The Gauss map and oriented area change

Try

On a sphere of radius two, the outward normal map sends p to p/2 on the unit sphere. Tangent lengths shrink by one half, so tangent areas shrink by one quarter. The Gaussian curvature is correspondingly 1/4.

The Gauss map of an oriented surface is N:M→S², assigning each point its unit normal. Its derivative measures how quickly normals change. Because the tangent plane of M at p and the tangent plane of S² at N(p) are both perpendicular to N(p), they can be identified as vector planes in R³.

Predict

Predict the Gauss-map image of a circular cylinder. Its normal is horizontal and independent of height, so the image lies in the equator. One tangent direction is collapsed, giving zero area derivative.

Predict whether the area of the image set always equals ∫|K|dA. Repeated normal directions can be covered several times. The integral counts multiplicity unless injectivity is established.

Experiment

Compare normal maps.

Surface	Normal-map behavior	K
Plane	Constant normal	0
Radius-R cylinder	One-dimensional equator image	0
Radius-R sphere	Scaling onto unit sphere	1/R²

For the cylinder, differentiate the normal along the vertical direction to exhibit an actual kernel vector. For the sphere, compute the derivative on two orthonormal tangent directions and take its determinant. These calculations distinguish rank zero, one and two.

Understand

Our convention is S=−dN. In dimension two, det(dN)=det(−S)=det S=K: the two sign reversals cancel. With the surface and unit-sphere tangent orientations both chosen by N, this determinant is the signed local area factor of the Gauss map. The unsigned area factor is |K|.

If K(p)≠0, dN is invertible between two-dimensional tangent spaces. The inverse function theorem in surface charts makes N a local diffeomorphism near p. This does not imply global injectivity. A distant point may have the same normal, and local orientation may vary when K changes sign.

For a cylinder N=(cos θ,sin θ,0). The height derivative is zero while the angular derivative is nonzero, so rank is one and determinant zero. For the sphere N=p/R, dN=(1/R)identity on each tangent plane, giving the positive factor 1/R² even though the corresponding S eigenvalues are negative with outward orientation.

Apply

Repair “Negative K means the Gauss map shrinks area by a negative amount.” Unsigned area is never negative. Negative K means the normal map reverses local orientation, while its area magnitude is |K|. A saddle can send an oriented tangent basis to one with reversed orientation.

For optional depth, on compact oriented surfaces the signed integral of K relates to the degree of the Gauss map. Establishing that relation needs the degree/integration theorem and must account for multiplicity and regular values. Later Gauss–Bonnet provides an independent topological interpretation of total curvature.

A small patch image or a finite normal plot can illustrate these facts but cannot establish global degree or rule out remote repeated coverage.

Continue

When interpreting a Gauss-map calculation, state rank, orientation and whether area is counted with multiplicity. Continue to graphs and minimal surfaces for explicit curvature formulas and a variational meaning of mean curvature.