Gauss map

Namely, given a surface X in Euclidean space R3, the Gauss map is a map N: X → S2 (where S2 is the unit sphere) such that for each p in X, the function value N(p) is a unit vector orthogonal to X at p. The Gauss map is named after Carl F. Gauss.

The Gauss map can be defined (globally) if and only if the surface is orientable, in which case its degree is half the Euler characteristic.

The Gauss map can always be defined locally (i.e. on a small piece of the surface).

The Jacobian determinant of the Gauss map is equal to Gaussian curvature, and the differential of the Gauss map is called the shape operator.

Gauss first wrote a draft on the topic in 1825 and published in 1827.

The Gauss map can be defined for hypersurfaces in Rn as a map from a hypersurface to the unit sphere Sn − 1  ⊆  Rn.

In this case a point on the submanifold is mapped to its oriented tangent subspace.

One can also map to its oriented normal subspace; these are equivalent as

Finally, the notion of Gauss map can be generalized to an oriented submanifold X of dimension k in an oriented ambient Riemannian manifold M of dimension n. In that case, the Gauss map then goes from X to the set of tangent k-planes in the tangent bundle TM.

The Gauss–Bonnet theorem links total curvature of a surface to its topological properties.

The Gauss map reflects many properties of the surface: when the surface has zero Gaussian curvature, (that is along a parabolic line) the Gauss map will have a fold catastrophe.

Both parabolic lines and cusp are stable phenomena and will remain under slight deformations of the surface.

The Gauss map provides a mapping from every point on a curve or a surface to a corresponding point on a unit sphere. In this example, the curvature of a 2D-surface is mapped onto a 1D unit circle.
A surface with a parabolic line and its Gauss map. A ridge passes through the parabolic line giving rise to a cusp on the Gauss map.