S^2. The whole power of the idea is that curvature becomes the study of how this map stretches and turns. On a flat plane the normal points the same way everywhere, so the Gauss map is constant — it collapses the whole plane to a single point of the sphere, and there is no curvature. On a sphere of radius R the normal at each point already points radially, so the Gauss map is essentially the identity (scaled), spreading the surface over the whole sphere. The derivative of the Gauss map, dN_p (a linear map of the tangent plane to itself, called the shape operator or Weingarten map), measures the rate at which the normal swings as you move — and that rate IS the bending of the surface.
This is Gauss's master stroke: it converts the vague notion 'how does the surface curve?' into the concrete, differentiable question 'how does N move?' The eigenvalues of dN_p (up to sign) are the principal curvatures; their product is the Gaussian curvature, their average the mean curvature; the second fundamental form is built from dN_p. A beautiful fact lurks here: the Gaussian curvature at p equals the local 'area magnification' of the Gauss map — the limit of (area swept on the sphere)/(area on the surface) as the region shrinks to p. One caution: the Gauss map needs an orientation (a consistent choice of 'outward'); on a non-orientable surface like a Mobius band you cannot make this choice globally, so a single global Gauss map does not exist.
On a sphere of radius R, the outward unit normal at a point p is N(p) = p / R — it points radially out. So the Gauss map sends p to p/R, which is exactly the corresponding point of the unit sphere: the map covers the whole unit sphere once. On a flat plane, N is the same constant vector everywhere, so the Gauss map crushes the entire plane to one point — visibly registering 'no curvature.' Comparing these two extremes already shows the Gauss map turning curvature into how widely the normal's image spreads.
The Gauss map records 'which way the surface faces' as a point on the unit sphere.
The Gauss map needs a global choice of unit normal (an orientation). On a Mobius band that choice cannot be made consistently — slide the normal once around and it returns reversed — so the band has no global Gauss map, only local ones.